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A. A. REREZIN et al. : Conductivity Mechanism of @-Rhombohedra1 Boron 447

phys. stat. sol. (a) 20, 447 (1973)

Subject classification: 14.3 and 14.3.4; 22.1

A.F. Ioffe Phyeico-Technical Institute, Academy of Sciences of the USSR, Leningrad

Studies of a Conductivity Mechanism of B-Rhombohedra1 Boron in a Strong Electric Field

BY A. A. BEREZIN, 0. A. GOLIKOVA, M. M. KAZANIN, E. N. TKALENKO, and

V. K. ZAITSEV

The dependence of electroconductivity of pure @-rhombohedra1 boron on the temperature and the intensity of a n electric field is presented. The results are discussed using three models: I) the Poole-Frenkel model for an isolated trapping centre and for a screened Coulomb centre, 11) the hopping models of Mott and Shklovskii, and 111) the small polaron model. The results of the electric field measurements may be satisfactorily interpreted in both the screened Poole-Frenkel and the hopping model giving for the dependence of the conductivity u on the electric field intensity E the relations (u/uo)1/2 In (o/uo) - E and In (u/uo) - E--114, respectively (uo is the conductivity in zero field). The model I11 alone gives an unreasonable result for the hopping length but nevertheless the essentiality of polaronic effect for @-boron seems sufficient from both the experimental and the theoretical point of view. It is supposed that the very complex crystal lattice of @-rhombohedra1 boron has a mixed conductivity of a polaron-Mott type. The magnetoconductivity of boron in the hopping region is also discussed.

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1. Introduction

The nature of the electric conductivity mechanism of p-rhombohedra1 boron is not known with definite clearness at present. There is strong evidence that the electroconductivity of P-boron is mainly determined by hopping processes [l] to [6]. The high density of trapping levels 14, 51 and low values of mobilities make

448 ,4. A. BEREZIN et al.

the usual theory of band-type conduction unapplicable to boron [7, 81 and lead to the supposition about hopping conduction [9]. Two principal types of hopping conduction, polaronic and impurity-like, seem to have some connections with boron [9, lOJ but up to now there is no unambiguous picture of hopping of carriers in the crystal lattice of P-boron.

In the present paper we represent the results of conductivity measurements of pure @-boron in strong electric fields. The measurement of current-voltage ( I - U ) characteristics in strong electric fields is one of the most effective toolr for the investigation of materials with hopping conduction [11 to 131. Here we study the I-U characteristics of boron at 77, 113, 205, and 293 OK in fields up to 60 kV/cm. As we know I-U characteristics of boron were studied by Pruden- ziati et al. [5] (P-rhombohedra1 boron) and Moorjani and Feldman [14] (amorphous boron). But in [5] and [14] the measurements of I-U characteristics were carried out for lorn fields only, where according to [a] and [14], the formation of I-U characteristics is determined primarily by space-charge-limited currents.

We shall try to demonstrate the possibility that the hopping nature of electric conductivity of P-boron is connected with the peculiarities of the structurc of its complex crystal lattice.

2. Experiment For thc measurements we used single crystals of pure P-rhombohedra1 boron

The I-U characteristics were measured in direct current, becausc for 7' which were prepared by zone melting.

300 OK thc resistivity of pure boron is large enough to allow sample heating to be neglected. The samples for the measurements were prepared in the form of plates with dimensions 0.1 x 4 x 4 mm3. At T = 300 OK their resistivity was about 4 x lo6 Qcm. The area of the silver contacts was 1 x 1 mm2.

To reduce the surface current1) the surface of the sample was etched in concen- trated nitric acid. After etching the surface current was reduced to 1 to 2% of the full current.

The I-U characteristics of pure zone-molten boron for 77, 113, 205, and 293 OK are shown on Fig. 1. As one can see from Fig. 1, the I-U characteristics of p-rhombohedra1 boron are essentially non-ohmic. In Section 3 we shall discuss the possible ways to iiitmpret these results.

Fig. 1. Current-voltage characteristics of pure zone- molten boron for 77, 113, 205, and 293 OK. For the

sample geometry see the text

') The surface current depends on the pre-history of the sample; for our case its value was up to 40% of the full current through the sample.

Studies of a Conductivity Mechanism of p-Rhombohedra1 Boron

3. Theoretical Models

449

Several possible physical mechanisms of the non-ohmic conductivity of semi-

The most familiar of these mechanisms are : 1. the Poole-Frenkel effect [15 to 171, 2. the specific non-activated hopping conduction in sufficiently strong elec-

3. the electric field dependence of the mobility of the small polaron considered

4. the space-charge-limited currents [5, 141. According to [5] and [la] the space-charge-limited currents play an essential

role for the formation of I-U characteristics of @- and amorphous boron in the region of comparatively small electric fields only. In the present paper we shall consider the applicability of the first three above-mentioned models to the case of @-boron only.

conductors in strong electric field are known a t present.

tric fields considered by Mott [l l , 121 and Shklovskii [13],

by Efros [18],

4. The Pooh-Frcnkel Model 4.1 The Poole-Frenkel effect for a single trapping centre

When the conductivity of a system is determined by thermal ionization of Coulomb-like trapping centres the applied electric field leads to the reduction of a potential barrier and consequently to the increase of carrier concentration in the conduction band. Simple statistical arguments give the following dependence of the conductivity a = a(E) versus the electric field E [19] :

u = a, exp (T) . 7 JIE

I n (1) a, = o(0) is the static conductivity of a semiconductor with trapping centres in zero electric field. The parameter q is determined by the effective charge of a Coulomb centre Z and the static dielectric constant x

If one takes into account the induced space anisotropy of the thermal ionization probability of a trapping centre then the dependence (1) becomes more complex r201:

When the potential of a trap has the form of a spherical rectangular potential well of radius b one can get the relation

instead of (1) and correspondingly the relation

- -_- a '' [exp(e$)-~]+; a, 2 e E b

instead of (3).

450 A. A. BEREZIN et al.

t 3.0 - 25 Q

2.0

1.5

1. a

05

I 50 J 60

E ihllcm-'J - Fig. 2 Fig. 3

Fig. 2. Dependence of Ig [u(E)/a(O)] on the first power of field intensity (Poole law) for P-rhombohedra1 boron a t 77 and 113 OK, For both temperatures three calculated curves corresponding to b = 200, 250, and 300 A are given. The left triplet of calculated curves corresponds to T = 77 OK, the right one to T = 113 OK. The curves are calculated by (5 ) , i.e. by taking into account the field-induced anisotropy of the probability of thermal

ionization

Fig. 3. Left side: I-U Characteristics of p-boron in Poole coordinates for 77, 113, 205, and 293 OK (see subscript to Fig. 2). Right side: dependence of Ig (um/uo - 1) on the inverse temperature. The experimental points correspond to 77, 113, 205, 220, 248, and 293 O K

The relation (1) with (Fin the exponent is known as Frenkel law [19] and (4) with E instead of d z i s known as Poole's law [21].

Dependences corresponding to the Poole law in P-boron are shown in Fig. 2 and 3 (left side). Comparing the theoretical curves cr = o ( E , b ) with experimental ones for T = 77 and 113 O K (Fig. 2) we found that b = 250 to 270 n, which seems to be too large because the linear dimensions of the unit cell of P-rhombohedral boron are 25 x 10 x 10 A3.

For the ease of the Frenkel law (1) similar considerations give that for T = 77 and 113 OK the value of 7 is near to 0.75 at. units. Then (2) gives for the effective charge 2 (for P-boron ~t x 10) the estimate 2 = q2 x/e3 = (5 to 6) which also seems to be unrealistically high in order to be the charge of a single Coulomb trapping centre.

Thus the Frenkel-Poole model for both types of isolated trapping centres (i.e. for the spherical rectangular well and the single Coulomb centre) seems to be insufficient for the interpretation of I-U characteristics of P-boron.

4.2 The Poole-Frenkel effect for the Coulomb centre screened b~ free carriers

If one takes into account the shielding of the Coulomb-like potential of a happing centre by free carriers the dependence cr = o ( E ) is described approxi- mately by a relation [15, 161 formally similar to the Poole law (4) :

Studies of a Conductivity Wxhanism of p-Rhombohedra1 Boron 451

77 1.82 x 1014 3.06 x 10-9 113 1-70 x 1015 3.06 X lo-" 205 1.49 x 10l6 3.77 x 10-5 293 5.96 x 10le 3.50 x 10-5

a

428 16.7 4.55 2.86

E IhV cm4--

Fig. 4. Dependence of (cr/cro)1/2 lg o/n0 on E (screened Poole-Frenkel law) for P-rhombohedra1 boron for a) T = 77 OK and b) T = 113 OK

I n (6) E is a value of the order of unity which weakly (logarithmically) depends on the parameters of the problem and As is the inverse screening length. For 1, we take the Debye-Huckel approximation:

(n is the carrier concentration) and neglect the influence of the field on the mobility of the carriers. Then we find the following form of a(E) [16] :

where the expression under the square root does not depend on E. In (8) n,,(T) is the concentration of the charge carriers on the current level, i.e. n,(T) = = a,(T)/e p ( T ) (p is the drift mobility).

The dependence corresponding to screened Poole-Frenkel law (8) is shown in Fig. 4a and b. These dependences are nearly linear and hence we can estimate n,,(T) and the corresponding values of the mobilities ,u( T ) = a,(T)/e no(?'') (see Table 1).

The increase of p( T) with temperature could be described by the activation law

,u( T ) N exp (- &) (9)

452 A. 8. BEREZIN et al.

At sufficiently strong fields before breakdown the I-U characteristics of p-boron show a tendency to saturation, i.e.

( T ( E ) I E + ~ + coo x const,

where the limited value om is only a function of temperature. The values of urn( T)/oo( T ) are given in the right column of Table 1 . The value of oaS/a0 is rapidly reduced when the temperature increases.

One may consider this saturation as a consequence of the depletion of the trapping levels by the electric field. Then we can estimate the value of the activation energy of the current level, i.e. the binding energy of the carrier on the trapping centre ( A ) , the concentration of traps ( N t r ) , and the concentration of current states ( N v ) .

Let nt,( T ) be the number of localized carriers, i.e. the carriers bound on trapping cent,res, and A the energy separation between the bound (currentless) and current level. In case of Boltzmann statistics we have

Writing (11) we imply that the growth of (T in the electric field is caused by the

Plotting lg (am/ao - 1) as a function of l jkT (Fig. 3, right side) we find the growth of carrier concentration only and not by the growth of mobilities.

following estimates for A and N,/Nt , in @-boron:

- (2 to 8) . A x 0 . 0 5 e V , - - ( 1 3 ) NC

Ntr The second value of (13) and N t , = no am/oo give the following estimate for

the concentration of trapping levels :

Nt, = (10'' to cm-3. (14) Kote that this activation energy A does not coincide with the activation ener-

gy of drift mobility A, given above.

5. The Hopping Model

In this section we shall consider the zero-field and high-field conductivity of P-rhombohedra1 boron from the point of view of the theory of hopping conduction. This interpretation is alternative to the Poole-Frenkel interpretation de- veloped above. Some hints on the possible integration of these two approaches will be given in thc discussion (Section 8).

5.1 The tempevntzcre dependent Mott law

The large unit cell of P-rhombohedra1 boron and the non-equivalence of its lattice atoms lead to some analogy of boron with amorphous substances [7].

Studies of a Conductivity Mechanism of P-Rhombohedra1 Boron 4 3

One may suppose that in boron as well as in amorphous or doped semiconductors the local (or quasilocal) trapping states have energy levels which are not degenerate but have different positions within a finite energy interval of width W . Thus it is natural to expect that the conduction of boron at comparatively low temperatures should have some similarities with the hopping conduc- .

-' ?7 -I3

3

u N exp [ - (37, which is known as Mott T-ll4 law [23 to 251. Ig o versus T-lI4 for $-boron is shown in Fig. 5. The slope of the curve corresponds to the value To = 109 OK. The corresponding value of kT ( ~ 0 . 0 4 eV) is probably near to the width of energy interval of local states W (see [26] where at least two trapping levels with energy separation 0.04 eV were reported for p-boron from thermally stimulated current measurements).

The characteristic temperature To in the Mott law (15) is connected with go, the density of localized states on the Fermi level. In the case of properly dis- ordered systems (e.g. doped or amorphous semiconductors) theory gives [25]

v @.3 90

kT, z-.

I n (16) a is the decay constant of the wave function y of the localized carrier

For oc the following estimation may be offered: (y IV ecUr) and v a dimensionless constant. According to [25] v = 16.

(17) &-1 - - ( 2 to 5) A . (17) agrees with the results of the calculations of the wave function of the localized polaronic carrier made for @boron in [27] in the polarizable point-atom lattice approximation in analogy with the calculations of charged electronic colour centres in alkali halides [28].

In [27] the trial wave function was taken in the form y N e-ar/T suitable for the bound state of a particle in delta-well potential. Optimal values .;;p't = 2.9, 2.8, and 2.7 A were obtained for carrier localization on atoms with coordination numbers 6, 8, and 9, respectively.2)

For To = lo9 OK and a given by (17) equation (16) gives go z (1.2 x lo1* to 2.4 x l O l 9 ) eV-1 cmP3. This yields for the density of localized levels x w go ( W z 0.04 eV) the following estimate :

N x (5 x 1016 to lo'*) emp3 . (18) ') In P-rhombohedra1 boron the unit cell contains 105 atoms. Among them 91 atoms

have the coordination number 6, 12 atoms 8, and 2 atoms 9.


The estimate (18) although near to (14) is perhaps too low to be explained by the localized levels connected with some characteristical atoms (or groups of atoms) in the unit cell of @-boron. (The concentration of unit cells in P-boron is near loz1 ~ m - ~ . ) Probably our value of To is overestimated (and consequently go and N are underestimated) and should bc reduced by taking into account the polaronic effect for @-boron (see below).

5.2 The hopping conductioii in strong electric field

According to Mott's idea [ l l , 121 hopping conduction in a sufficiently strong electric field can be realized by a series of successive hops of a carrier in which only phonon emission (and not absorption of phonons) takes place. This typc of hopping conduction needs no temperature activation and for field dependence Mott proposed the formula

where the characteristic field Eo is temperature independent and is given by 8 a4

go e E , = - .

According to Shklovskii [13] the exponential term of the field dependence of activationless hopping conduction differs from (19) by the power index and the urouer form of this law is L L

where the characteristic field E, is almost, the same as in (20), namely 16 (y4

e go E -

0 -

t I e - P 9

a

- EfkVcm'l 300 50 20 10 5 3 2 7

25 - f s= 22 h 1 20

1.5

7.0

0.5

0 02 06 0.6 0.8 10

- E(kVun4J 300 50 20 70 5 3 2

31,

25

2.0

75

7.0

05

0 02 04 06 0.8 1;

I

f ' ' ( hV"&~- - - c [" [ ~ ~ ~ ' " ~ l l + ) - Fig. 6 . Dependence of Ig u/q, on the electric field intensity in "hopping coordinates" for T = 77 and 113 OK. Here u,, is the zero-field conductivity plotted against temperature in Fig. 5. a) Ig ./ao = j(E-113) according t o Mott [ll, 121 and b) lg u/co = j(E-114) according

to Shklovskii [I31

Studies of a Conductivity Mechanism of P-Rhombohedra1 Boron 455

The experimental dependences corresponding to (19) and (21) for P-boron a t 77 and 113 OK are shown in Fig. 6a and b. The slopes of the linear parts of the curves correspond to the values E, = 6 x lo3 kV/cm for Mott’s E-lI3 law (19) and E, = 1.2 x lo5 kV/cm for Shklovskii’s E-1/4 law (21). According t o (17), (20), and (22) these values of E, give the following estimates for the concentration of hopping sites (trapping centres) N = W go ( W = 0.04 eV):

for Mott’s law and

for Shklovskii’s law. In contrast to (14) and (18) based on the Poole-Frenkel model and on the

temperature dependent Mott law, respectively, (23) and (24) may be well com- pared with the number of unit cells in P-boron crystals (

N = to 3 x 1OZ3) ~ m - ~ (23)

N =z (1021 to 3 x (24)

cm-s).

6. The Small-Polaron Model

The electric conduction by small polarons is a specific type of hopping conduction which occurs if quasilocalized states of a carrier are formed by polarization interaction of a charged carrier with the surrounding crystal lattice [29]. At very low temperatures the main contribution to the current comes from the narrow-band conduction of a small polaron, i.e. from the activationless tunneling of a small polaron as a whole3), while a t intermediate and high temperatures the thermally activated hops become dominant. The last type of polaron conduction, the only one we shall discuss here, is realized by hops of the carrier without its polarization cloud. Such hops need the formation of a polarization well a t the unoccupied crystal site by thermal fluctuations. Thus the expression for the conductivity u contains the exponential factor

o - e x p j - 2 k T 3 ) ’ (25)

where E, is the polaron shift [29]. The principal possibility for the polaronic effect in P-rhombohedra1 boron to

exist is caused by the ionicity of its lattice [30]. The dispersion of the dielectric constants for P-boron is noticeable ( x , = 8 and x, x 10). In [lo] the experimental value E , = (0.22 f. 0.04) eV was reported while the polarizable point- atom lattice calculation [27] gives E, = 0.25 to 0.30 eV.

It should be mentioned that the region of the polaron activation behaviour (25) is expressed much more weakly for p-boron [lo] than for aluminium dodeka- boride (a-AlB,,) [31] whicli has, as well as @-boron, the isocahedrons B,, as basic elements of its crystal structure.

The electric field dependence of small-polaron thermally activated hopping conductivity for constant carrier concentration has the form [18]

A (e E Z)2 e E 1 a(E) - ~

e E I [ - 1-1 sinh 2’ where A is field independent, E, = E,/2 [29], and 1 is the effective hopping length,

3, Small polaron means the combination of a carrier with a polarization well generated by the electric field of the carrier itself.


For the field range 2 k2’le 1 < E Q 8 E,/e 1 (26) is rather similar to (4) and for P-boron we get (at T =1: 77 OK) 1 x 250 A. For this value of I our fields are really within this interval [2 k T / e 1 = 5.4 kV/cm and 8 E,ie 1 = 380 kV/cm (for E, = 0.12 eV)] but the obtained value of 1 seems to be too high to be satisfactorily understood in the framework of the standard small-polaron theory alone.

7. Magnotoconductivity

The decrease of hopping conductivity in magnetic field is caused by shrinking of the wavc functions of the localized states of trapped carriers. If all localized states have the same value of a , i.e. the constant of the exponential decay of the bound state wave function, then the dependence of u on the magnetic field intensity H is given by [24]

a(H) = o(0) exp

Here s is a constant of the order of 0.1 [24], 1 is the magnetic length, and Rapt is the optimal hopping length. For the region of Mott conduction (15) we can equate the exponent of (15) to the tunneling exponent exp ( - 2 CX R,,t). Thus

Substituting (28) into (27) we get (for the case lAol/a < 1)

Quadratic magnetoconductivity was observed for doped boron in [2] for ficlds H = 6 to 15 kOe. According to [2] -Aa/a = 7 x for H = 1.4 x x lo4 Oe (at T = 300” K). Supposing the present consideration to be applicable to doped as well as to pure @-rhombohedra1 boron and taking for To the above- mentioned value (lo9 OK) we find from (29) a-1 x 10 il. This value is not very much different from (17) .

8. Discussion

We see that the low-temperature (77 and 113 OK) experimental dependence a( E ) for p-rhombohedra1 boron could be satisfactorily interpreted by the hopping model (Section 5 ) . It seems reasonable that the polaron effect (Section 6) for the conduction phenomena in P-boron should also be taken into account.

Perhaps in P-boron with its very complex crystal lattice we are faced with a mixed type of Conduction which we can call “polaron-Mott” (or polaron- hopping) conduction. The theory of such type of conduction is not yet develop- ed but some qualitative considerations were given by Austin and Mott [32] (see also [33]).

We can suppose that the temperature dependence of carrier mobility for po- iaron-Mott conduction can be determined as a product of the polaronic exponent (25) and the tunneling (hopping) exponent exp (-2 a Rapt) x exg [ - ( T0/T)’ /4] . It is evident that the separation of the observed a( 2’) dependence into polaronic and hopping components (for constant carrier concentrattion),

Studies of a Conductivity Mechanism of @-Rhombohedra1 Boron 457

should lead to a reduction of the effective value of To and consequently to an increasing effective density of trapping states (18).

We think that in the polaron-hopping conduction in P-boron the polaron effect has no essential influence on the magnetoconductivity (except the formation of the value 01 of the wave function of localized states where the polaron effect is sufficient [27]).

From our experimental investigations we cannot draw a t present a definite con- clusion if the Poole-Prenkel effect really takes part in the formation of I - /J characteristics of pure b-boron. Probably the strong electric field situation for P-boron is quite near to that of amorphous germanium, where the measured I-U characteristics were primarily explained by the Poole-Frenkel effect [34] while lately the hopping explanation was put forward [13, 351. Among the possible experiments which can throw more light on the understanding of the conduction mechanism of P-boron we may mention the following :

1. measurements of a( T), I-U characteristics, and magnetoconductivity for temperatures below liquid-nitrogen temperature;

2. measurements of I-U characteristics in the presence of a magnetic field; 3. investigations of current phenomena in other modifications of boron (i.e.

u-rhombohedral, tetragonal, and amorphous) and in such borides as u-AlB,, and P-AlB,, which have crystal structures similar t o those of various modifications of pure boron.

Acknowledgements

We are grateful to Dr. F. A. Chudnovskii and Dr. B. I. Shklovskii for valuable discussions. One of us (A.A.B.) acknowledges useful discussions with Dr. V. A. Ganin.

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(Received August 1, 1973)