[Modern Problems in Condensed Matter Sciences] Optical Orientation Volume 8 || Theory of Optical Spin Orientation of Electrons and Nuclei in Semiconductors

CHAPTER 2

Theory of Optical Spin Orientation of Electrons and Nuclei in Semiconductors

M.I. DYAKONOV and V.I. PEREL

A.F. Ioffe Physico-Technical Institute USSR Academy of Sciences

194021 Leningrad USSR

© Elsevier Science Publishers Β. V., 1984

11

Optical Orientation Edited by

F. Meier and B.P. Zakharchenya

Contents 1. Introduction 15

2. Band structure of GaAs 17

2.1. General picture 17

2.2. Conduction band 18

2.3. Bands of light and heavy holes 19

2.4. Split-off band and nonparabolicity of light-hole band 20

3. Optical orientation in interband transitions near k = 0 22

3.1. Selection rules 22

3.2. Density matrix of photoexcited electrons 24

3.3. Role of spin relaxation 26

3.4. Polarization of edge luminescence 27

4. Dependence of electron spin orientation on energy of exciting photons .* 29

4.1. Influence of split-off band 29

4.2. Spin relaxation in thermalization 30

5. Deformational effects 32

5.1. Circular polarization of luminescence in strained crystals 32

5.2. Optical orientation of holes under deformation 34

6. Optical orientation of electrons in η-type semiconductors 36

6.1. Optical pumping 36

6.2. Optical and electrical effects in a semiconductor with oriented electrons 37

7. The Hanle effect 39

7.1. Optical orientation in a magnetic field 39

7.2. Hanle effect in cascade processes 40

7.3. Hanle effect in η-type semiconductors 44

8. Circular polarization of luminescence in magnetic field for nonpolarized excitation 47

8.1. Polarization in equilibrium conditions 47

8.2. Polarization in case of departure from equilibrium 48

9. Dynamic polarization of lattice nuclei by optically oriented electrons 50

9.1. Nuclear spin system 50

9.2. Hyperfine interaction of electrons and nuclei 51

9.3. Nuclear spin relaxation due to electrons 52

9.4. Nuclear field 53

9.5. Dynamic polarization of nuclei by oriented electrons 54

9.6. Dynamic nuclear self-polarization 57

12

10. Optical manifestation of nuclear field

10.1. Nuclear effects in longitudinal magnetic field

10.2. Influence of nuclear polarization on Hanle effect in oblique field

10.3. Optical detection of N M R

11. Optical cooling of nuclear spin system

References

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60

60

61

63

66

69

1. Introduction

In the process of interband absorption of a photon in a semiconductor an electron in the conduction band and a hole in the valence band are generated, the total spin of the electron and the hole being equal to the angular momentum of the photon absorbed. Photons of right or left circularly polarized light have a projection of the angular momentum on the direction of the wave vector equal to + 1 or - 1 , respectively. This angular momentum is distributed between the photoexcited electron and hole according to the selection rules determined by the band structure of the semiconductor. The photoexcited carriers live some time τ before recombination. During this time the spin orientation of carriers decreases due to different relaxation processes. If the orientation has not entirely disappeared by the time of recombination, the recombination radiation will be partially circularly polarized.

Thus the process of optical orientation includes two stages: creation of spin-oriented carriers in absorption of circularly polarized light, and spin relaxation during the carriers' lifetime.

The degree of circular polarization of the recombination radiation serves as a useful and sensitive indicator of the carriers' spin state and its changes under the influence of external factors and relaxation processes, determining the kinetics of the nonequilibrium carriers in a semiconductors.

Along with the optical one, other methods of detection of the carriers' spin orientation are possible. Thus in the experiments of Lampel (1968a, b), who was the first to demonstrate the possibility of optical orientation of free electrons in a semiconductor, nuclear magnetic resonance (NMR) was used as a detection method. The 2 9Si nuclei in a silicon crystal were polarized due to their hyperfine interaction with optically oriented electrons. Optical detection was first used by Parsons (1969) in his studies of optical spin orientation in GaSb (see also Parsons 1971).

A typical experiment on optical orientation is schematically presented in fig.i.

The degree of spin orientation of photoexcited carriers is determined by the details of the band structure, the type of optical transition, the relaxation processes, as well as by the influence of different external factors. This is why optical orientation is an effective method of studying various physical processes in semiconductors. In the most detailed way optical orientation was studied in

15

16 M.I. Dyakonov and V.I. Perel

Fig. 1. Schematical drawing of a typical experiment on optical orientation. A circularly polarized beam of light (1) induces interband transitions in a semiconductor (2). The degree of circular

polarization of the recombination radiation (3) is measured.

GaAs and (GaAl)As solid solutions. The first experiments on optical orientation in these materials were performed by Ekimov and Safarov (1970) and Zakharchenya et al. (1971). Optical orientation reveals itself in all types of edge luminescence, in particular in recombination radiation of optically oriented excitons. Exciton orientation was discovered in hexagonal crystals of CdSe by Gross et al. (1971). Measurements of the polarization of hot luminescence allow to study the energy and momentum relaxation of hot electrons (see ch. 4).

A quite special kind of phenomena involves optical effects due to dynamic polarization of the lattice nuclei of a semiconductor. The nuclei, polarized by optically oriented electrons, create an effective magnetic field which, in turn, acts on the electron spins and thereby on the circular polarization of the luminescence (fig. 2). Studies of associated phenomena began since the work of Ekimov and Safarov (1972a, b) who detected NMR for the lattice nuclei of a (GaAl)As solid solution by resonance changes in the degree of circular polarization of luminescence.

The dynamic nuclear polarization under conditions of optical orientation of electrons is in fact a result of deep cooling of the nuclear spin system. Convincing experimental proof of such an interpretation was obtained by Fleisher et al. (1976). Optically oriented electrons and polarized lattice nuclei form a strongly coupled system. High sensitivity to small external magnetic fields, long relaxation times and typical nonlinear effects (hysteresis, self-sustained oscillations) are characteristic of this system. These phenomena are dealt with in ch. 5.

The present chapter is a review of basic theoretical ideas related to optical orientation of electrons and nuclei in semiconductors. Although most of the considerations are quite general, we shall, for definitiveness, consider semiconductors with the band structure of GaAs, for which processes accompanying optical orientation are best studied experimentally.

17

Fig. 2. Coupling between the electron (ESS) and nuclear (NSS) spin systems of a semiconductor. Optically oriented electrons polarize the lattice nuclei. Because of this an effective magnetic field arises which, in turn, influences the electron orientation and thereby the luminescence polarization.

2 . Band structure of GaAs

2.1. General picture

The band structure of GaAs is well known (see, for example, Madelung 1964). The bottom of the conduction band and the top of the valence band are located at the centre of the Brillouin zone (Γ-point). The band structure in the vicinity of this point is schematically shown in fig. 3. Without taking into account comparatively small terms caused by the absence of an inversion centre, the conduction band (c) is twice degenerate in spin, while the valence band consists of three twice degenerate sub-bands: the heavy-hole sub-band (vl), the light-hole sub-band (v2) and the split-off sub-band (v3). The band extrema belong to the representations Γ6 (c-band), Γ8 (vl- and v2-bands) and ΓΊ (v3-band). The absence of an inversion centre results in a small spin splitting of the bands and a small displacement of the top of the valence band from the point k = 0. These effects are not important for the following discussion of optical transitions; they may play a role, however, in spin relaxation processes. The energy surfaces in the light-hole and, particularly, the

Theory of optical spin orientation


e

c

ο Fig. 3. Band structure of gallium arsenide near the Γ-point. c denotes the conduction band, v l the

heavy-hole band (hh), v2 the light-hole band (lh), v3 the split-off band (sh).

heavy-hole band are warped, i.e. possess not spherical but cubic symmetry. Away from the Γ-point the bands are nonparabolic. The nonparabolicity of the conduction band is due to its interaction with the valence band and will not be taken into account below. On the contrary, the nonparabolicity of bands v2 and v3, due to their interaction with each other, is essential for the selection rules at corresponding energies of optical transitions.

2.2. Conduction band

An electron state in the conduction band is described by the Bloch wave function

Here S denotes the coordinate part of the S-type Bloch amplitude, i.e. a function invariant under symmetry transformations of the crystal.

Near the Γ-point the energy spectrum is parabolic and isotropic with an effective mass m e,

Vkm = UnPik'r> in = ± 1 / 2 ,

where the Bloch amplitudes u ± 1 /

Wl / 2 = & Τ ) W- l / 2 == & A *

1 / 2 correspond to different spin directions: (1)

(2)

e c (* ) = h2k2

(3) 2m e *

Theory of optical spin orientation 19

The absence of an inversion center results in the appearance of terms cubic in k in the Hamiltonian Η of an electron in the conduction band (Dresselhaus 1955). Taking these terms into account,

h2k2 h

where σ are the Pauli matrices, and Q{k) is a vector such that

Qx(k) = ah2(mcf^g)~lkx{k2y - k2)9 (5)

and β and Ωζ may be obtained from eq. (5) by cyclic permutation of indices x9 y, z. Here the axes JC, y9 ζ are directed along the crystal axes [100], [010], [001]. The dimensionless coefficient a determines the value of the spin splitting. For GaAs a = 6 x l 0 ~ 2 (see ch. 3). Thus, the absence of an inversion center leads to an effective magnetic field acting on the electronic spin and depending on the quasimomentum. Note that from eq. (5) it follows that the vector Ω is always perpendicular to k.

2.3. Bands of light and heavy holes

Near the Γ-point at energies small compared to the spin-orbit splitting Δ the four states of an electron in the bands of light and heavy holes are described for a given k by the wave functions

^ 2 ) = e i A-'ExW,(*K3 / 2 ) , M = ± l / 2 , ± 3 / 2 , (6)

where Μ = ± 3/2 corresponds to the heavy-hole band (hh), and Μ = ± 1/2 to the light-hole band (lh). The index μ assumes the values μ = ± 1 / 2 , ± 3 / 2 , the corresponding Bloch amplitudes w£3 / 2) transform as eigenfunctions of the z-component of the angular momentum Jz. The index μ gives the component of the angular momentum (equal to 3/2) along the direction [001] (z-axis). Explicit formulae for the Bloch amplitudes may be given as

~-^(X + iY)U u%\ =\yi2{X-iY)i,

« C ^ - | i / 3 [ - i V 5 " ( J f + i r ) i + ^ Z T ] , (7)

« ? $ - i V 3 [ i ^ ( A - - i y ) T + i / 2 Z l ] .

Here X, Y, and Ζ are the p-type coordinate parts of the Bloch amplitudes, which transform as the coordinates x, y, z.


The energy spectrum and the coefficients χΜμ are to be found by solving the equation

HXM = %XM> (8)

where χΜ is a column of coefficients χΜμ with different μ, Η is the Hamiltonian of Luttinger (1956). For simplicity the so-called spherical approximation is often used, in which the warping of the energy surfaces is neglected. In this approximation Luttinger's Hamiltonian has the form

"=£r(yi + iiy)k2-&y(k'J)2< γ = ( 2 γ 2 + 3 γ 3 ) Α (9)

where m0 is the free electron mass, Yi,Y2>Y3 are Luttinger's parameters, and Jx, Jy, Jz are 4 x 4 matrices corresponding to the value of angular momentum being equal to 3/2.

The energy spectrum of light and heavy holes is isotropic in this approximation,

_h2k2 _ . . ε ν ΐ " ε ± 3 / 2 - 2mh > εν 2 - ^ ± ι / 2 2mx ' 1 }

where mh = m0/(yl — 2y) and ml = m0/(yl + 2γ) are the effective masses of heavy and light holes. In spherical approximation the coefficients xMlL(k) depend only on the direction of k and are related to the finite rotation matrices (see Edmonds 1957) by

ΧΜ μ = ^ / 2 )( < ρ , 0 , Ψ ) , ( i i )

where θ and φ are the polar angles of vector k with respect to some fixed coordinate frame, and ψ is an arbitrary angle defining the phase of the wave function (one may choose, for example, ψ = 0). In spherical approximation the index Μ denotes the value of the component of the electron angular momentum in the valence band along the direction of the wave vector k.

2.4. Split-off band and nonparabolicity of light-hole band

In the vicinity of the Γ-point the electron states in the split-off band are described by functions analogous to (6). In spherical approximation:

Ψ & = β " · ' Σ ^ > ( φ , <?, Ψ ) < 1 / 2 )· (12)

The projection Μ of the angular momentum on the direction of k assumes the values Μ = ± 1 / 2 . The index μ also assumes the values ± 1 / 2 , the corresponding Bloch amplitudes transforming as eigenfunctions of Jz with the value of


angular momentum equal to 1 / 2 . They have the form

uiyv = -^[(x+iY)i + z<t], « < _ ^ ) = i / 3 [ - ( ^ - i r ) T + z i ] .

(13)

If the energy is comparable to the spin-orbit splitting 4 , mixing of states in the light-hole band and the split-off band occurs (Kane 1956). Such mixing results in nonparabolicity of the energy spectrum in these bands.

The Hamiltonian describing all the three sub-bands of the valence band in spherical approximation may be written in the form

H=-{A+2B)k2+lB{k-L)2-\A{6-L) + \A. (14)

The constants in eq. (14) are chosen in such a way that A, B, and A coincide with the constants defining the energy spectrum in Kittel (1963). The quantities A and Β are related to the effective masses of light and heavy holes by

4 \ml mh) 4 \ m l m h )

In eq. (14), Lx,Ly,Lz are 3 x 3 matrices corresponding to an angular momentum equal to 1, and A is the spin-orbit splitting.

The wave functions may be found as linear combinations of functions (6) and (12) with coefficients au)(k):

**Η-*"''ΣαυΚ*)Σ»$(ψΛ*Κ'\ (16) j P-

where index j assumes the values y = 1 / 2 , 3/2. The component of the angular momentum along the wave vector is conserved, so that functions with different Μ are not mixed.

For heavy holes Μ = ± 3/2, and so the wave function (15) reduces to (6), i.e. αγ/2) = 1, a[1/2) = 0 for all k, while the energy spectrum is described as before by the first formula of eq. (10).

For Μ = ± 1/2 the Schrodinger equation with Hamiltonian (14) leads to the following system of equations for the coefficients a(J):

-(A + B)k2aW±y/2Bk2aW = £ a ^ 2 \

±y[2Bk2aW -(Ak2 -A)aW = m ^ 2 \ (17)

Hence we find the light-hole spectrum,

£Y2 = -(A + \B)k2 + \A-\{9B2kA+2BAk2 + 4 2 ) 1 / 2, (18)

as well as the split-off band spectrum,

e y 3 = -(A + \B)k2 + \A + \(9B2k* + 2BAk2 + A2)l/1. (19)


- 3 - 2 - 1 0 I 2 (-B/A) κ

/7 1 \ \

/ I / I 2 \

3 \

ε/Δ

Fig. 4. Energy spectrum of the valence band of GaAs. The nonparabolicity is the result of interaction between the light-hole and split-off bands.

The corresponding values of the coefficients a(^/2) and a£/2) (n = 2,3) are defined by eq. (17) combined with the normalization condition

| f l( l / 2 , | 2 + | a( 3 / 2 ) | 2 = 1

We recall that the energy range ε ̂ E% is considered and the condition Δ Eg is supposed to be valid. The results obtained obviously coincide with the results of Kane (1956) for this case. Neglecting of the warping of the energy surfaces allowed us to obtain explicit formulae for the energy spectrum and wave functions (Dyakonov and Perel 1971a).

The energy structure of the valence band of GaAs calculated with the help of eqs. (18), (19) is depicted in fig. 4.

3. Optical orientation in interband transitions near k = 0

3.1. Selection rules

The most vivid description of direct interband transitions gives the correspondence principle. According to this principle, there is a correspondence between


a quantum transition from state a to state b and a classical dipole with frequency ωα1? = (Ea - Eh)/h, the amplitude of the dipole moment being equal to the transitions matrix element

emitted or absorbed in quantum transitions (in the dipole approximation) are the same as for emission or absorption of light by the classical dipole (20).

Let us find out what kind of dipoles corresponds to direct interband optical transitions in cubic semiconductors A i n B v (GaAs type). Near the Γ-point the electron wave functions in the conduction band and in the valence band are described by eqs. (2), (6) and (12). If we direct the quantization axis (z-axis) along the wave vector k, the valence band wave functions (6) and (12) will coincide [apart from a factor exp(i£*r)] with functions (7) and (17) respectively. Now it is easy to find the dipole moments corresponding to transitions between the conduction band and different sub-bands of the valence band (Keldysh et al. 1969, Baumgardner and Woodruff 1968). One must bear in mind that (1) the spin functions (denoted by arrows) are normalized and orthogonal for different spin directions, (2) the quasimomentum is conserved in the transition (dipole approximation), (3) the only matrix elements not equal to zero are

The matrix elements of the dipole moment are presented in table 1 (Zakharchenya et al. 1982), see also fig. 5.

The following conclusions may be drawn from this table. Two dipoles rotating clockwise and counterclockwise in the plane perpendicular to k correspond to transitions hh-c. Transitions lh-c are represented by two dipoles oscillating along the direction of k and two dipoles rotating in the

(S\Dx\X) = (S\Dy\Y) = (S\Dz\Z).

Table 1 Matrix elements of dipole moment Dvc/D for different interband transitions.*

c

Band ν 1/2 - 1 / 2

hh + 3 /2 - 3 / 2

- y i 7 2 ( A + i , i )

0 0

lh

sh + 1/2

- 1 / 2

+ 1/2

- 1 / 2

\/273""

ν / Ϊ Τ 6 ( λ - ί μ )

- ^ Ϊ 7 3 " ( λ - ί μ )

- , / ϊ 7 6 ( λ + ϊ μ )

γ/273""

-γ/ΐΤΤίλ + ίμ)

ν/ϊ73~"

* Symbols: λ and μ are unit vectors in a plane perpendicular to the momentum, ν is a unit vector along the momentum.

where D is the dioole moment onerator. Intensity and nolarization of light

(20)


i Φ 1 φ

.1

hh- th-i ά .

Fig. 5. Dipoles corresponding to various interband transitions. The numbers near the arrows representing the dipoles indicate the relative transition intensities. The numbers near the levels correspond to the projection of the total angular momentum on the quantization axis directed

along the quasimomentum.

plane perpendicular to k. The same is true for transitions sh-c, however the relative intensities are different.

3.2. Density matrix of photoexcited electrons

The momentum and spin distribution of electrons generated in the conduction band by absorption of light is characterized by the density matrix F(k),

(21) Μ

where e is the unit polarization vector, m, m' are spin indices of the electron in the conduction band assuming the values ± 1 / 2 , and the summation index Μ denotes two degenerate states in each sub-band of the valence band.

If the exciting light is circularly polarized, we have

e={ + ex-\ey)/yil,

where ex and ey are unit vectors in the plane perpendicular to the direction of the light beam z. Using eq. (21) and table 1, it is possible to obtain the following expression for the density matrix of electrons at the instant of photogeneration (Dymnikov et al. 1976)

F=F„ 3(n-v)2

+ 2 ( σ · 5 0 ) ± Α )

3(σ·ρ)(η·ν)—(σ·η)

(22)


Table 2 Intitial values of parameters a 0, S 0, βο m e <3- (22).

Transition « 0 So βο

h h - c - 1 - 1 / 4 - 1 l h - c 1 - 1 / 4 1 s h - c 0 1 / 2 0

Here F0 depends on energy only, / is the unit 2 x 2 matrix, σ are Pauli matrices, and η, ν are unit vectors in the direction of the exciting beam and momentum, respectively. The anisotropy parameters a 0 , /?0, and S0 = + S0n depend on the type of transition and are given in table 2. The parameters a 0 , /?0, and 5 0 characterize, respectively, the alignment of momenta, the correlation between spin and momentum, and the average electron spin at the instant of photocreation,

5 0 = KSpaF>/<SpF> ) (23) where brackets denote averaging over the momentum directions (directions of v).

The average spin S0(v) of an electron with a given direction of momentum at the instant of creation may be found from eq. (22) with the help of the relation

S0(r) = \Sp(OF)/SpF. (24) Thus one finds for various transitions (under excitation by right circularly polarized light)

hh-c: 1 + (ι>·/ι)

, Λ 3v(vn)-2n lh-c: 5 0 ( r ) = — — — ,

5-3(vn)

sh-c: S0(v) = ±n. (25) These formulae show that the absolute value and the direction of the average spin depend on the angle between the momentum and the light beam. Electrons with momentum directed, at the moment of excitation, along the light beam, are 100% oriented for each transition (along the direction of photon angular momentum for transitions lh-c and sh-c, and opposite to that direction for transition hh-c). For hh-c transitions the average spin is always collinear with momentum, whereas for sh-c transitions it is collinear with the beam.

The spin-momentum correlation arising in the process of photoexcitation is essential for the polarization properties of hot luminescence (see ch. 4). However, this correlation vanishes during momentum relaxation, so that for


thermalized electrons, responsible for edge luminescence, such correlation is practically absent, their density matrix having the form

where S is the average spin of thermalized electrons, and F(e) describes their energy distribution.

3.3. Role of spin relaxation

The average spin S of thermalized electrons under stationary excitation differs from the value 5 0 of the average spin of electrons at the instant of creation. η-type semiconductors, in which equilibrium electrons are present in the conduction band besides the photoexcited ones, are considered in sect. 6. Here we consider p-type semiconductors, for which the majority of experiments on optical orientation were performed. In this case only photoexcited electrons exist in the conduction band, and the difference between S and S0 is caused by spin relaxation processes. (This difference may also be due to the action of magnetic fields, see sect. 7).

In usual conditions, when the electron lifetime τ is much greater than the energy relaxation time, the majority of electrons, existing in the conduction band under stationary excitation, is thermalized. It was already mentioned in sect. 3.2 that the momentum alignment and spin-momentum correlation arising at the instant of excitation are completely lost during thermalization. The average spin may be also partially lost during this stage (see sect. 4.2). If, however, the initial energy of photoexcited electrons is not large enough, this practically does not happen.

The spin relaxation of thermalized electrons may be described by a characteristic time T s . Obviously if T s » τ the average spin value in stationary conditions is equal to the initial one: S = S0. Thus, the values of S0 given in table 2 represent limiting values of S corresponding to the case when spin relaxation during the lifetime may be neglected. It may be seen from table 2 that for excitation with ft ω < £ + 4 , when only transitions hh-c and lh-c are possible, this limiting value is — 0.25 (the minus sign corresponds to orientation of the average electron spin in the direction opposite to the photon angular momentum). This result, obtained above in spherical approximation, is actually valid even if the real, cubic, symmetry of the crystal is taken into account (Dyakonov and Perel 1971a).

If T S <c τ the spin orientation is practically completely destroyed during the electron lifetime. In the general case the relation between S and 5 0 is given by

F = F ( e ) [ / + 2 ( * - S ) ] , (26)

S = 1 + T / T S *

(27)


The ratio T / T s may vary in a wide range, depending on temperature and concentration of impurities, defects and majority carriers. For a discussion of different mechanisms of spin relaxation see ch. 3.

3.4. Polarization of edge luminescence

Consider first band-to-band transitions. Edge luminescence is due to c-hh and c-lh transitions. In p-type semiconductors it occurs as a result of recombination of photoexcited electrons with equilibrium holes which are nonpolarized. For high excitation levels, or in η-type material, nonequilibrium holes may also participate in the recombination. At the instant of photocreation the holes, like the electrons, are oriented. The average angular momentum of light and heavy holes at the instant of excitation by circularly polarized light is +1.25, so that it adds up with the average electron spin (-0.25) to give unity angular momentum of the absorbed photon. However, because of the strong coupling between the hole's angular momentum and quasimomentum in GaAs-type crystals, the orientation of holes is quickly destroyed (during a period on the order of the momentum relaxation time τρ). Thus, although hot holes are oriented, the orientation of thermalized holes is practically absent. The situation is different in strained cubic or uniaxial crystals (see sect. 5.2).

The total intensity of radiation with polarization e is given by

where F is the density matrix of thermalized electrons. The index Μ assumes the values ± 3/2 or ± 1/2 for c-hh or c-lh transitions. It is taken into account in eq. (28) that the holes are nonpolarized.

The degree of circular polarization, p, is defined in the usual way,

where I+ and I_ are the intensities of right- and left-polarized radiation. Using eq. (26) and the matrix elements of the dipole moment listed in table

1, we find that for both types of transition

where nx is the unit vector in the direction of observation of the luminescence. This simple formula shows that the degree of circular polarization of the

luminescence is equal, apart from sign, to the component of the average electron spin along the direction of observation. The minus sign means that the angular momenta of the emitted photons are directed opposite to the average electron spin.

(28)

P = i + - i -

p = -S-n (29)


Thus, measuring the degree of circular polarization of the luminescence, we measure, in fact, the average spin of electrons in the conduction band. The maximum value of ρ for edge luminescence is 0.25.

The results obtained in this and the previous sections for the degree of orientation of thermalized electrons, as well as for the degree of polarization of recombination radiation, coincide with the results obtained by considering the selection rules for optical transitions between two atomic levels with angular momenta j = 1/2 and j = 3/2 (see fig. 6). This coincidence is related to the fact that the momentum distribution of thermalized electrons is isotropic. The anisotropic momentum distribution and spin-momentum correlation arising under photoexcitation reveal themselves in hot luminescence. In particular, the degree of circular polarization of hot luminescence is different for c-hh and c-lh transitions and exceeds the value 0.25 (see ch. 4).

Equation (29), derived for band-to-band transitions, remains true for luminescence caused by transitions from a shallow donor level to the valence band or from the conduction band to a shallow acceptor level. This is because the wave function of an electron localized on a shallow impurity level is formed by the corresponding band wave functions.

The polarization of exciton luminescence needs special consideration. A ground state exciton in GaAs may have angular momentum y = 1 or y = 2 (in spherical approximation). These states have different energy due to exchange interaction. The luminescence polarization depends essentially on the relation between the energy of exchange splitting, δ , and the quantity h / τ Η , where rh

is the relaxation time of the holes' angular momentum. If δ » ft /rh the coupling of the electron's and hole's angular momenta is strong and the relaxation of the hole's angular momentum results in electron spin relaxation with the same time r h . The exciton luminescence in this case will not be polarized if, as is usually the case, rh is much shorter than the exciton lifetime. In the opposite case, δ « : ft / T h , the electron spin cannot react to the fast rotations of the hole's angular momentum. Then the radiation will be polarized according to eq. (29).

Fig. 6. Selection rules and relative intensities of transitions between levels with j x = 3 / 2 and Λ = 1 / 2 .


More about optical orientation of excitons can be found in Pikus and Ivchenko (1982).

4. Dependence of electron spin orientation on energy of exciting photons

4.1. Influence of split - off band

The results of sect. 3 are valid for transitions in the vicinity of the point k = 0, i.e., when the energies of the exciting photons Α ω only slightly exceed the band gap Eg. With increasing Α ω - Eg the degree of electron orientation decreases, because states of light and heavy holes, participating in the transition, begin to be mixed up with states in the split-off band. At Ηω - Eg> 4 , transitions from the split-off band are allowed. One may see from table 2 that for such transitions electrons generated in the conduction band have an average spin S0

opposite in sign to the average spin of electrons excited from the light-hole and heavy-hole bands.

The electric field of the exciting beam directly influences only the electron orbital motion, not its spin. Thus spin orientation in interband absorption occurs only because of spin-orbit interaction. For Ηω - Eg^> Δ this interaction is of no importance, so the electron optical orientation is absent.

Let us calculate the average spin S0n of electrons excited from the nth sub-band of the valence band, assuming an arbitrary relation between the quantities Λ ω - Eg and Δ (Dyakonov and Perel 1971a). According to table 2 the values of S0 for transitions from states with j = 3/2 and j = 1/2 are, respectively, - 1 / 4 and 1/2. The probabilities for an electron of the nth sub-band to be in these states are defined by the coefficients a(

nJ)(k) in the

expression (16) for the wave function. Thus it is obvious that

where the coefficients a{nj) should be calculated for values of k defined by the

energy-conservation law

5 0 „ = - i k 3 / 2 ) i 2 + i K / 2 ) i 2 , (30)

hu-E = ev„(k) + ec(k). (31)

Using eqs. (17)-(19) one obtains


where the following notations are introduced:

4 m;l + 3m[1/4+m^/4 6 = 7 Z[ Γ ϊ > (33)

g = [36x 2-12x(f t + 3)+(3fr + l ) 2 ] 1 / 2, (34)

J - 1 - (35)

If in the process of thermalization the orientation of electrons, excited from different sub-bands of the valence band, does not decay, the average spin of thermalized electrons is

5Ο = Σ Ά „ / Σ > « > ( 3 6 ) η η

where rn are coefficients proportional to the joint densities of states in the Aith valence sub-band and the conduction band at values of k defined by eq. (31):

r i = (2xY/2(3b-3)-i/2, (37)

r 2x-lbK2-\ ' 2 · 3 Κ3 6 ( 3 * - 3 / > κ 2 - 1 ) + 9 κ 2 - ΐ ' ( 3 8)

±g + 6bx-3b-l 9(b2-l) · ( 3 9)

In eq. (39), as well as in eq. (32), the upper sign corresponds to the light-hole band (n = 2), the lower sign to the split-off band (n = 3).

According to eqs. (32) and (33) the frequency dependence of the average spin S0 is defined by a single parameter b. For gallium arsenide b~3.

The theoretical dependence of the degree of circular polarization of luminescence on the energy of exciting photons is depicted in fig. 7 together with experimental data of Ekimov and Safarov (1971b). The most distinctive feature of this dependence is the characteristic kink at the absorption threshold corresponding to transitions from the split-off band. This property was used by Hermann et al. (1980) and Arsentiev et al. (1980) for measuring the values of spin-orbit splitting Δ in various solid solutions.

4.2. Spin relaxation in thermalization

One can see from fig. 7 that at high impurity concentrations the experimental and theoretical curves are in good agreement, displaying the characteristic kink at Aco = Eg + Δ. The small discrepancy between curves 1 and 2 is obviously caused by spin relaxation of thermalized electrons.


Fig. 7. Dependence of the degree of circular polarization of the luminescence in GaAs on the photon energy of the exciting light (Ekimov and Safarov 1971b). (1) theory not taking spin relaxation into account, (2,3) experiment. Impurity concentration ( c m - 3) : (2) 4 X 1 0 1 9, (3)

7 .8ΧΙΟ 1 6. Τ = 4 . 2 Κ.

Quite different is the character of the experimental curve 3 corresponding to small impurity concentration. The main feature of this curve is the change in sign of the circular polarization at excitation energies slightly exceeding Eg + Δ. Such behavior was predicted beforehand (Dyakonov and Perel, 1971a), and is explained by a considerable decay of electron spin orientation in the thermalization process.

In noncentrosymmetric crystals (similar to GaAs) an effective mechanism of spin relaxation exists which is due to the band structure itself, and whose role rapidly increases with increasing electron energy. The spin splitting of the conduction band in such semiconductors [eq. (4)] may be viewed as a result of a magnetic field acting on the spins, whose magnitude and direction depend on the quasimomentum. The precession of the electron spin in this field occurs with frequency Ω defined by eq. (5). Changes of the electron quasimomentum due to collisions result in rotations of the precession axis. The time interval between successive rotations is of the order of the momentum relaxation time Tp. For Ωτρ <cl the angle of spin precession during the time interval between collisions is small and so dynamical averaging of the effective magnetic field takes place. This effect is analogous to the dynamical narrowing of magnetic resonance lines (Bloembergen et al. 1948). The spin relaxation time in this case is T S ~ (Ω2τρ)~ι where Ω2 is Ω2 averaged over the quasimomentum directions. The peculiar features of this mechanism lie in its being the more effective the cleaner the crystal (i.e., the longer τρ) and the higher the electron energy (since Ω2 is proportional to the cube of energy). Due to its strong energy dependence this mechanism is particularly important for photoexcited electrons generated high in the conduction band.


Fig. 8. Dependence of the degree of circular polarization of the luminescence in GaAs on the photon energy of the exciting light (Garbuzov et al. 1972). Solid curve: theory taking into account spin relaxation during thermalization; dots: experiment. Impurity concentration 3 x l 0 1 8 cm 3,

Τ = 4.2 Κ.

Let us return to the discussion of fig. 7. At Ηω- Eg> Δ there are two groups of photoexcited electrons: (1) electrons with large initial energy, excited in hh-c and lh-c transitions, (2) electrons excited in sh-c transitions whose initial energy is much less. The signs of the orientation of these two groups of electrons are opposite.

If the spin orientation does not noticeably decay during thermalization, as is the case in samples doped strongly enough (curve 2), then, as a result of mutual compensation, the average spin of all electrons tends to zero for Ηω - Eg » Δ. In sufficiently clean samples (curve 3) the orientation of the electrons of the first group is almost destroyed during thermalization, due to the mechanism discussed above. The resulting average spin of thermalized electrons is therefore determined by the orientation of electrons of the second group. This brings about the change in sign of the luminescence circular polarization.

Experimental results of Garbuzov et al. (1972) are presented in fig. 8 together with a theoretical curve calculated on the basis of the above considerations.

5. Deformational effects

5.1. Circular polarization of luminescence in strained crystals

The states in the valence band of a nondeformed cubic crystal are characterized by the projection of the angular momentum on the direction of the quasimomentum (see sect. 2.3).


Under uniaxial deformation, the four-fold degeneracy of the top of the valence band is lifted and the coupling between angular momentum and quasimomentum for states near the top of the band is destroyed. The direction of the quantization axis is now determined not by the quasimomentum, but by the axis of deformation, so that states are defined by the projection of the angular momentum on this axis.

The influence of deformation is essential for states with energies less than the deformational splitting, ΔΕ, at k = 0 (for GaAs ΔΕ «10 meV/kBar).

The selection rules for optical transitions, involving holes in this energy interval, change. This circumstance is essential in the excitation process only if the photon energy is close enough to £ , so that excited holes have the energy ε<ΔΕ. In edge luminescence the influence of deformation manifests itself at ΔΕ>Τ, where Τ is the temperature in energy units. (For transitions from the conduction band to the acceptor level Δ Ε should stand for the deformational splitting of the acceptor ground state.)

Let us consider the circular polarization of the edge luminescence in case Δ Ε > Γ, assuming that the energy of holes generated in the excitation process is much greater than ΔΕ. Then the thermalized holes are not polarized, and the luminescence polarization is completely determined by the average electron spin, S.

In order to find the connection between S and the degree of circular polarization, p, it is necessary to use eq. (27) and to take into account the influence of deformation on the valence-band wave functions in calculation of the matrix elements. The wave functions have the form given by eq. (7) in which, however, the z-axis should be chosen along the deformation axis.

Calculation gives the following result (Dyakonov and Perel 1973b)

where the vector Sx depends on the direction of observation as well as that of the deformation axis and also on the type of transition.

Under uniaxial compression the top of the light-hole band happens to be higher than the heavy-hole band, so that the recombination radiation is connected with c-lh transitions. For such transitions

ρ = 4 5 · 5 ] (40)

5 , = 3 f ( f - i i 1 ) - 2 n 1

5 - 3 ( ? · Λ ι ) 2 ' (41)

where ξ and nx are unit vectors in the directions of the deformation axis and observation, respectively.

Under uniaxial stretch, on the contrary, the top of the heavy-hole band is


higher, and c-hh transitions occur. Then

JitlhL (42) ι+α·«ι)2

If the luminescence is observed along the deformation axis, p = 2S*nl for compression, whereas for stretch ρ = -2S*nv so that the degree of polarization is, in absolute value, two times larger than for a nondeformed crystal, and has the opposite sign in case of compression [cf. eq. (28)].

Note that eqs. (41) and (42) coincide with eqs. (25) if one changes ν -> ξ, if -> nv

The dependence of the degree of circular polarization on deformation was used by Dzhioev et al. (1974) and by Vekua et al. (1975) for studying the deformation of GaAs epitaxial films. The effect of deformation on optical orientation in GaSb was studied in detail experimentally by Berkovits et al. (1976). These authors convincingly demonstrated the difference in sign of circular polarization for the transition lh-c and hh-c in the conditions when the deformation axis is parallel to the light beam.

Their results for the dependence of the degree of polarization of luminescence on deformation when the deformation axis is perpendicular to the beam (ξ ±n,nx) are presented in fig. 9. At maximum stress only lh-c transitions were excited, while recombination occurred in transitions of an electron to the upper (M = ± 1 / 2 ) of the ground state acceptor sublevels split by deformation. In these conditions the average spin of the photoexcited electrons is defined by eq. (27) in which 5 0 is given by eq. (25) for lh-c transitions, with the change ν-*ζ. For f i n we have S 0 = —2#i/5. According to eq. (41) (which is to be used in the case of transitions to the acceptor sublevel with Μ = ± 1 / 2 ) Sx = -2nx/5. In the experiment nx = - /i, thus we find from eq. (40) the value of the degree of circular polarization at large deformations,

ρ = -0.64(1 + T / T S ) _ 1.

In the absence of deformation

ρ = -0.25(1 + T / T S ) - 1.

Thus for large stress ρ must increase by a factor of about 2.5. Just this was observed experimentally.

5.2. Optical orientation of holes under deformation

It was mentioned in sect. 3.3 that in a nondeformed cubic crystal of the GaAs type the spin relaxation of holes is very fast—on the time scale of the momentum relaxation time τρ. This is caused by the strong spin-orbit interaction in the valence band, resulting in a rigid coupling between the hole's angular momentum and its quasimomentum. Deformation destroys this coupling for holes with energy e<AE. As a result, the spin relaxation of such

Theory of optical spin orientation

o.s

0.6

0.4

0.2

I 0 10 ^ 0

P , k^/mm2 .

Fig. 9. Dependence of the degree of circular polarization of the luminescence in GaSb on strain applied to the crystal (Berkovits et al. 1976). The deformation axis is perpendicular to the directions of excitation and observation. (1) Measured values of p; (2) measured values corrected by the factor (1 + T / T s) determined in the absence of deformation; (3) pressure dependence of the linear polarization J±/I\\> The value / ± / / | | = l / 4 at large deformations shows that the holes

occupy only the upper sublevels of the split acceptor ground level.

holes is slowed down. If photoexcitation produces holes with sufficiently small energies, then under deformation stationary orientation of thermalized holes is possible.

Let us estimate the spin relaxation time for holes in a deformed crystal (Dyakonov and Perel 1973b). Consider, for example, uniaxial stress. At the point k = 0 the wave functions of light holes correspond to projections of the angular momentum on the deformation axis equal to ± 1 / 2 . For finite values of quasimomentum these states are mixed up with states corresponding to projections equal to ± 3 / 2 . If the energy of the light hole, ε, is small compared to the deformational splitting ΔΕ, the amplitude of states ± 3/2 is of the order of ε/ΔΕ. Thus the amplitude of scattering, accompanied by flipping of the light-hole angular momentum, is proportional to (ε/ΔΕ)2, while the probability of such a process is proportional to (ε/ΔΕ)4. Therefore we have the following estimate for the hole spin relaxation time at ε<ΔΕ:

rs~rp(AE/e)4. (43)

35


Thus for sufficiently large deformation, in the case when holes with low energy are excited, the spin relaxation rate may be slowed down to such an extent that optical orientation of holes would be possible. Equation (43) describes the relaxation time of the projection of the angular momentum on the deformation axis. A detailed calculation of the hole spin relaxation in a deformed crystal was made by Rebane (1981). He also discussed the possibility of orientation of holes in optical transitions between sub-bands of the valence band, induced by circularly polarized light.

In noncentrosymmetric crystals an additional mechanism of hole spin relaxation exists, due to terms linear in k in the Hamiltonian, connected with the absence of Kramers degeneracy (a mechanism analogous to that discussed in sect. 4.2 for electrons in the conduction band).

Optical orientation of holes in GaSb crystals under deformation was observed by Lampel et al. (1978). These authors observed also a small orientation of holes in the absence of deformation.

The above considerations are as well applicable to uniaxial crystals, which may be regarded as strongly deformed cubic ones. Optical orientation of holes in the hexagonal crystal CdSe was first observed by Gross et al. (1971) by means of studying the polarization of luminescence due to annihilation of excitons bound at neutral donors.

6. Optical orientation of electrons in η-type semiconductors

6.1. Optical pumping

In p-type material it is the electrons photoexcited by circularly polarized light that become oriented. As was pointed out in sect. 3.4, orientation of holes in the valence band practically does not occur. Thus only spins of nonequilibrium carriers are oriented. Their number is proportional to the light intensity, however the degree of orientation is intensity independent. In this sense the phenomenon is analogous to optical orientation of excited atoms in a gas (see Cohen-Tannoudji and Kastler 1966).

We shall show in this section that in η-type semiconductors optical orientation of the majority carriers may be reached (Dyakonov and Perel 1971b)—an effect analogous to optical pumping of the gas atoms in the ground state.

Circularly polarized light generates electrons in the conduction band predominantly with a certain spin direction. Assume for simplicity that the electron recombination rate is spin independent. It is clear than that accumulation of oriented electrons in the conduction band will occur. If the spin relaxation rate is slow enough then in the stationary state the equilibrium electrons will be oriented. We stress that this may happen at such intensities of


excitation when the excess electron concentration is still small compared to the equilibrium one.

The average electron spin is given by a formula analogous to eq. (27):

where Tj has the meaning of the lifetime of an electron in the conduction band. In the situation considered the vanishing of an electron from the conduction band is due exclusively to its recombination with photoexcited holes. Under stationary conditions the recombination rate is obviously equal to the photoex-citation rate, thus

where η is the electron concentration, and G is the number of electron-hole pairs generated by light in a unit volume per unit time. Equations (44) and (45) show that the stationary value of the average electron spin is proportional to the intensity for small intensities Grs<^n, and saturates at the level S0 at GTs> n. The excess electron concentration is An = Gry where τ is the lifetime of a nonequilibrium hole. If T « T s , the saturation of orientation will occur at such values of G when still An<zn. Thus orientation occurs not by adding oriented photoexcited electrons to the equilibrium ones but by replacing nonoriented electrons by oriented ones.

Usually the light is absorbed in a thin layer near the crystal surface, the experimental conditions determining not G, but the number of electrons / generated per unit surface area per unit time. In this case excess carriers penetrate to a distance determined by the diffusion length L = (Dhr)1/2 where Dh is the hole diffusion coefficient. The orientation penetrates to a larger distance determined by the spin diffusion length L s = ( Z ) ET S) 1 / 2, De being the electron diffusion coefficient. The time Tj now defines the time of renewal of electrons in a layer of depth L s , so that the intensity / is distributed over this layer: G = J/Ls9 Tj = nL%/J.

Optical orientation of electrons in η-type semiconductors was observed by Ekimov and Safarov (1971a) and by Vekua et al. (1976).

Experimental data for the intensity dependence of the degree of circular polarization in η-type samples of G a 0 8 A l 0 2 A s are presented in fig. 10. The dependence is linear at small intensities and exhibits saturation at large intensities. The saturation occurs however at values of ρ less than the theoretical value 0.25. This difference may be related to additional electron spin relaxation by photoexcited holes, the rate of which grows with increasing intensity.

6.2. Optical and electrical effects in a semiconductor with oriented electrons

The nonequilibrium orientation of electrons arising in optical pumping may manifest itself in a number of optical and electrical effects. If electrons are

S = S 0( 1 + T , / T S ) -1 (44)

Tj = n/G, (45)


β

0.1S Λ a

ο 10 20 30 4 0 50 60

Fig. 10. (a) Luminescence spectrum of the n - G a 0 8A l 0 2 As crystal at 4.2 K. (b) Dependence of the degree of circular polarization of the luminescence on the intensity of the exciting light for the bands A (1) and Β (2). The solid curves were calculated with the help of the relation

ρ = C / / ( l + DJ), for C = 0.0145, and D = 0.115 (1) and D = 0.245 (2). (Vekua et al. 1976).

degenerate, their orientation leads to a difference in the position of Fermi levels ζ± for electrons with spins " u p " and "down". The values of ζ± are given by

where ζ0 is the Fermi energy for the same total electron concentration in the absence of orientation. Such a difference in Fermi levels should result in circular dichroism as well as in rotation of the plane of polarization of a probe light beam (Aronov and Ivchenko 1973).

Peculiar electrical effects in a degenerate system of spin-oriented electrons are connected with the energy dependence of the mobility. Because of this dependence and the difference in Fermi levels the mobilities as well as the diffusion coefficients of electrons with "up" and "down" spins should be different. When a gradient of spin orientation exists a specific "spin emf" arises, analogous to the Dember emf.

A number of effects is due to spin-orbit interaction in scattering of oriented electrons (Dyakonov and Perel 1971c, d). It is well known that, if a polarized electron beam is scattered by an unpolarized target, the spin-orbit interaction

f ± = f 0 ( l ± 2 S ) 2/3 (46)


leads to asymmetry of scattering relative to a plane containing the directions of spin and initial momentum. Such an asymmetry leads to the so-called anomalous Hall effect. (Review of literature on this subject in Maranzana 1967.) The expression for additional electron flow q due to the existence of spin orientation and its gradient has the form

q = - / ? r t £ x S - 5 r o t S , (47)

where the coefficients β and δ are connected in absence of degeneracy by the Einstein relation: β = e8/T. The first term is responsible for the anomalous Hall effect, while the second describes the peculiar effect of a current appearing as a result of nonhomogeneity in spin density.

In the usual conditions of experiments on optical orientation a spin-density gradient exists due to the surface character of excitation. This does not however produce an electrical current since the gradient is normal to the surface and parallel to S, so that rot(flS) = 0. With the help of a magnetic field applied parallel to the surface one may create a component of S along the surface. This would lead to the appearance of a current along the direction of the magnetic field (Averkiev and Dyakonov 1983). This effect may be used for electrical detection of the electron optical spin orientation.

We note that an inverse effect is also possible: a current flow must lead to spin orientation at the sample surface (Dyakonov and Perel 1971c, d).

Equation (47) is written for a cubic crystal with an inversion center. In the absence of an inversion center an additional term, proportional to 5 , may be written in eq. (47), i.e., homogeneous nonequilibrium orientation by itself results in the appearance of a current (Ivchenko and Pikus 1978, Belinicher and Sturman 1980). Such an effect was observed in optical orientation in Te (Asnin et al. 1978).

7. The Hanle effect

7.1. Optical orientation in a magnetic field

Depolarization of luminescence by a transverse magnetic field is called the Hanle effect. This effect was discovered and studied in the resonance fluorescence of gases (Hanle 1924). It is effectively employed in experiments on optical orientation in semiconductors. The first observation of the Hanle effect in a semiconductor was made by Parsons (1969).

The essence of this effect lies in the precession of spins of photoexcited electrons in a magnetic field Β with the Larmor frequency Ω = /zBg2?/A where μ Β is the Bohr magneton, and g is the g-factor. If the magnetic field is perpendicular to the exciting light beam, i.e., Β JL 5 0 , the component of the spin along the direction of the beam changes periodically with frequency Ω.


Let us introduce the "spin lifetime" Ts,

As ' 'S

where τ is the electron lifetime, and T s is the spin relaxation time. Equation (48) reflects two reasons for the average electron spin to vanish: recombination and spin relaxation. If during the time Ts the electron spin makes many revolutions around B, then in stationary conditions the component of the average spin along the initial direction (S 0) will be very small. It is just this component that defines the degree of circular polarization of the luminescence in the usual experimental conditions when nx\\n (see eq. (29)).

At time t after excitation the component S2 of the electron spin will be 50cosi2iexp(- t/rs). Its average value may be obtained by averaging this expression with the distribution of lifetimes W(t)\

Sz = S0 Γ at W(t)exp(- t/rs)cosQt. (49)

Under usual conditions for thermalized electrons we have W(t) = T - 1e x p ( - t/τ). Then eq. (49) gives the well known expression for the Hanle curve

S^B) = - ^ - 2 , SZ(0) = T^7-- (50) l + ( i2r s) 2 1 + T A S

Equations (50) give the basis for determining the times τ and T s by the method of optical orientation. By measuring the degree of polarization in zero field and the halfwidth of the magnetic depolarization curve (the Hanle curve), ^ 1 / 2 = ^ / g ^ B r s , one can find τ and T s (if the g-factor is known). Such measurements were done in numerous works. The values of Bl/2 vary from several G to several kG, depending on material, impurity concentration and temperature.

Equations (5) may also be derived from the equation of motion of the average spin in a magnetic field, taking into account spin relaxation, generation and recombination:

^ = fix5-^-^. (51) dt T s τ

For the stationary state, putting dS/dt = 0 and assuming that Ω ± 5 0 , it is not difficult to obtain eqs. (50) from eq. (51). It follows also from eq. (51) that the angle φ between S and 5 0 depends on the magnetic field as tan φ = ΩΓ8.

7.2. Hanle effect in cascade processes

In the previous section the simplest case of electron depolarization in a magnetic field was described. More complicated situations may exist in which


the Hanle curve does not have the simple Lorentz form given by eq. (50). One of these situations is considered in this section. Suppose there are two electron states, 1 and 2, the pumping being done in state 1 while the luminescence is observed from state 2. The electron gets to state 2 from state 1 via a transition defined by a characteristic time τ 1 2. Equations for the average electrons spins 5 ( 1 ) and S(2) in states 1 and 2, respectively, have the form

« _ _ β , χ 5 « , _ * _ + & « . . β , χ ^ . ^ Ι . ( 5 2 )

where Ts /, and τ, are respectively, the Larmor frequency, "spin lifetime" and lifetime for the ith state. Assuming as before that β, ± 5 0 , we find for the stationary state:

s-c > w°5 C > ( 0 >%+.Wa+.^,r (53)

S B ( 0 ) , i l i l s 0 . (54)

Equation (53) may be reduced to

The shape of the Hanle curve calculated from eq. (55) for ΩχΤ^ - ^ 2 ^ s 2 ^ depicted in fig. 11. A characteristic feature of this curve is the change of sign of orientation. It may be explained as follows. In the usual Hanle effect the maximum angle of rotation of the average spin in a magnetic field is 90°, this value being reached for i2!Ts->oo. In the cascade process the spin of an electron coming to state 2 is already rotated so that the value φ = 90° is reached at a finite value of B.

The roles of states 1 and 2 could be played, for example, by electron states in the conduction band and at an impurity level. A Hanle curve with change in sign was observed by Razbirin et al. (1975) in resonantly induced exciton luminescence in GaSe crystals. This effect was interpreted as a result of a cascade transition, the role of states 1 and 2 being played by exciton states with momenta parallel to the exciting beam and to the luminescence beam, respectively (Ivchenko et al. 1977).

A cascade process also occurs in emission of optical phonons by hot electrons (see ch. 4).

Equations (53) and (54) may easily be generalized to the case when the cascade consists of several steps:

S O O ( j ) - S < " > ( 0 ) R e n 1 + io r . S<«>(0) = Son ^ • (56) k=l l^n£kA^k k=l Tk

42 Μ. I. Dyakonov and V.L Perel

If the number of steps, «, is large eqs. (56) may be simplified by using the identity l + = expln(l + ii2^rsA;) and expanding the logarithm in terms of ®kT&k. Keeping only the first term of the expansion and assuming, for simplicity, all i2k identical, we obtain from eq. (56):

SZ(B) = S(0)cosO/0, S(0) = 5 0exp( - r 0 / r s ) , (57)

where t0 = Ll=irk is the total time of transition of an electron from the initial state to the final one. In deriving eq. (57) we put Tsk = (l/rk + 1 / T s ) _ 1 and assumed that T S » rk.

Thus if the cascade consists of a large number of steps the average spin in the final state oscillates as a function of the magnetic field. This is due to the fact that the total transition time t0 is well defined. Indeed the rms fluctuation At0 is of the order of t0/yfn . Equation (57) may be derived from eq. (49) if one puts W(t) = 8(t — / 0 ) . Fluctuations of t09 though small, lead to damping of oscillations which may be taken into account if one keeps the term quadratic in QkTsk in the expansion of the logarithm. Then an additional factor exp( - Ω2θ2/2) will appear in eq. (57) for SZ(B), where Θ2 = ZUrf.

The effect of oscillations of the average spin of photoexcited electrons in a magnetic field due to the Larmor precession during the energy relaxation time was considered theoretically by Fishman and Hermann (1974).

Fig. 11. The Hank curve for a cascade transition calculated according to eq. (55) for SilTsl̂ i22Ts2 = aTs.


A specific effect of spin oscillations in a magnetic field in a graded-band structure was observed by Volkov et al. (1977). Theoretically this effect was considered by Volkov and Tsarenkov (1977). The graded-band semiconductor (a semiconductor in which the band gap depends on the coordinate) p-(GaAl)As was studied, in which the band gap decreased inside the sample from the surface value 1.98 eV with gradient 160 eV/cm. The sample was illuminated from the side, where the gap is wider, by circularly polarized light with quanta of energy #<oexc =1.96 eV. A magnetic field was applied perpendicular to the beam. Circular polarization of recombination radiation, coming in the direction opposite to that of the exciting beam, was detected. The energy of radiation quanta determines the depth at which the recombination occurs, Ηω = Eg(x). Thus, to each part of the luminescence spectrum a definite drift time corresponds for an electron to come from the point of photoexcitation x0(Eg(x0) = #(oe x c) to the point of recombination x. The spin precession in the magnetic field occurring during this time reveals itself in oscillations in the luminescence spectrum.

Fig. 12. Spectral dependence of the luminescence circular polarization of a graded-band semiconductor in a transverse magnetic field (Volkov et al. 1977). Magnetic field, kG: curve 1,0; 2,

1; 3, 2.7. The ordinates of curves 2 and 3 are amplified 3 and 14 times respectively.


Experimental data of Volkov et al. (1977) are presented in fig. 12. The decrease in amplitude of oscillations at the low frequency end of the spectrum is due to electron diffusion accompanying their drift motion.

Similar oscillations are observed in the magnetic field dependence of ρ for fixed hω.

7.3. Hanle effect in η-type semiconductors

It was shown in sect. 6.1 that in η-type material under usual conditions of surface excitation the electron spin orientation penetrates into the sample to a depth of the order of the spin diffusion length L s = ( Z ) eT s) 1 / 2. This length is much larger than the depth at which the light is absorbed as well as the diffusion length L = (Dhr)1/2. Specific features of the Hanle effect in n-type samples are due to the fact that one of the processes determining the "spin lifetime" in the surface layer, where recombination occurs, is spin diffusion. In the main region where orientation exists (in a layer of depth L s) there are no holes and recombination is prohibited. Depolarization of electrons in this region leads to a changing gradient of the degree of orientation, and so the spin diffusion rate becomes magnetic-field dependent. It is clear that in such conditions the shape of the Hanle curve must be unusual and that it should depend on the intensity of the exciting light.

The following equation is valid for the average electron spin in the region where excess carriers are absent:

f« D t 0_ | + O x s . (58) Assume that the magnetic field is directed along the x-axis and is parallel to

the surface. The boundary condition to eq. (58) follows from consideration of processes in a narrow surface layer where recombination and generation of excess carriers takes place. It has the form

JS0 = -nD^ + JS a t z = 0. (59)

Here the left-hand side represents generation of the electron spin by light, the first term in the right-hand side gives the flow of diffusion spin inside the sample, while the second one describes the vanishing of spin in recombination (notations are the same as in sect. 6.1).

The solution of eq. (58) with boundary condition (59) gives the following formula for the dependence on transverse magnetic field of the average component of the spin, 5 Z, at the surface (Dyakonov and Perel 1976):

SZ{B) = SMHB), 5,(0) = ^ , (60)

(ΐ + α)ίΐ + α/(ΐ + η ) / 2 ) F(B) = - ^ y, ' ; / \ (61)

1 + α2τ, + 2α/(1 + η ) / 2


where the following notations were introduced:

η = (ι + Ω \ ψ 2 . (62)

The expression for S,(0) coincides with eq. (44). Equation (61) may be simplified in the limits of large (a<zl) and small

(a » 1 ) intensity of the exciting light. At small intensities the Hanle curve is intensity independent and is described by

At large intensities, when in zero field the average spin is practically equal to S0, we have

1 + (Ωτ*)ι/2

1 + 2 ( Ω τ * ) 1 / 2+ β τ *

where τ* = a\/2, i.e.,

... n2D„ 2J2

(65)

Thus at large intensities the Hanle curve should be broadened proportional to the square of the intensity. The dependences given by eqs. (63) and (64) are presented in fig. 13. A characteristic feature of both curves is the relatively slow decrease of orientation at high magnetic fields. In the usual Hanle effect at high field S(B) ~ l/B2, while eqs. (63) and (64) give S(B) - l/B1/2.

Let us discuss the physical meaning of the result obtained. The halfwidth of the Hanle curve is determined by the shortest of the times T s and T Y which characterize, respectively, spin relaxation and the vanishing of spin as a result

Fig. 13. The shape of the Hanle curve in an η-type semiconductor in case of surface excitation (Dyakonov and Perel 1976): (a) for small intensity [eq. (63)]; (b) for large intensity [eq. (64)].

(64)

46 Μ.I. Dyakonov and V.I. Perel

of recombination of electrons with photoexcited holes. At small intensities, when Tj » T s , the half width of the Hanle curve is intensity independent, being determined by the time T s [eq. (63)]. The degree of orientation in zero field is small and proportional to the intensity.

At large intensities, when T ; « T s , recombination is the most essential process. The degree of orientation in zero field is saturated and is intensity independent. In this case the characteristic depolarizing field is such that β τ δ » 1 . It is important that in such fields the width of the layer where oriented electrons exist is reduced, becoming equal to (De/Q)1/2 instead of Ls = (Ders)1/2. In these conditions the time for spin vanishing because of recombination is f7 = n(De/Q)1/2/J. Thus at large intensities the depolarizing field is determined from the condition β τ 7 ~ 1 . This is equivalent to the condition Ωτ* - 1 , where τ* is given by eq. (65).

We note that the theory presented above becomes invalid if Β is so large that ( D e / i 2 ) 1 / 2 becomes less than the diffusion length L. In such fields the region where orientation exists coincides with the region where there are excess carriers. The far wing of the Hanle curve is described by the asymptotics of the Lorentz curve, F= (QnL/J)~2.

Experimental studies of the Hanle effect in η-type (GaAl)As solid solution were performed by Vekua et al. (1976).

Typical experimental results, together with curves calculated with the help of eqs. (63) and (64), are presented in fig. 14. One can see that the experimental

3

0 1 2 3 4 S 6 7 8 Β / β ι / 2

Fig. 14. The Hanle effect in an n - G a 0 8A l 0 2A s crystal at 4.2 Κ for the Α-band presented in fig. 10 (Vekua et al. 1976). Curves 1 and 2 were calculated according to eqs. (63) and (64) respectively, curve 3 is the Lorentz contour with halfwidth Bl/2 = 6 G equal to the halfwidth of the

experimental Hanle curve.


curve is essentially different from the Lorentz one, as well as that it has a characteristic tail in the high-field region. Vekua et al. (1976) also observed broadening of the Hanle linewidth with increasing intensity of excitation. This broadening, however, was slower than the theory predicts.

8. Circular polarization of luminescence in magnetic field for nonpolarized excitation

8.1. Polarization in equilibrium conditions

Orientation of the carriers by a magnetic field will, obviously, lead by itself to circular polarization of the recombination radiation. The origin of nonequi-librium carriers is insignificant: they may be created, for example, by interband absorption of nonpolarized light or by injection through a contact.

The polarization of the recombination radiation in a weak magnetic field (μΒΒ <^T) for GaAs-type semiconductors was calculated by Dyakonov and Perel (1972a).

Assume, to begin with, that both electrons and holes are, separately, in thermodynamic equilibrium. In linear approximation in Β the contributions of electrons and holes to the luminescence polarization, p, are additive. It may be shown that these contributions are equal to the average angular momenta of corresponding carriers taking part in recombination. Thus

P = Pe + Ph> Pe = <S 2> , P h = < ^ > > ( 6 6)

where Sz and Jz are the components of the electron and hole angular momenta along the direction of the magnetic field. The symbol ( · · · ) in eqs. (66) denotes the averaging over k:

fd3kJ(k)f(k)<p(k) (J) = ~ , (67)

/ d 3 * / ( * M * )

where f(k) and φ(£) are the electron and hole distribution functions in the absence of magnetic field.

We will present here the results following from eqs. (66) and (67) for the case when both electrons and holes are nondegenerate. In this case the average electron spin, 5 7, does not depend on the quasimomentum, so that

(68)


Calculation of p h gives

Ph = 3 μΒΒ 2 Τ

+ ^ ( κ - Υ ι + γ ) ( / 4 / 2 - μ 1 / 2 ) (μΥ2 + μΥ2Υ (69)

where j u h = mcmh(me + mh)~l is the reduced mass of the electron and the heavy hole measured in units of the free-electron mass m 0 , μ} is the same for the electron and the light hole, γ 1 ?γ , κ are the constants entering Luttinger's Hamiltonian.

Consider now the case when the recombination radiation is emitted in transitions from the conduction band to the acceptor level. The acceptor ground state is four-fold degenerate and p h coincides with the average angular momentum of a particle with total angular momentum 3/2 in a magnetic field,

Ph = 5 μ^ΑΒ

(70)

where g A is the g-factor of a hole bound to an acceptor. Calculation in the frame of spherical approximation for mh^> m] gives the following expression for g A (Gelmont and Dyakonov 1973):

£ Α = !κ + § Ϊ Ι - 2 γ . (71)

Measurements of the luminescence circular polarization in a magnetic field allow one to define the g-factor of a hole bound to an acceptor, if the electron contribution to the polarization may be neglected. This is just the situation in an η-type semiconductor if the electrons are degenerate. In this case, instead of eq. (68) we have

pe = 3/iBg£/8f « p h ,

where ζ is the electron Fermi energy. Such measurements were done for GaAs by Dzhioev et al. (1973).

8.2. Polarization in case of departure from equilibrium

We assumed in the previous section that the orientation of electrons and holes in a magnetic field corresponds to thermodynamic equilibrium. In reality the spin relaxation may be not fast enough for the equilibrium orientation to be reached during the lifetime. Let us find the stationary electron orientation assuming the relaxation of holes to be fast enough so that their orientation may be considered as having its equilibrium value.


We consider a p-type semiconductor and restrict ourselves for simplicity to the case when recombination is due to transitions from the conduction band to the acceptor level. The following balance equations may be written (if the excitation light is nonpolarized):

dn+ n+ n+ — n_— 2STn G ~ Ϊ Γ = ~77 2 ^ + ~29

dn_ n_ n+-n_— 2STn G , , - τ - = + — ~ — + ΊΓ . (72)

at τ_ 2TS 2 where n+ and n_ are the concentrations of electrons with spins directed along and opposite to the magnetic field, τ + and τ_ are their lifetimes, η = n+ + n_, ST = μ^Β/4Τ is the equilibrium value of the average electron spin, G is the number of electrons generated in a unit volume per unit time.

The difference between the lifetimes τ + and τ_ is due to the difference in occupation of the sublevels of the acceptor ground level. Thus we may write:

1 1 4 a . Λ

7 Γ - " 3 7 ^ ( 7 3 )

where 2 / τ = τ+ 1 + τ ΐ 1 , p h coincides with the average angular momentum of a hole bound to an acceptor and is defined by eq. (70), and α is a dimensionless coefficient depending on the recombination mechanism. The numerical factor in eq. (73) is chosen in such a way that when the recombination is purely radiative, the coefficient a is equal to unity. In this case the difference τ " 1 - τ ! 1 is easily calculated if one uses the ratio 3 :2 :1 :0 of the probabilities of transitions of an electron with spin m = 1/2 to acceptor sublevels with Μ = 3/2, 1/2, - 1 / 2 , and - 3 / 2 , respectively.

From eqs. (72) one finds the stationary value of the average electron spin, S = (n+ — n_ )/2(n+ + n_), which gives the electron contribution to the circular polarization of the luminescence. Taking eq. (73) into account we obtain

r S R - « T s P h/ 3 E Τ + T C

The first term in eq. (74) represents the equilibrium electron orientation reduced by the factor τ / ( τ 4- T s) . The second term describes the peculiar effect of electrons being polarized due to the fact that orientation of holes leads to different recombination rates for electrons with spins directed along and opposite to the magnetic field.

For T S « T eq. (74) gives the same result as eq. (68). In the opposite limiting case pe = - a p h / 3 .


9. Dynamic polarization of lattice nuclei by optically oriented electrons

9.1. Nuclear spin system

We recall in this section the basic properties of the nuclear spin system in crystals (see Abragam 1961). First of all, this system is to a large extent isolated from the lattice: the spin-lattice relaxation time Tx is on the order of a few seconds to a few days, depending on lattice temperature and content of paramagnetic impurities. At the same time interactions inside the nuclear spin system are characterized by much shorter times. These interactions include in the first place dipole-dipole interaction of the nuclear magnetic moments. Each nucleus experiences a fluctuating local magnetic field BL created by its neighbours, the value of BL being on the order of / X j / a 3 , where μγ is the nuclear magnetic moment and a is the lattice constant. This field is of the order of a few G. The period of the nuclear spin precession in this field, Ηα3/μ\9 which is on the order of 10~ 4 s, defines the time T2 characteristic of the interaction in the nuclear spin system. Along with dipole-dipole interaction, an important role is often played by interactions due to virtual electron transitions (indirect exchange).

Owing to internal interactions the nuclear spin system reaches thermodynamic equilibrium during a time of the order of T2. This state is described by a single parameter, the spin temperature Θ which may, generally speaking, differ strongly from the lattice temperature. The rate of equilibration of the spin and lattice temperatures is defined by the large time of spin-lattice relaxation Tv

Dipole-dipole interaction does not conserve the total nuclear spin. Thus in the absence of an external magnetic field the average nuclear spin is equal to zero. In the presence of an external field Β the average nuclear spin, / a v , is defined by the usual thermodynamic formula

/ a v = H/ + l ) / M W (75)

where / is the value of nuclear spin and Θ is the spin temperature in units of energy. Here, as well as in the following, it is assumed that μ^/Θ^Ι (high-temperature approximation). For Β = 1 G this approximation is valid if Θ > 1 0 ~ 7 Κ. The high-temperature approximation implies also the condition μιΒι^/Θ 1. If this condition is violated, i.e., at temperatures Θ < 1 0 ~ 7 - 1 0 - 8

Κ the spin-spin interactions will result in a phase transition: ferro- or antiferromagnetic ordering of nuclear spins.

The energy of the nuclear spin system in an external magnetic field consists of two parts: Ji? = Jfz + Jfd, where

^ζ=(μι/Ι)'Β·Σΐη


is the Zeeman energy (the sum is over all the nuclei in the crystal), and J^a is the energy of spin-spin nuclear interactions. An effective energy exchange between the two parts on a time scale defined by Γ2, leading to the establishment of a common spin temperature, occurs only when Β < BL. If, however, Β greatly exceeds BL, the exchange time grows significantly and may become even greater than Tv In this situation the Zeeman and spin-spin reservoirs may be described by different temperatures. The Zeeman reservoirs for nuclei of different species may also have different temperature (for more details see Goldman 1970).

9.2. Hyper fine interaction of electrons and nuclei

The electron and nuclear spins in a semiconductor are coupled through hyperfine interaction. For electrons in the conduction band described by s-type Bloch functions the contact hyperfine Fermi interaction prevails,

ν=ψμ»μιμ(*)\2ί'έ, (76)

where ψ(/?) is the electronic wave function at the nucleus. The hyperfine interaction of nuclear spins with holes in the valence band is of dipole-dipole nature (since the holes are described by p-type Bloch functions) and is considerably weaker.

The nuclear spin relaxation rate due to interaction with holes was calculated by Gryncharova and Perel (1977).

The wave function in a crystal ψ(/?) is a product of the Bloch amplitude u and the modulating wave function Ψ(Ε): \p(R) = Y(R) u(R). We assume the following normalization conditions for u and Ψ:

(\u(R)\2d>R = v0, / d 3 l ? | * ( * ) | 2 = l , (77)

where the first integral is taken over the volume u0 of the unit cell, and the second is over the entire volume of the crystal.

Equation (76) may be rewritten in the form

V=Ai-SMR)\2v0, Λ = ^ « η , (78)

where η = |w(0)|2, and w(0) is the Bloch amplitude at the site of the nucleus. If the electron density were homogeneously smeared over the unit cell, η would be equal to unity. In fact, the electron density has a sharp maximum at the nucleus. Thus, for InSb Η Ι Η = 6.3 XlO3, T J SB = 1 0 . 9 x l 0 3 (Gueron 1964). For GaAs Paget et al. (1977) estimated T ] GA = 2.7 Χ 1 0 3 , T J As = 4.5 Χ 1 0 3 .


9.3. Nuclear spin relaxation due to electrons

Hyperfine interaction is a result of the magnetic field created by the electron magnetic moment and acting on the nucleus. Fluctuations of this field lead to nuclear spin relaxation. The order of magnitude of the relaxation time Tlc due to interaction with electrons can be estimated using the familiar formula of dynamic averaging

y - = * 2 T E , (79) M e

where ώ is the precession frequency of the nuclear spin in the fluctuating field created by electrons and T E is the correlation time of these fluctuations.

An external magnetic field lowers the nuclear spin relaxation rate. This is described by an additional factor (1 + β 2 τ ε

2 ) - 1 in eq. (79), Ω being the electron Larmor frequency. The following estimate holds for ώ2, if the electron is localized in a region rQ:

*7~(m)'F- <80> where Ν ~ r0

3/*;0 is the number of nuclei in the localization region and F is the probability for an electron to be in this region. The quantity A/hN in eq. (80) is the precession frequency of a nuclear spin in the magnetic field created by a single electron. In the case of free, nondegenerate electrons, r0 stands for the wavelength λ - h(mT)~l/1, so that F ~ ηλ3 (η is the electron concentration). The correlation time T c in this case is just the time needed for an electron to travel a distance of the order of λ, hence re~h/T. Thus we obtain the following estimate for the time Tle in the case of interaction with free electrons:

^--A2ulnm3^Tl^h-4. (81) M e

An accurate calculation by Bloembergen (1954) (with correction of Abragam 1961) gives a numerical factor in eq. (81) equal to 2 / ( 2 π ) 3 / 2.

An estimate of Tlc in GaAs for η = 1 0 1 5 cm" 3, Γ = 4 Κ, A = 1 0 " 1 6 erg gives a value of the order of 10 4 s.

For electrons localized at donor centers r0 ~ aB (aB being the Bohr radius), F= η/ΝΌ is the filling factor (now η is the donor concentration), and T E is the lifetime of an electron in the bound state, determined by the processes of thermal ionization, hopping to neighbouring donor sites, as well as recombination. It may also be determined by the electron spin relaxation time, if this time is shorter than the characteristic times of the processes mentioned above. Thus we obtain the following estimate for nuclei located in the vicinity of the donor center:

(82)


Quantitative calculation gives the following result for a nucleus located at a distance r from the donor site (Lampel 1968b, Dyakonov and Perel 1973a):

For a shallow donor in GaAs for r<^aB, η = 0ΛΝΌ, T e ~ 1 0 s one obtains Tlc ~ 0.1 s. These estimates demonstrate that localized electrons are much more effective in nuclear spin relaxation than free ones (Bloembergen 1954, Lampel 1968b).

Under the conditions of optical orientation of electrons, hyperfine interaction leads to dynamic polarization of the lattice nuclei. The steady-state value of this polarization is calculated in sect. 9.5. This stationary value is reached during the time 7\ e.

Nuclear polarization is usually achieved due to electrons bound to donors. One can see from eq. (83) that the characteristic time Tlc increases greatly with increasing distance from the nucleus and the donor site. This effect was observed by Kalevich et al. (1982a, b).

The direct polarization of nuclei located far enough from the donor center becomes ineffective. These peripheral nuclei are polarized due to nuclear spin diffusion, which may be described by a diffusion coefficient D - 1 0 " 1 3 cm 2/s (Abragam 1961). The propagation of nuclear polarization over a distance r, due to diffusion, occurs during a time on the order of r2/D. If the time of diffusion over the distance aB is greater than 7 \ c ( a B) , the "diffusion radius," S, may be introduced (see Khutsishvili 1965) as a solution of the equation 82/D = 7\e(S). Nuclei for which r < δ ("near" nuclei) are polarized directly, while nuclei in the peripheral region, τ > δ, are polarized by spin diffusion. We note that optical manifestations of nuclear polarization (see sect. 10) are due to the near nuclei.

Direct experiments demonstrating the presence of nuclear spin diffusion under the conditions of optical orientation were recently done by Paget (1982). Here is a brief description of one of his experiments. As a result of prolonged illumination by circularly polarized light, near nuclei as well as peripheral ones were polarized. A subsequent brief illumination by linearly polarized light washes out the polarization of the near nuclei. During the following dark period this polarization is restored, due to spin diffusion from the peripheral region. The polarization of the near nuclei was registered by optical detection of NMR.

9.4. Nuclear field

The interaction (76) plays a dual role. On the one hand, it leads to relaxation and dynamic nuclear polarization, described above. On the other hand, in the presence of spin orientation it leads to the appearance of effective nuclear (1*N)

(83)


and electronic (Be) magnetic fields acting, respectively, on electronic and nuclear spins. These fields are known to produce the Overhauser shift in electron spin resonance and the Knight shift in NMR.

An electron moving freely in the conduction band or localized on a shallow donor interacts simultaneously with a large number of nuclei. Thus for a shallow donor in GaAs the Bohr radius is aB «100 A , so that about 105 lattice nuclei exist in the localization region of the electron. Under these conditions the concept of a mean nuclear field is quite satisfactory.

In order to obtain the expression for the nuclear field, let us find the average (with respect to nuclear state) interaction energy of an electron spin with all nuclei (of a certain species), supposing their mean spins / a v to be equal. To do that we sum the expression (78) for V over all the nuclei. Replacing the sum over nuclear positions by an integral, and using the normalization condition (77) we find the average energy to be ^ f S * / a v. Regarding this quantity as an electron spin energy in an effective magnetic field 2fN, we obtain the expression for the nuclear field of a given species:

= b^hJL bN = j ^ i ^ ' ( 8 4)

where ξ is the number of nuclei of the species considered in the unit cell. The quantity 6 N is the nuclear field, which would appear in the case of

complete nuclear polarization. If the electron density were smeared homogeneously over the unit cell (η = 1), this field would be on the order of 1 G, i.e., of the same order of magnitude as the real average magnetic field, created by the nuclear magnetic moments. In fact, due to the factor η, for heavy nuclei the field bN is much greater. Thus for lattice nuclei in GaAs, taking into account the natural abundance of isotopes, the following estimate may be given (Paget et al. 1977):

fcN(69Ga) = -13 .7 kG, bN(nG<i) = -11 .7 kG,

ftN(75As) = - 2 7 . 6 k G . (85)

The sign of bN (at μι > 0) is defined by the sign of the electron g-factor. In GaAs g = -0 .44 (Weisbuch et al. 1974).

Thus, even if the nuclear polarization is comparatively small, the electron spins in a semiconductor may be subjected to the action of quite large fields. It is these fields that lead to optical manifestations of nuclear polarization (see sect. 10).

9.5. Dynamic polarization of nuclei by oriented electrons

As was already mentioned in the foregoing sections, hyperfine interaction between lattice nuclei and optically oriented electrons results in nuclear


polarization. The characteristic time of this process is defined by the value of 7\ e, discussed in sect. 9.3. Here we consider the steady-state value of the nuclear polarization in an external magnetic field, which is strong enough: Β » BL. In this case spin-spin interactions may be neglected, and the interaction of each nucleus with electrons may be considered separately. Nuclear polarization in weak fields will be considered in sect. 1 1 .

Moreover we will not discuss effects, due to quadrupolar splitting of nuclear spin levels, which is highly important in crystals of non-cubic symmetry, as well as in solid solutions (see ch. 5).

The contact hyperfine interaction (76) conserves the total spin of the electron-nuclei system, hence the balance equation for the steady-state populations Νμ of the nuclear spin levels has the form

%9ΐί-ιΝμ»- = %-ι.Λ-ιη + > (86) where μ is the projection of the nuclear spin on the direction of B, and η ± = 1 / 2 ± S are the populations of electronic states with spins up and down. The transition probabilities W μ_χ and W x are connected by the thermodynamic relation

» ; . M - i = « p ( / i B g B / r ) » ; _ l f | l. (87)

The exponent contains the energy difference between states ( μ , - ) and μ - l , + ) of the system nucleus + electron, in which we have neglected the Zeeman energy of the nucleus. The temperature Τ should be understood as the temperature of the reservoir from which the energy needed for the transition is taken. We will consider it equal to the lattice temperature.

From eqs. (86) and (87) we find

Νμ _ ( l + 2 S ) ( l - 2 S r ) ( g 8)

Νμ_, ( l - 2 S ) ( l + 2 S r ) '

where

S r = 4 t a n h ( ^ f ) (89) is the equilibrium value of the average electron spin.

The average nuclear spin / a v = ΣμμΝμ can be expressed in terms of the Brillouin function,

Iw = IB,(x\ (90)

where

Υ Λ (1 + 2S \-2ST\ * = / l nl T ^ T T 2 s ; J - ( 9 1)


In the particular case / = 1/2 we have

= s-sT av l-4SST

v }

Any deviation from equilibrium in the electron spin system results in dynamic nuclear polarization. When electrons are artificially maintained in a disordered spin state (5 = 0), eqs. (90) and (91) describe the Overhauser effect (Overhauser 1958). As distinct from the usual Overhauser effect, such a disordered state may be reached by optical pumping by nonpolarized (or linearly polarized) light (Lampel 1968a, b). Under the conditions of optical orientation, as a rule, S is much greater than ST and the quantity ST may be neglected in eqs. (91), (92).

Very often one needs only retain the first term in the expansion of the Brillouin function in terms of S. In this case, neglecting ST, we obtain

/ a v = | / ( / + l ) S . (93)

The average electron spin of optically oriented electrons is not necessarily parallel to the external magnetic field. The dynamic nuclear polarization, however, is caused only by the component of S longitudinal with respect to the magnetic field. At the same time under stationary conditions the nuclear spin is parallel to B. Thus, the vector generalization of eq. (93) is

/ . - l , ( / + l)i££2l (94)

So far we have considered the interaction with oriented electrons as the only process defining the nuclear spin state. If other processes of spin-lattice relaxation (characterized by the time 7\) are taken into account, a leakage factor, f = Tl/(Tl + T l e) , should be introduced into the right-hand side of eq. (94).

It was pointed out in the previous section, that nuclear polarization brings about the appearance of an effective field 2?N, acting on electron spins. According to eqs. (84), (94) the expression for this field may be presented in the form

(BS)B B2

* = ! Σ ( ' « + 1)6Ν«/«> (96)

' ^ b ^ ^ 9 (95)

where the sum is over all the nuclei in the unit cell of the crystal. All the results of this section were obtained under the assumption that the

contact Fermi interaction (76) is the dominant interaction between the electronic and nuclear spins. It is well known (see Abragam 1961) that when the electrons are bound to impurity centers, the nuclei situated outside the


localization region of the electron are polarized due to dipole-dipole interaction with electrons and to spin diffusion. A detailed investigation of different processes of nuclear polarization by localized electrons, as well as of the competition of these processes, depending on the impurity species and the magnetic field value, was carried out by Bagraev et al. (1976, 1977, 1979) and Bagraev and Vlasenko (1979). By special doping of silicon crystals these authors succeeded in substantially enhancing the degree of nuclear polarization under optical pumping.

Buishvili et al. (1974) pointed out that effective nuclear polarization may be caused by paramagnetic impurities, whose orientation arises as a result of exchange interaction with oriented conduction electrons.

9.6. Dynamic nuclear self-polarization

In this section we consider a phenomenon, predicted theoretically (Dyakonov and Perel 1972) but so far not observed experimentally. Namely, if the nuclear spin system interacts with electrons artificially maintained in a disordered spin state, a substantial nuclear polarization must spontaneously arise. Its appearance has the features of a phase transition of the second kind with a critical crystal temperature Tc. In other words, for Τ < Tc the disordered state of nuclear spins in such a system is unstable. The stable state corresponds to practically complete nuclear polarization. In this state a large nuclear field exists, acting on the electron spins. The nuclear polarization, in its turn, is maintained due to the Overhauser effect in this effective magnetic field.

We will assume that an external magnetic field is applied, exceeding the local field BL (for example, Β ~10-100 G). The only role of this field is to suppress nuclear spin-spin interactions, while equilibrium electron orientation in this field is negligible.

With regard to such conditions, a question may arise concerning the meaning of the words " the electrons are artificially maintained in a disordered spin state". They mean the following. Though the external field is negligible, as a result of a fluctuation some nuclear polarization may locally appear, together with the corresponding effective magnetic field. In the absence of external influences an equilibrium spin orientation would be achieved in the electron system, corresponding to the value of the fluctuational nuclear field. This would happen on the time scale of T s , very short compared to the characteristic transient time of nuclear polarization. It is important for the effect considered here, that some external agent should permanently disorder the electron spins, thus preventing the electron spin system from adjusting to the fluctuations of the nuclear field. The role of such an agent may be played by optical pumping by nonpolarized light. If the spin relaxation time T s is much larger than the lifetime τ, the photoexcited electrons will be stationary maintained in a disordered spin state.


Usually, in considering the Overhauser effect the action of the nuclear field on the electron spin is not taken into account. This is permissible for small nuclear polarization, i.e., at high temperatures or if the "leakage" of nuclear spin is substantial. Dynamic nuclear self-polarization is caused by the Overhauser effect not in an external field, but in the effective magnetic field created by the nuclei themselves.

Consider for simplicity a crystal lattice of identical nuclei with spin 7 = 1/2, which interact with non-oriented electrons by means of the contact Fermi interaction. From eq. (92) we find 7 a v = - ST. One should now replace the quantity Β by 7?N = 2£>N7av in expression (89) for ST. Then we obtain the following equation for 7 ·

(97)

(98)

where the critical temperature Tc is introduced as follows

In eq. (98) A is the hyperfine interaction constant. For T> Tc eq. (97) has a single solution 7 a v = 0. However, for T<TC, there

is also a nonzero solution describing nuclear self-polarization. It is just this state that is stable, while the state with 7 a v = 0 is unstable.

The generalization of eq. (98) to the case when the unit cell contains several species of nuclei with different spins is given by

where the sum is over all the nuclei in the unit cell, and the abundance of different isotopes should be taken into account.

In order to make a numerical estimate of Tc one must know the values of η entering the expression (78) for the constant A. The values for InSb and GaAs were given in sect. 9.2. Using these we find Tc « 9 Κ for InSb and Tc «1.3 Κ for GaAs. These values of Tc> as well as eqs. (98) and (99), do not take into account the possible "leakage" of nuclear spin. If this is taken into account, the leakage factor / appears in the right-hand side of eqs. (98) and (99), leading to a decrease in the critical temperature.

Consider now the possibility of achieving dynamic nuclear self-polarization and its detection by means of optical pumping (Dyakonov 1974). It was pointed out above, that the disordered electron spin state may be maintained by optical pumping of a semiconductor by nonpolarized light. The most favourable for self-polarization is the situation when the disordering is complete, which is achieved at τ «c T s . In this case, however, the recombinational radiation will be nonpolarized, and nuclear self-polarization will not be revealed in the luminescence. For optical detection of self-polarization it is important that the ratio T / T s should not be too small. Then the appearance of the nuclear field due to self-polarization will be accompanied by a noticeable

(99)


electron spin orientation and hence by appearance of circular polarization of the luminescence. In such conditions, however, the critical temperature, as may be shown, decreases proportionally to the factor T S / ( T + T S ) .

The temperature dependences of the average nuclear and electronic spins, calculated by Dyakonov (1974) for / = 1 / 2 and different values of the parameter T / ( T + T s) , are presented in fig. 15. One can see that for sufficiently large values of this parameter the build-up of self-polarization has the character of a first-order phase transition.

Fig. 15. Nuclear self-polarization under optical pumping. Theoretical temperature dependences of (a) average nuclear spin and (b) equilibrium (in the nuclear field) electron spin (Dyakonov 1974). / = 1 / 2 . The degree of circular polarization of the luminescence is determined by the average spin S = STT/(T + T s) . Values of the parameter τ / ( τ + rs): curve 1, 0; 2, 0.33; 3, 0.6; 4, 0.8. Dashed

curves correspond to unstable states.


10. Optical manifestation of nuclear field

10.1. Nuclear effects in longitudinal magnetic field

The effective magnetic field of polarized nuclei influences the electron orientation, thus providing the means for its detection by optical methods. The nuclear field acts on the electron spins in the same way as an external magnetic field. Thus it should reveal itself in any spin effect which is sensitive to external magnetic fields. There are several reasons why the nuclear field may influence the spin orientation of photocreated electrons. One of them is related to the influence of the magnetic field on the electron spin relaxation rate.

Figure 16 presents experimental results of Berkovits et al. (1973) on optical orientation in a longitudinal (with respect to pump light) magnetic field. One can see that with increasing field the degree of circular polarization of the luminescence increases and saturates in strong fields at the limiting level ρ = 0.25. This shows that the electron spin relaxation rate decreases, being suppressed by the action of magnetic field. The most interesting feature in fig. 16 is the difference in the p(B) dependences for opposite signs of the circular polarization of the exciting light (the corresponding directions of the average electron spin are also opposite). This difference is related to the existence of the nuclear field (Dyakonov and Perel 1973a), the direction of which is defined by the direction of the electron spin. Either enhancing or weakening the influence of the external magnetic field on electron spin relaxation, the nuclear field influences the steady-state orientation of the electron spins.

Fig. 16. The longitudinal-magnetic-field dependences of the degree of polarization of luminescence in (AlGa)As solid solution (Berkovits et al. 1973). (1,2) steady-state values for right and left circularly polarized excitation (for the same direction of field B)\ (3) values obtained immediately after the excitation is switched on. Solid curves: theory taking account of the nuclear field, dashed curve: without taking into account the nuclear field. Parameters used in calculations: T /T s( 0 ) = 24,

VsMBTC = 600 G, b = BN/S = 16 kG.


Such an interpretation is supported by the observed long (on the order of tens of seconds) transient times connected with the inertness of the nuclear spin system. The values of ρ observed immediately after switching on the excitation, while the nuclear polarization is not yet built up, are denoted in fig. 16 by squares. The most convincing proof of the influence of polarized nuclei on electron orientation was the optical detection in these conditions of NMR (see sect. 10.3).

The dependence of the electron spin relaxation time T s on the magnetic field is given by the expression (see Abragam 1961)

where Ω is the precession frequency of the electron spin in the field B, and T C

is a characteristic time of the order of the correlation time of the fluctuating magnetic field responsible for the spin relaxation. Equation (100) is valid for T S » T c . The T c values for different mechanisms of spin relaxation of electrons bound to shallow donor centers were discussed by Dyakonov and Perel (1973a). If the nuclear field BN is present, the quantity β in eq. (100) is defined by the total field acting on the electron spin,

In order to obtain the steady-state value of the average electron spin 5, taking into account the nuclear field, one must solve together eqs. (27), (94), (100) and (101). The dependences p(B) obtained in this way are presented in fig. 16.

10.2. Influence of nuclear polarization on Hanle effect in oblique field

In this section we describe a more general phenomenon due to the nuclear field, which manifests itself independently of electron spin relaxation (Dyakonov et al. 1974). In a transverse (with respect to the exciting beam) magnetic field depolarization of the electron spins occurs (Hanle effect, see sect. 7). The nuclear field may participate in this depolarization. However, under the conventional geometry of the Hanle effect, when the magnetic field is perpendicular to the average electron spin, nuclear polarization is practically impossible. (This statement does not apply to the domain of very small external fields, where an essential role is played by the electronic field acting on nuclear spins. See ch. 5 for details.) Indeed, it was shown in sect. 9.5. that the nuclear field arising in dynamic polarization by oriented electrons is proportional to the scalar product Β · S.

The situation changes if the angle between the external field and the exciting light beam differs from 90° (Hanle effect in oblique field). Then a nuclear field will appear, directed along or opposite to the direction of the external field and participating in depolarization of electrons. The nuclear polarization is created

T s ( 2 0 = T t ( 0 ) ( l + O Y ) . (100)

Ω=(8μΒ/Λ)\Β + Β„\. (101)


by the longitudinal (with respect to the external field) component of the electron spin. The arising nuclear field either helps or hinders (depending on its sign) the destruction of the transverse component of the electron spin by the external field.

The steady-state value of the average electron spin component S2 along the direction of excitation (direction of 5 0) in an oblique field Β may easily be obtained from eq. (51)

S2 (Β) « S2 (0) ( ™ 2 \ 2 + cos2 Λ (102)

where θ is the angle between the field and the exciting light beam (z-axis). Equation (102) is a generalization of eq. (50), describing the incomplete Hanle effect in which the component of the electron spin transverse to Β decreases in the usual way with increasing field, while the longitudinal component is field independent.

The nuclear field being taken into account, the quantity β is given by eq. (101). The dependence SZ(B) given by eq. (102) for BN = 0 is presented in fig. 17 (dotted curve). The solid curve represents the dependence given by eq. (102) if the nuclear field is taken into account (under the assumption that BN does not depend on B). The possible magnetic-field dependence of the spin relaxation time, discussed above, is not taken into account. The Hanle curve turns out to be displaced, so that the maximum orientation corresponds to the value Β = - 2?N, at which the total field is zero.

Our assumption about the nuclear field field being independent of the external one is obviously false in the region of small external fields Β < BL>

/ 1 \ / ' / 1 \ ·" ·

é · ,

/ ' \ ·'" '

\ ' ·., 1 Γ ι 1

- β * ο Fig. 17. Oblique Hanle effect. Dependence of the average electron spin projection SZ(B) along the direction of excitation on external magnetic field Β for θ = 45°, following from eqs. (101) and (102) (Dyakonov et al. 1974). Dotted and solid curves correspond to absence and presence of magnetic field, respectively. The dashed curve schematically depicts the maximum in the weak field

region where the nuclei are not polarized.


o.s J ^ \ J ^ \

0 - 4 - 2 0 2 4 - 4 - 2 0 2 4 - 4 - 2 0 2 4

Fig. 18. Oblique Hanle effect. Dependence of the degree of circular polarization in G a 0 7A l 0 3A s on external field Β for several values of θ (Dyakonov et al. 1974). Τ =42 Κ. Broken lines represent the Hanle curves calculated for a given angle from experimental data at θ = 90° (without

taking into account the nuclear field): (a) θ = 90°; (b) θ = 63°; (c) θ = 45°.

where BL is the local field (see sect. 9.1). In this region the nuclear polarization essentially depends on the magnetic field (see sect. 11.1) and is equal to zero at Β = 0. Therefore one should expect the oblique Hanle curve to have the form of a displaced Lorentz curve with an additional narrow peak at Β = 0.

The experimental data of Dyakonov et al. (1974) presented in fig. 18 qualitatively support this conclusion. The broadening of the Hanle curve as well as the shrinkage of the displaced maximum are due, probably, to inhomo-geneity of the nuclear field. The sample used in this experiment did not exhibit any appreciable dependence of T s on B.

In fact, the peak at Β = 0 has some fine structure due to the influence of the electronic field. Detailed investigation of the oblique Hanle effect disclosed a number of interesting features related to the electronic field and nuclear quadrupole effects. For the results of these studies see ch. 5.

10.3. Optical detection of NMR

The influence of the nuclear spin system on the optical orientation of electrons in semiconductors was directly proved by Ekimov and Safarov (1972a, b), who discovered resonance changes in the degree of circular polarization of luminescence under NMR. Figure 19 presents their experimental results, demonstrating optically detected NMR signals for all species of nuclei in the solid solution (GaAl)As.

Changes in the degree of polarization of the luminescence under NMR are due to variations of the absolute value and the direction of the nuclear field during the passage of resonance. Since various mechanisms of nuclear field action are possible, as discussed in the previous sections, the shape of the curve of optically detected resonance essentially depends on the experimental condi-


0,08

0.06

0 .04 5As

10

69QA 27KI Xa

1A f MHz

Fig. 19. Degree of circular polarization of the luminescence ρ in (GaAl)As solid solution versus frequency / of the transverse RF field (Ekimov and Safarov 1972b). Longitudinal field Β = 1.5 kG.

Arrows indicate resonance N M R frequencies.

tions. We will restrict ourselves to the case of NMR in a longitudinal field. It is in this case that the data in fig. 19 were obtained, the amplitude of the RF field being small enough. With increasing RF field amplitude the lines assumed doublet structure, which may be seen in fig. 20. Such shape of the curves may be explained (Dyakonov and Perel 1973a) on the basis of the concept of nuclear spin temperature in the rotating frame (Redfield 1955, see also Abragam 1961, Goldman 1970).

Consider the case of slow (compared to 7\) passage and assume that the RF field amplitude is large compared to the local field BL. Then eqs. (95) and (96) may be used for U N , where one should replace Β by B*, the field in the rotating frame. The components of B* along Β and in the perpendicular plane are respectively 2?* = (ω 0 — ω ) / γ and B\ = Bl9 where Bx is the amplitude of the rotating component of the RF field, γ is the gyromagnetic ratio. Thus the nuclear field has the following components:

( ω - co0) + Ω{

( 1 0 4 )

(ω — ω 0) + Ω{

where Ωχ = γΒν and Bm is the nuclear field value away from the resonance. The nuclear field BN rotates together with the field B* with the frequency

ω. However, this frequency being small compared to the characteristic frequencies of electronic processes, the nuclear field field acts on the electron spins as a static one. If we ignore the possible dependence of T s on B9 then the only reason for the electron spin orientation to change under NMR is partial electron depolarization due to the appearance of the transverse component of the nuclear field 2?N ± . For this reason the degree of electron orientation in the vicinity of the resonance must decrease. However, at exact resonance Β* ± 5 , hence BN = 0, so that this effect is absent. This explains the doublet structure of lines.


0.08

0.06

0 . 0 4

Fig. 20. Dependence of circular polarization of the luminescence in a G a 0 8A l 0 2 As crystal on the frequency of applied RF field (Ekimov and Safarov 1972a). Longitudinal field Β = 2.6 kG. Arrows

indicate resonance N M R frequencies. The RF field amplitude is greater than in fig. 19.

Figure 21, taken from Berkovits et al. (1973), presents the resonance line for 7 1 Ga nuclei obtained in a longitudinal field on a sample with a strong rs(B) dependence. The directions of S and Β were opposite. One can see the characteristic doublet line structure. However the degree of polarization observed at exact resonance was higher than its value away from resonance. This is explained by complete switching off of the field of7 1 Ga nuclei at ω = ω0 [see eqs. (103), (104)]. The consequent increase of the total field acting on electron spins leads to an increase of rs and a corresponding increase of the degree of polarization p.

The nuclear field concept permits to explain also a number of subtle effects in transient processes, accompanying the switching on, or off of the RF field (see for details Berkovits et al. 1973, Dyakonov and Perel 1973a).

? a.u.

— B Fig. 21. Variation of circular polarization of the luminescence in the resonance region for 71 Ga

nuclei at large RF power (Berkovits et al. 1973). S and Β are antiparallel, Β ~ 3.6 kG.


11. Optical cooling of nuclear spin system

This section is devoted to dynamical nuclear polarization, accompanying optical orientation of electrons in weak magnetic fields (of the order of or less than the local field BL). The appearance of nuclear polarization is, in fact, a result of the cooling of the nuclear spin system due to its interaction with nonequilibrium electrons. Indeed, an arbitrary state of the nuclear spins relaxes towards the state of thermodynamic equilibrium (described by the nuclear spin temperature Θ) during the time of transverse relaxation T2. This time is very short compared to the characteristic times Tx and Tle of the change of Θ due to spin-lattice relaxation and due to interaction with oriented electrons. Therefore in the optical orientation process the nuclear spin system can be regarded, with good approximation, as being in thermodynamic equilibrium (the deviation from equilibrium is of the order of T2/Tu). The entire action of photoexcited electrons reduces thus to the changing of the nuclear spin temperature.

Such an approach allows one to extend the theory of nuclear polarization by optically oriented electrons, described in sect. 9.5, to the domain of weak external magnetic fields Β < BL (Dyakonov and Perel 1975). It may appear at first glance that at Β < BL the nuclear polarization should be proportional to the ratio T2/Tu <cl, inasmuch as in weak field the time T2 characterizes the rate at which nonequilibrium nuclear polarization is destroyed due to the transfer of the Zeeman energy into the energy of spin-spin interactions. In fact, such reasoning is incorrect, since it pertains only to the nonequilibrium part of the nuclear polarization. The oriented electrons lower the temperature of the nuclear spin system in such a way that the equilibrium nuclear polarization (75) corresponding to this temperature and to the external magnetic field is much higher than the nonequilibrium polarization. Only in a very weak field Β ~ BL(TC/Tlc)l/2 does the equilibrium polarization become smaller than the nonequilibrium one.

Consideration of the energy balance of the nuclear spin system, interacting with oriented electrons, on the basis of the Master equation for the density matrix describing this system leads to the following equation for θ :

where Tlc is the nuclear spin relaxation time due to electrons. The expression for the steady-state nuclear spin temperature Θ 0 has the

form

(105)

(106)


where Δζ is the difference in the chemical potentials of electrons with spins along and opposite to the field. The quantity BL is of the order of the local field BL. Explicit definitions of these quantities are

^ = 3 S p ( « ) / E ^ ( / „ + l)(2/„ + l),

H = l Σ AnAmSp[^ *A /ΣΑ2

ημ]η^Ρ-(2Ι„ +1), (107)

where Jfd is the Hamiltonian describing spin-spin interactions and the sum is over all the nuclei in the crystal. When jfea reduces to dipole-dipole interaction, we have B\ = 3BL (Abragam 1961).

If the electron spin system is in thermodynamic equilibrium then Δξ = 0. In this case it follows from eq. (106) that Θ 0 = οο. The reason is that in the derivation of eq. (106) the Zeeman nuclear energy was neglected compared to T. In fact, as Δξ -> 0 the value of Θ0 should, of course, tend to the lattice temperature T.

The quantity Δζ in (106) can be connected with the mean value of the electron spin projection Sz on the magnetic-field direction. In particular, for nondegenerate electrons we have

- 4 = 7 ^ · <10S> where ST is the equilibrium value of the electron spin given by eq. (89). Thus for nondegenerate electrons eq. (106) takes the form

41 Β Sz - ST

©ο μι B2 + B2L 1 - 4 5 2 5 T

(109)

As seen from eqs. (106) and (109), the most effective cooling of the nuclear spin system occurs at Β ~ BL. The temperature Θ0 can be positive, as well as negative.

The nuclear polarization is defined by eqs. (75), (109). In the conditions of optical orientation in weak field the quantity ST in eq. (109) may be neglected. Then for the average nuclear spin we obtain the expression (Dyakonov and Perel 1975)

+ (no)

This formula is a generalization of eq. (94) to the weak-field case. It was independently obtained by Paget (1975) on the basis of previous results of Solomon and Ezratty (1962).


Thus the average nuclear spin is always directed along the magnetic field or opposite to it, depending on the sign of the spin temperature (the sign of B*S). Now an expression for the nuclear field can be obtained which is valid for weak external fields. With the help of eqs. (84) and (110) we find

•»-5M/+.)f£g. ο..» Equation (111) shows that under optical orientation conditions the nuclear

spin system acts as a powerful amplifier of external magnetic fields. For example, at bN=10 kG, 7 = 3/2, S = 0.25 and B = BL = 3 G we have BN = 300 G. Thus in an external field equal to 3 G the electron spins are acted upon by a nuclear field that is 100 times larger.

Possible leakage of nuclear spin is not taken into account by eqs. (109)-(111). Allowance for this would lead to the appearance of the leakage factor in the right-hand sides of these formulae.

If several nuclear species are present, a common spin temperature is established at weak fields. The generalization of eqs. (109)—(111) for this case can be found in Dyakonov and Perel (1975).

Optical cooling of the nuclear spin system at weak fields and, as a consequence of this, the effective amplification of the applied external field was studied in a large number of experimental work. For details see Zakharchenya (1978) and ch. 5.

Of principal interest is the question, whether it is possible to cool the nuclear spin system optically down to the temperature of phase transition to a magnetically ordered state. It follows from eq. (109) that at Β ~ BL ~ 3 G and Sz = 0.25 the nuclear spin temperature may be lowered to the value Θ 0 ~ μ\Β^ - 1 0 " 7 K. However, this formula was obtained in the framework of the high-temperature approximation (see sect. 9.1) and is, strictly speaking, not valid in the domain of such low temperatures. In this domain it can give only a correct order-of-magnitude estimate for Θ0, which is not enough for deciding whether the phase transition is possible.

Merkulov (1982) studied this question for the special case when the dominating nuclear spin-spin interaction is the scalar interaction of the lnIm type. Such an interaction conserves the total nuclear spin and hence does not contribute to the quantity B\ (if only one nuclear species is present), so that BL <c BL. The local field 7?L, on the contrary, is defined just by the scalar interaction. It happens in this case that at Β » BL formula (109) is valid, this result being obtained without using the high-temperature approximation. Hence it is applicable, as well, in the temperature region Θ0 < 7?L, where one expects the phase transition to occur. Thus, at least in the special case considered, optical cooling of the nuclear spin system down to the temperature of the phase transition is, in principle, possible.


References

Abragam, Α., 1961, The Principles of Nuclear Magnetism (Clarendon Press, Oxford). Altukhov, P.D., G.E. Pikus and A.A. Rogachev, 1978, Fiz. Tverd. Tela 20 , 489. Aronov, A.G., and E.L. Ivchenko, 1973. Fiz. Tverd. Tela 15, 231. Arsentiev, I.N., B.P. Zakharchenya, V.K. Kalevich, V.D. Kul'kov, V.D. Rumyantsev and V.G.

Fleisher, 1980, Fiz. Tverd. Tela 22 , 3378. Asnin, V.M., A.A. Bakun, A.M. Danishevskii, E.L. Ivchenko, G.E. Pikus and A.A. Rogachev,

1978, Zh. Eksp. Teor. Fiz. Pis'ma 28, 80; 1979, Solid State Commun. 3 0 , 565. Averkiev, N.S., and M.I. Dyakonov, 1983, Fiz. Tekhn. Poluprovodn. 17, 629. Bagraev, N.T., and L.S. Vlasenko, 1979, Fiz. Tverd. Tela 210 , 120. Bagraev, N.T., L.S. Vlasenko and R.A. Zhitnikov, 1976, Zh. Eksp. Teor. Fiz. 7 1 , 952. Bagraev, N.T., L.S. Vlasenko and R.A. Zhitnikov, 1977, Zh. Eksp. Teor. Fiz. Pis'ma 25, 207. Bagraev, N.T., L.S. Vlasenko and R.A. Zhitnikov, 1979, Zh. Eksp. Teor. Fiz. 76, 552. Baumgardner, C.A., and T.O. Woodruff, 1968, Phys. Rev. 174, 746. Belinicher, V.I., and B.I. Sturman, 1980, Usp. Fiz. Nauk 130, 415. Berkovits, V.L., A.I. Ekimov and V.I. Safarov, 1973, Zh. Eksp. Teor. Fiz. 65 , 346. Berkovits, V.L., V.I. Safarov and A.N. Titkov, 1976, Izv. Akad. Nauk SSSR Ser. Fiz. 40 , 2346. Bloembergen, N. , 1954, Physica 20, 1130.

Bloembergen, Ν., E.M. Purcell and R.V. Pound, 1948, Phys. Rev. 7 3 , 679. Buishvili, L.L., N.P. Giorgadze and A.I. Ugulava, 1974, Fiz. Tverd. Tela 16, 3043. Cohen-Tannoudji, C , and A. Kastler, 1966, in: Progress in Optics, Vol. 5, ed. E. Wolf (North-

Holland, Amsterdam) p. 33. Dresselhaus, G., 1955, Phys. Rev. 100, 580. Dyakonov, M.I., 1974, Zh. Eksp. Teor. Fiz. 67 , 1500. Dyakonov, M.I., and V.I. Perel, 1971a, Zh. Eksp. Teor. Fiz. 60 , 1954. Dyakonov, M.I., and V.I. Perel, 1971b, Zh. Eksp. Teor. Fiz. Pis'ma 13, 206. Dyakonov, M.I., and V.I. Perel, 1971c, Zh. Eksp. Teor. Fiz. Pis'ma 13, 657. Dyakonov, M.I., and V.I. Perel, 1971d, Phys. Lett. 35A, 459. Dyakonov, M.I., and V.I. Perel, 1972a, Fiz. Tverd. Tela 14, 1452. Dyakonov, M.I., and V.I. Perel, 1972b, Zh. Eksp. Teor. Fiz. Pis'ma 16, 563. Dyakonov, M.I., and V.I. Perel, 1973a, Zh. Eksp. Teor. Fiz. 65 , 362. Dyakonov, M.I., and V.I. Perel, 1973b, Fiz. Tekhn. Poluprovodn. 7, 2335. Dyakonov, M.I., and V.I. Perel, 1975, Zh. Eksp. Teor. Fiz.. 6 8 , 1 5 1 4 . Dyakonov, M.I., and V.I. Perel, 1976, Fiz. Tekhn. Poluprovodn. 10, 350.

Dyakonov, M.I., V.I. Perel, V.L. Berkovits and V.I. Safarov, 1974, Zh. Eksp. Teor. Fiz. 67 , 1912; Proc. XII Int. Conf. on the Physics of Semiconductors, Stuttgart, 1974, ed. M.H. Pilkuhn (Teubner, Stuttgart) p. 771.

Dymnikov, V.D. , M.I. Dyakonov and V.I. Perel, 1976, Zh. Eksp. Teor. Fiz. 7 1 , 2374. Dzhioev, R.I., B.P. Zakharchenya and V.G. Fleisher, 1973, Zh. Eksp. Teor. Fiz. Pis'ma 17, 244. Dzhioev, R.I., V.G. Fleisher, E.L. Ivchenko, V.L. Vekua and B.P. Zakharchenya, 1974, Proc. XII

Int. Conf. on the Physics of Semiconductors, Stuttgart, 1974, ed. M.H. Pilkuhn (Teubner, Stuttgart) p. 776.

Edmonds, A.R., 1957, Angular Momentum in Quantum Mechanics (Princeton Univ. Press). Ekimov, A.I., and V.I. Safarov, 1970, Zh. Eksp. Teor. Fiz. Pis'ma 12, 293. Ekimov, A.I., and V.I. Safarov, 1971a, Zh. Eksp. Teor. Fiz. Pis'ma 13, 251. Ekimov, A.I., and V.I. Safarov, 1971b, Zh. Eksp. Teor. Fiz. Pis'ma 13, 700. Ekimov, A.I., and V.I. Safarov, 1972a, Zh. Eksp. Teor. Fiz. Pis'ma 15, 453. Ekimov, A.I., and V.I. Safarov, 1972b, Proc. XI Int. Conf. on the Physics of Semiconductors,

Warsaw, 1972, ed. M. Miasek (PWN Polish Scientific Publ., Warsaw) p. 1351. Fishman, G., and C. Hermann, 1974, Phys. Stat. Sol. (b) 6 3 , 307.


Fleisher, V.G., R.I. Dzhioev and B.P. Zakharchenya, 1976, Zh. Eksp. Teor. Fiz. Pis'ma 23 , 22. Garbuzov, D.Z., R.I. Dzhioev, L.M. Kanskaya and V.G. Fleisher, 1972, Fiz. Tverd. Tela 14, 1720. Gelmont, B.L., and M.I. Dyakonov, 1973, Fiz. Tekhn. Poluprovodn. 7, 2013. Goldman, M., 1970, Spin Temperature and Nuclear Magnetic Resonance in Solids (Clarendon

Press, Oxford). Gross, E.F., A.I. Ekimov, B.S. Razbirin and V.I. Safarov, 1971, Zh. Eksp. Teor. Fiz. Pis'ma 14,

108. Gryncharova, E.I., and V.I. Perel, 1977, Fiz. Tekhn. Poluprovodn. 11, 1697. Gueron, M., 1964, Phys. Rev. 135, A200. Hanle, W., 1924, Z. Phys. 30 , 93. Hermann, C , G. Lampel and T.P. Pearsall, 1980, J. Phys. Soc. Japan 49 , Suppl. A, p. 631 (Proc.

XV Int. Conf. on the Physics of Semiconductors, Kyoto, 1980). Ivchenko, E.L., and G.E. Pikus, 1978, Zh. Eksp. Teor. Fiz. Pis'ma 27, 640. Ivchenko, E.L., G.E. Pikus, B.S. Razbirin and A.I. Starukhin, 1977, Zh. Eksp. Teor. Fiz. 72 , 2230. Kalevich, V.K., V.D. Kul'kov and V.G. Fleisher, 1982a, Zh. Eksp. Teor. Fiz. Pis'ma 3 5 , 17. Kalevich, V.K., V.D. Kul'kov and V.G. Fleisher, 1982b, Izv. Akad. Nauk SSSR Ser. Fiz. 46 , No. 2. Kane, E.O., 1956, J. Phys. Chem. Solids 1, 82. Keldysh, L.V., O.V. Konstantinov and V.I. Perel, 1969, Fiz. Tekhn. Poluprovodn. 3 , 1042. Kittel, C , 1963, Quantum Theory of Solids (Wiley, New York). Khutsishvili, G.R., 1965, Usp. Fiz. Nauk 87, 211. Lampel, G., 1968a, Phys. Rev. Lett. 20 , 491. Lampel, G., 1968b, These de Doctorat d'Etat, Orsay. Lampel, G., A.N. Titkov and V.I. Safarov, 1978, Proc. XIV Int. Conf. on the Physics of

Semiconductors, Edinburgh, 1978, ed. B.L.H. Wilson (Conf. Ser. no. 43, The Institute of Physics, Bristol) p. 1301.

Luttinger, J.M., 1956, Phys. Rev. 102, 1030. Madelung, O., 1964, Physics of III-V Compounds (Wiley, New York). Maranzana, F.E., 1967, Phys. Rev. 160, 421. Merkulov, I.A., 1982, Zh. Eksp. Teor. Fiz. 82 , 315. Overhauser, A.W., 1958, Phys. Rev. 92 , 411. Paget, D. , 1975, These, Orsay. Paget, D. , 1982, Phys. Rev. B25 , 4444. Paget, D. , G. Lampel, B. Sapoval and V.I. Safarov, 1977, Phys. Rev. B15 , 5780. Pikus, G.E., and E.L. Ivchenko, 1982, Optical Orientation and Polarized Luminescence of Exitons

in Semiconductors, in: Excitons, eds. E.I. Rashba and M.D. Sturge (North-Holland, Amsterdam).

Parsons, R.R., 1969, Phys. Rev. Lett. 23 , 1132. Parsons, R.R., 1971, Can. J. Phys. 49 , 1850. Razbirin, B.S., V.P. Mushinskii, M.I. Karamyan, A.N. Starukhin and E M . Gamarts, 1975, Zh.

Eksp. Teor. Fiz. Pis'ma 22, 203. Rebane, Yu.T., 1981, Fiz. Tekhn. Poluprovodn. 15, 1503. Redfield, A.G., 1955, Phys. Rev. 98 , 1787. Rogachev, Α.Α., 1980, Prog. Quantum Electron. 6, 141. Solomon, I., and J. Ezratty, 1962, Phys. Rev. 127, 78. Sundfors, R.K., 1969, Phys. Rev. 185, 458. Vekua, V.L., R.I. Dzhioev, E.L. Ivchenko, V.G. Fleisher and B.P. Zakharchenya, 1975, Fiz. Tekhn.

Poluprovodn. 9, 169; Fiz. Tverd. Tela 17, 1096. Vekua, V.L., R.I. Dzhioev, B.P. Zakharchenya and V.G. Fleisher, 1976, Fiz. Tekhn. Poluprovodn.

10, 354 Volkov, A.S., and G.V. Tsarenkov, 1977, Fiz. Tekhn. Poluprovodn. 11, 1709.


Volkov, A.S., A.I. Ekimov, S.A. Nikishin, V.I. Safarov, B.V. Tsarenkov and G.V. Tsarenkov, 1977, Zh. Eksp. Teor. Fiz. Pis'ma 25, 560.

Weisbuch, C , C. Hermann and G. Fishman, 1974, Proc. XII Int. Conf. on the Physics of Semiconductors, Stuttgart, 1974, ed. M.H. Pilkuhn (Teubner, Stuttgart) p. 743.

Zakharchenya, B.P., V.G. Fleisher, R.I. Dzioev, Yu.P. Veshchunov and I.B. Rusanov, 1971, Zh. Eksp. Teor. Fiz. Pis'ma 13, 195.

Zakharchenya, B.P., 1978, Proc. XIV Int. Conf. on the Physics of Semiconductors, Edinburgh, 1978, ed. B.L.H. Wilson (Conf. Ser. no. 43, The Institute of Physics, Bristol) p. 31.

Zakharchenya, B.P., D.N. Mirlin, V.I. Perel and I.I. Reshina, 1982, Usp. Fiz. Nauk 136, 459.

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[Modern Problems in Condensed Matter Sciences] Optical Orientation Volume 8 || Theory of Optical Spin Orientation of Electrons and Nuclei in Semiconductors