Embed Size (px)
Citation preview
PHYSICAL REVIEW' B VOLUME 46, NUMBER 12 15 SEPTEMBER 1992-II
oscillator strength for intersubband transitions in strained n-type Si„Ge, „quantum wells
S. K. Chun and K. L. WangDeuice Research Laboratory, 66-1478 Engineering IV, Department of Electrical Engineering,
Uniuersity of California, Los Angeles, California 90024(Received 31 January 1992; revised manuscript received 20 April 1992)
The oscillator strength for conduction intersubband transitions in a Si„Ge& potential well is calcu-lated using an inverse mass tensor for any arbitrary growth direction and various strain conditions. Theoceupaney of the specifi conduction valleys and their associated effective masses are shown to dependon the strain condition, and these dependences are taken into consideration in the calculation of the os-cillator strength. The result illustrates that the intersubband transition can occur for normal incidentlight (or for differently polarized electric field) if the ellipsoidals in the conduction valleys are tilted. Abiaxial tensile strain does not affect the oscillator strength; however, the biaxial compressive straincauses an increase of the oscillator strength for the z-polarized electric field, but a decrease for the xy-polarized electric field, independent of the nature of conduction minima being Si- or Ge-like. This is due
to the change of the occupied valleys under strain. The calculated result for a waveguide structure isalso obtained to compare with a recent experimental result.
I. INTRODUCTION
Recently, electron intersubband transitions have beensuccessfully demonstrated for an infrared detector appli-cation using Al„Ga
&„As/GaAs and Si„Ge
&„/Si
quantum-well structures. ' For clarity of discussion, wedefine the xy polarization for which the field is on thequantum-well plane, whereas for the z polarization thefield is parallel to the quantum-well growth direction.For GaAs, the isotropic effective mass of the I conduc-tion band makes the selection rule such that intersubbandtransitions are allowed only for incident light with theelectric field polarized along the quantum-well growthdirection (z polarization). For Si, the ellipsoidals fororientations other than [001] are tilted and there are off-
diagonal elements in the effective-mass tensor. ' In theinvestigation of the intersubband spectroscopy of themetal-oxide semiconductor (MOS) inversion layer, Nee,Claessen, and Koch observed an indication of a transi-tion for both perpendicular and parallel incident lightdirections for [110]MOS inversion layers. They used theeffective-mass tensor formulated by Stern and Howardand showed that the xy-polarized electric field inducesthe intersubband transition through the off-diagonalterms of the inverse mass tensor for [110] samples. Oneof the problems of using the inversion layer is the lowquantum efficiency for IR detector application. Anotherproblems is that the transition energy usually occurs inthe far-IR range (10 meV) because of its large effectivemass and thus making it impossible for application in thewavelength range of 8—12 pm ( —100 meV}. With the re-cent advent of molecular beam epitaxy (MBE) and low-temperature chemical vapor deposition (CVD}, the suc-cessful growth of the strained Si„Ge, layer makes pos-sible to explore quantum-well structures usingSi/Si Ge, „heterostructures and/or 5-doped layers.Park et al. have demonstrated the transition near a 10-
pm wavelength using p-type 5-doped Si„Ge& „/Siquan-tum wells. Similar results have been reported by n-typemultiple quantum-well structures on (100} orientedSi. ' However, all the results are for samples grown in[001] and thus show a similar polarization dependence asthat of GaAs/Al„Ga, „Asquantum wells. Theoretical-ly, Yang and co-workers"' have calculated the absorp-tion coefficient of conduction intersubband transitions ofthe Si„Ge, /Si quantum wells grown in the [110] and
[111]directions for the two xy and z polarizations. Theabsorption coefficient for the xy polarization is shown tobe comparable to that of GaAs/Al„Ga, „Asin this case.In their calculation, no strain effect was included and allindirect conduction valleys were assumed to be degen-erate. However, as we will show, the strain can affect thepopulation of each valley and the occupancy of differentvalleys will in turn determine the polarization behavior ofthe intersubband transitions. Further, the absorptioncoefficient changes according to an appropriate sum ofthe effective masses of the occupied valleys projectedonto the growth direction. Thus the strain effect is animportant factor to be taken into account for understand-ing the conduction intersubband transition physics ofSi„Ge&
„
layers.In this paper, the oscillator strength of the conduction
intersubband transition for both Si„Ge& strained layersand/or 6-doped structures is calculated for growth onvarious orientations and under different strain conditions.In the calculation, the one-band effective-mass approxi-mation is used and the Hamiltonian used is a tensor form.The results of the calculated oscillator strength areshown for different growth orientations and for compres-sive and tensile strain conditions. The polarization be-havior is different from that of the Al„Ga, „As/GaAsintersubband transition, where the conduction minimaoccur at the I point. Our results are quite general andcan be applied to other material combinations whose con-
46 7682 1992 The American Physical Society
OSCILLATOR STRENGTH FOR INTERSUBBAND TRANSITIONS. . . 7683
duction minima occur away from the I point. The oscil-lator strength is also obtained for a waveguide structurein order to compare with a recent experimental result.The results of the oscillator strength can be easily used topredict the actual absorption coe@cient or absorptionstrength. '
II. OSCILLATOR STRENGTHFOR INTERSUBSAND TRANSITIONS
In the one-band approximation, the Hamiltonian canbe modified in the effective-mass approximation as fol-lows
[001]AZ Growth
Direction
8=—,'P W P+ V(z),
where P is a vector momentum operator and W is a 3 X 3inverse mass tensor (instead of the mass tensor) to ac-count for the anisotropic mass. The coordinate system ischosen as shown in Fig. 1 to have the growth directionalong the z axis throughout the paper. For a givengrowth direction, the inverse mass tensor for each in-direct conduction valley can be obtained using a coordi-nate transformation in which the Euler's angles are usedto describe the relation between the direction of a con-duction valley and the growth direction. The potentialenergy, V(z}, of the Si Ge, „quantum well may comefrom an inversion layer, the conduction-band offset of theheterostructure, ' or the self-consistent potential due to a5-doped layer. The above expression describes themotion of an electron in the potential well and is valid ifthe well width is much larger than the atomic dimension.Then the motion of an electron can be effectively de-scribed by the envelope function. Within this framework,the envelope function can be written as
F(x,y, z )=f(z )exp[ —jz(k„w„»+k»w„,)/w ]
Xexp[j(k„x+ky)],where k, 's and w; 's are the wave vector and elements ofthe inverse mass tensor, respectively. The amplitudefunction f(z ) is a function of z only and does not vanishat the interfaces for a finite potential well. To simplify
the calculation, we use the boundary condition that thewave functions vanish at the interfaces, i.e., assuming aninfinite potential well case. This assumption makes f (z)simply an even or odd function. The approximation ofthe above boundary condition gives the calculated ab-sorption coefBcient of the GaAs system within a factor oftwo of experimental data. ' So is the oscillator strength,as expected.
The oscillator strength for the direct conduction valley(conduction-band minimum at I'} has been calculated byWest and Eglash' and can be modified for the more gen-eral indirect conduction valley when considering the opti-cal electric field of the incident light at an angle, i.e., witha polarization vector e,
where y is an index of indirect conduction valleys, mo isa free-electron mass, e is the angular frequency of sub-band transition, g, and g are the initial and final states,respectively, consisting of the envelope function de-scribed as above and an approximate Bloch wave func-tion. The oscillator strength for any direction of the in-cident light can be written as
2m 0ff 1= (6 w +e»w» +e w") )(FJ~P, ~F;)[ . (4)
Unlike the direct conduction valley case, the xy-polarizedoptical electric field can cause the electronic motion inthe growth direction through the oF-diagonal terms ofthe inverse mass tensor, w~ and w„~ and thus induce theintersubband transitions. Obviously, for the z polariza-tion transition, the electric field acts on the envelopefunction through w and the behavior is similar to thedirect gap ( 1 band) case.
We will evaluate the oscillator strength of both para-bolic and square potential-well cases. For the parabolicpotential well, the momentum operator P, may be ex-pressed in terms of a creation and an annihilation opera-tors' and the matrix element is nonzero only for thetransition between two immediate adjacent states(hn = l). The oscillator strength for the transition fromn to n+1 is given in terms of elements of the inversemass tensor w;~J as
= [010]
mofr „+,= (e„wr,+E»w»r, +e,w") (n+l) .W~y
[100]
FIG. 1. Coordinate system chosen to have the growth direc-tion along the 2 axis. The polar angle represents the angle be-tween the [001] direction and the growth direction and the az-imuth angle between the [100] axis and the projected directionof the growth direction on the (001) plane.
For the special isotropic conduction valley at the I point,the off-diagonal elements of the inverse mass tensor, w~
and w~„are zero, and the above oscillator strengthreduces to that for an infinite parabolic potential derivedby Karunasiri and Wang. ' For an infinite square poten-tial well, the oscillator strength can be written as (forcomparing with the above expression of the parabolic po-tential well)
7684 S. K. CHUN AND K. L. WANG
64 (n+1} (n+2)m (2n+3)
(6)
1fn~n+1= g fn~n+1
n,
where n, is number of the occupied valleys (six for Si-type and four for Ge-type conduction valleys) and g is
[001] [001]which is the same form as that for the isotropic conduc-tion band with an infinite square potential well' if theoff-diagonal terms vanish. As shown in Eqs. (5) and (6),the same mass tensor term in the brackets of the expres-sions is independent of the type of the potential well.That is, the oscillator strengths have the same behavior.
Since the effective mass of the occupied valleys deter-mines the transition first, we need to analyze the occu-pancy of the conduction valleys with and without strainas follows. For a relaxed Si Ge, , alloy, the energy gapcan be approximately obtained by the virtual crystal ap-proximation, i.e., a linear interpolation of the lowest bandedge at the 5 and L points of Si and Ge, ' respectively.It has been shown that the relaxed Si„Gei „alloy hasSi-like minimum conduction valleys for the Ge mole frac-tion up to 85%, and beyond 85%, they change to Ge like.In this unstrained case, the minima of Si- or Ge-like con-duction valleys are all degenerate. In quantum wells(even without strain), however, due to the different direc-tional masses, the subband energy levels are split for thedegenerate conduction valleys. In the latter case, theconduction valley with the largest directional mass givesthe lowest subband energy level and is occupied as shownin Figs. 2(a} and 2(b) (dotted lines) for the [001] and [110]growth directions, respectively. Here, for the [001]growth direction as shown on the top of Fig. 2(a), the twoshaded ellipsoidals in the [001] and [001] directions areoccupied due to the higher mass projected to the [001]growth direction. On the other hand, the four equivalentellipsoidals are occupied for the [110]growth direction asshown at the top of Fig. 2(b). In the strained case asshown in solid lines, the occupancy changes dependingon the strain condition. If the well is strained, the con-duction valleys of the Si„Ge, „atthe 5 or L point areno longer degenerate and the determination of occupiedvalleys cannot be assessed in terms of the directionalmass alone. For example, a biaxial compressive strain ispresent in the Si„Ge1
„
film grown directly on Si. Thebiaxial compressive strain can be decomposed into dila-tion and uniaxial components; the dilation term simplyshifts all the band edges together while the uniaxial termlifts the degenerate conduction valleys. On top, the solidlines of Figs. 2(a) and 2(b) illustrate the energy of the sixequivalent 6 valleys split into twofold and fourfold de-generate valleys. ' ' ' The heavy-hole band shifting isalso shown in the bottom part of the figures. The energyof the conduction valleys under strain can be readilydetermined from the known deformation potential. '
The occupancy of these split sets of the valleys will thusdepend on the strain and their effective masses projectedto the quantization (or growth) direction. In order toevaluate the oscillator strength, we need to sum the con-tribution of each of the occupied valleys according totheir occupancies as follows:
[010]
[100]
(a)li E
X4 X2
------ Unstrained
Strained Heavy
k [100][010]
[001] k
[110]fk
[110]
[110] [110]
[001]
ii E
X4 V'
UnstrainedHeavy
k [100][010]
[001] k
FIG. 2. Change of band structure due to the biaxial compres-sive strain when the growth direction is chosen along (a) [001]and (b) [110]directions. The dotted lines indicate the positionsof the conduction and heavy-hole bands for the unstrained casewhile the solid lines indicate the shifted positions of bands un-
der the biaxial compressive strain. The growth directional cur-vatures are drawn for X2 and X4 conduction valleys (bottom).
46 OSCILLATOR STRENGTH FOR INTERSUBBAND TRANSITIONS. . . 7685
the fractional occupancy of all equivalent valleys. In gen-eral, when the strain and the difference of the projectionmass from different equivalent valleys are small, the ener-
gy separation of each set of the equivalent valleys is smalland consequently all the valleys can be occupied. Forsimplicity and for clarity, however, we will assume that,initially, only one set of the equivalent valleys is occupiedand the others are empty. In this case, g equals unity.The oscillator strength is used to calculate the absorptioncoeScient, which is simply proportional to the product ofthe oscillator strength and the two-dimensional density ofstates. ' Here the two-dimensional density of states for allthe occupied equivalent valleys can be evaluated from theprojected effective masses onto the growth plane.
III. RESULTS AND DISCUSSION
4
2
o
Relaxed Ge well
(gp 4 ) 1
Cl'p~) ~& p~P
xy polarization (normal incidence)
A. Si and Ge 5-doping quantum well
First, we discuss the unstrained 5-doped quantum-wellcase. The oscillator strength for the intersubband transi-tions between the ground state and the first excited stateis calculated for a parabolic potential well and the resultsare shown in Figs. 3-6. The two angles shown in the
S
p.S
O.6
o o'
Q,2
Relaxed Si wellf5
0 ~gj&
z polarization
FIG. 4. Oscillator strength for the transition between theground state and the first excited state in the parabolic well: (a)
xy polarization and (b) z polarization. The parabolic quantumwell is in the relaxed Ge layer.
(4/@, ~gg
~ .~ll+p,g
xy polarization (normal incidence)
z polarization
FIG. 3. Oscillator strength (OS) for the transition betweenthe ground state and the first excited state in the parabolic well:(a) xy polarization and (b) z polarization. The parabolic quan-tum well is in the relaxed Si layer.
figures indicate the growth direction in relation to [001].For example, the [110] growth direction has a 90' polarangle and an azimuth angle of 45'. Figures 3(a) and 3(b)show the oscillator strength of Si 5-doped quantum wells(unstrained) for the xy- and z-polarized electric fields, re-spectively.
For the xy polarization shown in Fig. 3(a), the electricfield is polarized on the plane and induces the electronmotion along the z direction through the coupling of theoff-diagonal elements w~ and w~. This coupling ispresent for the growth directions other than [001] in Siand thus can produce the intersubband transitions for thenormal incident light. In this normal incidence case, theoscillator strength is averaged over all the angles on theplane, as it is dependent of the optical-field direction onthe quantum-well plane. By varying the Euler's angle(growth direction), the average oscillator strength forother growth directions can be calculated and the max-imum value of normal incidence, 0.8243, is found to benear the [023] growth direction. In contrast, for the [001]growth direction, all the off-diagonal terms of the inversemass tensor are zero as discussed above and the intersub-band transition for the xy polarization (i.e., for normal in-cident light) is forbidden, identical to the case for the
7686 S. K. CHUN AND K. L. WANG 46
direct conduction I valley.For the z-polarized electric field, the oscillator strength
for any growth direction is shown in Fig. 3(b}. It is clearfrom Eqs. (4) to (7) that the oscillator strength is propor-tional to w for this case. For a given growth direction,different valleys will have different inverse mass parame-ter w ~. Although the maximum oscillator strength is ex-
pected for the largest ~,~ valleys, these valleys are usually
empty due to their high subband energies. Thus theycannot contribute to the absorption. For the [111]growth direction, in which all conduction valleys at the b
point are degenerate, all the valleys contribute equally.In this case, the highest oscillator strength peak of 3.8489at the [111] direction is obtained, which is four timessmaller than that of a GaAs quantum well (f&,~, =15).But the higher doping level in Si may make the absorp-tion strength even higher than the GaAs case.
For unstrained Ge quantum wells, the oscillatorstrength for the xy and z polarizations are shown in Figs.4(a) and 4(b}, respectively, for the L and (111) valleys.For the xy polarization, Fig. 4(a) illustrates that the inter-
subband transition is forbidden when grown in [111]. Inthis case, similar to the [001] case in Si, the two heavierconduction valleys in the [111]and [1 1 1] directions arepopulated (and others are empty) and the inverse mass
tensor for these valleys has only diagonal terms. For the
xy polarization, the maximum oscillator strength of3.6755 is obtained when the growth direction is near[144]. For the z polarization, the dependence of the oscil-lator strength as a function of the growth direction looksdifi'erent from that of the Si quantum well [see Figs. 3(b)and 4(b)]: when the optical field is polarized in the [001]growth direction, the oscillator strength is maximum fora Ge quantum well, but minimum for Si. For the samez-polarized field, the [111] growth direction, however,produces an opposite picture which can be understoodfrom the fact that Ge( 111) is equivalent to Si(001).
B. Strained quantum wells
Since the strain will affect the energy and thus the oc-cupancy, we have calculated the oscillator strength forseveral strained quantum wells. First, we discuss the re-
sult for the strain-symmetrized Si, „Ge„/Siquantum-
Ge on Si»Ge08 (Ge-like)
0.gw
0.6
o"0,2
gg 2
O
(g ii is
xy polarization (normal incidence)
p P
((p 8 i i g+Cl'p~) ~ pgj%
xy polarization (normal incidence)
gg 2
o
g5
&@~ sg
z polarization
FIG. 5. Oscillator strength for the transition between the
ground state and the first excited state in the parabolic quantum
well: (a) xy polarization and (b) z polarization. The biaxial
compressive Sio 6Geo 4 layer grown on Sip SGeo & is the quantum
well, where the Si-like conduction valleys are occupied.
g5
Qp)0
z polarization
FIG. 6. Oscillator strength for the transition between the
ground state and the first excited state in the parabolic quantum
well: (a) xy polarization and (b) z polarization. The biaxial
compressive Ge layer grown on Si02Geo & is the quantum well,
where the Ge-like conduction valleys are occupied.
46 OSCILLATOR STRENGTH FOR INTERSUBBAND TRANSITIONS. . . 7687
well case. In a strain-symmetrized case, a thick relaxedSi, Ge buffer layer is used, where y =x/2 (Ref. 20) as-
suming that the layer thicknesses of Si and Si, „Ge„lay-ers are the same. For example, Si and Sio 6Ge04 layersare grown on a thick relaxed Si08Ge02 layer. In thiscase, the Si layers serve as the conduction-band wells andexperience a biaxial tensile strain on the plane and acompressive strain along the growth direction. In thiscase, for any growth direction, the conduction-band val-leys closer to the growth direction have a lower energythan the others due to the in-plane tensile strain ( or thecompressive strain along the growth direction), and thesevalleys are thus occupied. In the calculation, only theseoccupied valleys are used. In addition, the inverse masstensor of the strained Si layers is assumed to be the sameto the first order as that of unstrained Si. '
In the case when the Si06Ge04 layers serve as thequantum wells achieved by 5 doping, rather than by theband offset, the wells experience a biaxial compressivestrain. The biaxial compressive strain puts the valley en-ergy higher for those conduction 6 valleys closer to thegrowth direction (via deformation potential). In thisSi„Ge& „well case, the effective mass for the Si„Ge&alloys is needed for the calculation of the oscillatorstrength. However, since the curvature of the Ge 6 val-ley is not known, we cannot use a virtual-crystal approxi-mation to predict the curvatures of the alloys. Theoreti-cally, Cardona and Pollak ' have shown that the 6effective masses of Si and Ge are very close (within 20%),
although their calculated data did not agree with the gen-erally accepted measured values. In view of the lack ofthe data, we assumed the 6 valley masses for Si„Ge,alloys be the same as those of Si. This assumption willgive some errors quantitatively but it is expected to bequalitatively correct. For the case of the biaxial compres-sive Sio 6Geo 4 well, the calculated results of the oscillatorstrength for the xy and z polarizations are shown in Figs.5(a) and 5(b), respectively. When compared with that ofrelaxed Si shown in Fig. 3, the oscillator strength issmaller for the xy polarization and larger for the z polar-ization. This is due to the fact that the occupied valleysunder the biaxial compressive strain change from those ofrelaxed Si and thus different mass tensors for the newlyoccupied valleys affect the oscillator strength. However,for the [11 1] growth direction, all six conduction valleysat the 6 point are shifted all together with the same ener-
gy and thus remain degenerate. The oscillator strengthof Si„Gei „wells for the [111]growth direction is exact-ly the same as that of the relaxed Si well for both the xyand z polarizations discussed above.
Next, the structure of Ge/Si04Geo 6 layers grown onSic zGeos (for strain symmetrization) is discussed. Forthe quantum wells in the strained Ge layers grown onSic 2Geo „Figs.6(a) and 6(b) show the oscillator strengthfor the xy and z polarizations, respectively. For any arbi-trary growth direction, some of the L valleys will still bethe minimum under the biaxial compressive strain (com-pared with the X valley). Generally, the oscillator
TABLE I. Oscillator strength for the parabolic potential well for different growth reactions andstrain conditions. The values in the parentheses is the oscillator strength for the square potential well.+ indicates the maximum values for that polarized field.
MaterialGrowthdirection
Number ofoccupied valley
Normalincidence
Parallelincidence
relaxedSi
[100][110][111][023]
0.0000(0.0000)0.7162(0.6880)0.5197(0.4992)0.8243(0.7919)
1.0204(0.9803)3.1418(3.0182)3.8489(3.6975)2.3259(2.2344)
relaxedGe
[100][110][111][144]
1.7896(1.7192)3.3352(3.2040)0.0000(0.0000)3.6755(3.5310)
8.3333(8.0056)4.4715(4.2957)0.6098(0.5858)2.7162(2.6094)
Si onSip 8Gep 2
same asrelaxed Si
Sip. 6Gep. 4 onSip 8Gep &
[100][110)[111]
0.0000(0.0000)0.0000(0.0000)0.5197(0.4992)
5.2632(5.0562)5.2632(5.0562)3.8489(3.6975)
Sip.4Gep. 6 onS~p 2Gep 8
same asrelaxed Si
Ge onS~p. 2Gep. 8
[100][110][111]
1.7896(1.7192)0.0000(0.0000)0.6077(0.5838)
8.3333(8.0056)12.1951(11.7155)10.9079(10.4789)
7688 S. K. CHUN AND K. L. WANG
strength in the biaxial compressive Ge well is smaller forthe xy polarization and larger for the z polarization whencompared with the relaxed Ge well case shown in Fig. 4.This is due to the different valley occupancy as discussedabove. As shown in Fig. 6, there are abrupt changes inoscillator strength as a function of the growth direction.The abrupt changes of the oscillator strength are also dueto the change of the occupied valleys. For example, atthe growth direction of 50' of polar angle and 45' of az-imuth angle from [001], the Ge I. valleys at the [111]and [111] directions are occupied. However, as thegrowth direction changes to 55' of polar angle and 45' ofazimuth angle, the Ge valleys in the [111],[11 1], [111],and [111]directions are occupied instead.
Some of key features of the results for several orienta-tions are summarized in Table I, which gives the occu-pied valleys and the xy and z polarization results. Thevalues in the parentheses are the oscillator strength forthe square potential well, which are very close to thosefor the parabolic potential well; the dependences of theoscillator strength on the growth direction for the squareand parabolic potential wells are the same. From the re-sults, it is clear that the magnitude and angular depen-dence of the oscillator strength is nearly independent ofthe shape of the potential well (within a few percent).Our results also show that for a given growth direction,the oscillator strength for the z polarization is larger thanthat for the xy polarization. The relative ratio of the twofor the relaxed case is in agreement with the calculatedabsorption coefBcient for the inversion layer of MOS.
o 5e =90 y=90
0=0
AZ[110]
X[001]
Gh
o 3
0
90
solid line) is quite different and at the 90 polarization an-gle, there is a Gnite contribution and this contributionyields transition for normal incident light. For the casewhen the x is along [110],Fig. 7(b) shows the polariza-tion angle dependence for all six valleys, identical to thatof the [001]growth case.
The calculated results are compared with the experi-mental data as shown in Fig. 8(a); the data points arefrom an experiment' and the lines are the calculated re-sults. For comparison, our calculation uses the samestructure as that used in the experiment shown in Fig.8(b). ' For such a structure, the four X4 valleys of the Siwells are supposed to be occupied. In Fig. 8(a), the cir-cles are the experimental data for the case when the xdirection is [001] and the square data are for x along
C. Waveguide structure
For convenience of comparison with the experimentaldata, we also calculated the oscillator strength for a mul-tipass waveguide. The waveguide structure shown in theinset of Fig. 7(a) is used for increasing the absorptionwith multiple passes in experiments. We will discuss onlythe [110]growth directions. As shown in the inset of Fig.7(a), the sample has a bevel angle of 8 and the z and xaxis are chosen to be along the [110]and [001]directions,respectively. In the structure, for the incident light nor-mal to the polishing plane, the polarization angle (t isdefined as /=0 for the zx polarization having theelectric-field components along the x axis (on the plane)and the growth direction. For /=90, however, the fieldis along the y direction only. The results for general Si-like valleys are shown in Figs. 7(a) and 7(b) and the con-tributions from both X2 and X4 valleys are given sepa-rately, as functions of the bevel angle 0 and the polariza-tion angle P, assuming a parabolic potential. Figure 7(a)shows the oscillator strength for the case when the x axisis along the [001] direction. In the figure, the oscillatorstrength (in dotted lines) from the two X2 valleys in the[001] and [001] directions decreases as the polarizationangle increases, showing the same dependence as the Ivalley case (-sin Ocos P). Similarly, as the bevel angledecreases, the oscillator strength also decreases and be-comes zero for the normal incidence (8=0). On the oth-er hand, the dependence of the polarization angle of theoscillator strength for the other four X4 valleys (shown in
15 30 45 60
Polarization Angle It) (degree)
90
6
(b) AZ [110]
e =90 /=90 X[110]
4bQ
C/J
o 3
0
X2x4
15 30 45 60 75
Polarization Angle $ (degree)
90
FIG. 7. Oscillator strength for the transition between theground state and first excited state when the minimum conduc-tion valley is Si like. The growth direction is along [110]direc-tion and the x axis is chosen along (a) [001] and (b) [110]direc-tions.
46 OSCILLATOR STRENGTH FOR INTERRSUBBAND TRANSITIONS. . . 7689
110. Inusin an
t e calculation, we adjust th te ransition ener y'ng an depolarization and excitonlike shifts ' to fi
e ito 'esiA
o et esameasthede old this magnitude is assumed
p arization shift factor a, wherea and are Ando's coefficients. ' In obtainin a
'n epen ence data shown in Fig. 8(a),
we rotate the sample plane along the z[110~ directi
a s ig t error in making a cut alon the xmixed contribution ma c
ong e x axis. The'
u ion may come from a small population in
1.2
z axis x axis
[001] [100]
[010]
Oscillator strength
f»=1.0234sin Ocos Pfx4=5.2632sin H cos Pf x3=1.0204si nH3c os Pfx4 =5.2632 sin'O cos P
x=y x&y x(yfxz fx4 fx3
fx3 fx4 fx3
[110] [001] fx3=5.2632sin'Hcos Pfx4= l.4324 sin'P+3.1415sin3Hcos P
[110] fx3=5.2632sin Ocos Pfx4= 1.4324 cos'H cos P+3.1418sin Ocos P
fx4 fx3 fx4
fx4 fx3 fx4
TABLE II. An
e eve and polarization an les randy are the G
g s, respectively. The xe e contents of the well an
tively.an buffer layers, respec-
lt
cj
o 0.8~~z
Pg 0.6CJ5
O
E; 04-O
0.2
15
0 0
'Q
CI
'la
X[001]:experiment
X[001]:calculation
X[110]:experiment
X[110]:calculation
30 45 60 75
Polarization Angle P (degree)
[112] fx6=0.5196cos Ocos P+0.5196sin3$+3.8489 sin3H cos3$
[110] fx6=0.5196cos Ocos P+0.5196sin'P+3.8489 sin3H cos3$
fx6 fx6 fx6
fxe fx6 fxe
the X2 valleys in that exy'
at experiment, in addition to the X4
For other growth directions, [001] and [111,the re-sults are listed in Tables II and III.and 6 -lk 11e- i e va eys are given.
an . Both cases for the Si-
Si Sip 3Gep 7 SiTABLE III. AnAngular dependence of the oscillator s
when the minimum cond'
1con uction valley is Ge hke.
17meV
I
128meVI
II
273meV
26meVT ))I
85meVI
I
]i
I
IIIIIIII
X4-----. X2rII
I
I
III ~
I
I
I
IIIIIIII
2' axis x axis
[001]
[110]
Oscillator strength
[100] fL4=1.7896cos Hcos P+1.7896sin IP)
+ 8.3333 sin3H cos3$[010] f« = l.7896 cos'O cos'P
+ l.7896 sin3(()
+8.3333 sin3H cos P
[001] fL3=6.6704cos Hcos p+4.4715 sin Ocos IIfL3=12.1951sin Ocos II
[110] fL, =6.6704 sin'II+4.4715 sin Ocos PfL3 = 12.1951 sin O cos I)
x=y x&y x&y
fL4 fL4 fL4
fL~ fL4 fL4
fL3 fL2 fL3
fL3 fL3 fL3
FIG. 8. Absor tionp strength as a function of the olangle II for the 8-do ed S'(110)
e po arization
and squares are thepe i 0) quantum well. (a) The circles
e e experimental data (Ref. 10) whenho to b 1o [001 [[ o], p 4ong and &1103 r
o e ines are the calculated valuesn '„respectively. (b) Band dia ram
io 7 eo 3/Si layers used in the experiment (Ref.
[111] [112] fL|=0.6098sin OcosfL3=0.6076cos Ocos P+0.6076 sin I)+10.9079sin Ocos (()
[110] fL, =0 6098 sin'H c.os'IIfL3 0.6076 cos O cos+0.6076 sin P+10.9079sin Hcos II
f« fL3 fL|
fL1 fL3 fLl
7690 S. K. CHUN AND K. L. WANG 46
IV. CONCLUSION
We have calculated the oscillator strength of the inter-subband transition in a strained Si„Ge, potential wellfor any arbitrary growth direction and several strain con-ditions. The occupied conduction valleys are determinedby both the strain condition and the growth directional(quantization) mass. The intersubband transitions forboth xy and z polarizations are possible if the ellipsoidalsof the conduction valleys are titled. When the minimumconduction valleys are Si like, the oscillator strength forboth xy and z polarizations under the biaxial tensilestrain is the same as that in the relaxed Si well. For thelatter, only the subbands of the two X2 valleys are occu-pied.
However, under the biaxial compressive strain in Si-like valleys, the oscillator strength for xy polarization issmaller, while the oscillator strength for z polarization islarger than those in the relaxed Si we11. On the otherhand, for Ge-like wells under the biaxial compressivestrain, where the minimum conduction valley is at the L
point, the oscillator strength is sma11er for xy polariza-tion and larger for z polarization as compared with thosein the relaxed Ge. These changes of the oscillatorstrength are due to the different valley occupancies, thatis, different mass tensors under strain. The growth direc-tions for the normal incident light to have a large oscilla-tor strength have been identified for several strained andrelaxed wells. For the [110] growth direction, thedifferent angular dependence of the oscillator strength onthe direction of the polishing plane in the waveguidestructure has been worked out in order to analyze experi-mental data. The improved crystal-growth and process-ing technology appear attractive for application ofSi,Ge, „quantum wells for normal incidence detectors.
ACKNOWLEDGMENTS
The authors acknowledge helpful discussions with R.P. G. Karunasiri. This work was in part supported bythe Office of Naval Research and Army Research Office.
'L. C. West and S.J. Eglash, Appl. Phys. Lett. 46, 1156 (1985).~B. F. Levine, K. K. Choi, C. G. Bethea, J. Walker, and R. J.
Malik, Appl. Phys. Lett. 50, 1092 (1987).3K. S. Yi and J.J. Quinn, Phys. Rev. B 27, 2396 (1983l.4S. M. Nee, U. Claessen, and F. Koch, Phys. Rev. B 29, 3449
(1984).5F. Stern and W. E. Howard, Phys. Rev. 163, 816 (1967).J. S. Park, R. P. G. Karunasiri, Y. J. Mii, and K. L. Wang,
Appl. Phys. Lett. 58, 1083 (1991).7H. P. Zeindl, T. Wegehaupt, I. Eisele, H. Oppolzer, H. Reis-
inger, G. Tempel, and F. Koch, Appl. Phys. Lett. 50, 1164(1987).
SG. Tempel, N. Schwarz, F. Muller, and F. Koch, Thin SolidFilms 184, 171 (1990).
9F. Miiller, G. Tempel, and F. Koch, in Proceedings of the International Conference on the Physics of Semiconductors, edited
by E. M. Anastassakis and J. D. Joannopoulos (WorldScientific, Thessaloniki, 1990), Vol. II, p. 1254.Chanho Lee and K. L. Wang, J. Vac. Sci. Technol. B 10, 992(1992).
"C.L. Yang and D. S. Pan, J.Appl. Phys. 64, 1573 (1988).' C. L. Yang, D. S. Pan, and R. Somoano, J. Appl. Phys. 65,
3253 (1989).' R. People, Phys. Rev. B 32, 1405 (1985).'4S. Wieder, The Foundation of Quantum Theory (Academic,
New York, 1973).' R. P. G. Karunasiri and K. L. Wang, Superlatt. Microstruct.
4, 661 (1988).' R. Braunstein, A. R. Moore, and F. Herman, Phys. Rev. 109,
695 (1958).I. Balslev, Phys. Rev. 143, 636 (1966).
tsR. People, J. C. Bean, and D. V. Lang, in Proceedings of theInternationai Conference on the Physics of Semiconductors,edited by O. Engstrom (World Scientific, Stockholm, 1986),Vol. 2, p. 767.Author, in Numerical Data and Functional Relationships inScience and Technology, edited by O. Madelung, Landolt-Bornstein, New Series, Group III, Vol. 17 (Springer-Verlag,New York, 1982).E. Kasper, H.-J. Herzog, H. Jorke, and G. Abstreiter, Super-latt. Microstruct. 3, 141 (1987).M. Cardona and F. H. Pollak, .Phys. Rev. 142, 530 (1966).
2T. Ando, Z. Phys. B 26, 263 (1977).
