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Ga1-xInxSb superlattices for long-wavelength IR applications
Superiattlces and Microstructures, Vol 16, No 1, 1994
77
TAILORING InAs/Gal-.rIn.l'Sb SUPERLATTICES FOR LONG-WAVELENGTH IR APPLICATIONS J. H. Roslund and T. G. Andersson Department of Physics, Chalmers University of Technology and the University of Goteborg, S-412 96 Goteborg, Sweden (Received 22 August 1994)
A systematIc study has been carned out on the design of long-wavelength Infrared detector matenals from InAs/GaSb and InAs/GalnSb superlatllces with absorpl1on energies below 250 meV. i.e .\ > Slim. The influence from layer thicknesses and alloy composition on cut-off wavelength and opllcal matnx element has been analysed. A three-band envelopefunction model Including strain effects was used to calculate conduction-band electron and valence-band light-hole states, while an effective-mass approximation wa~ used to descnbe heavy-hole states. In order to achieve useful absorption coeffiCIents. the period of such superlattices must be less than typically 20 monolayers. Calculations revealed that the absorptIon can be increased Wllh almost unaffected cut-off wavelength by redUCing the GalnSb thIckness. The effects of including a small amount of In In the GaSb in order to reach longer wavelengths were studied. One conclusion is that although it makes longer wavelengths pOSSible. It also makes the cut-off wavelength much more sensitive to monolayer and compositIOn fluctuations.
1. Introduction In 1970, Esaki and Tsu introduced the concept of semiconductor superlattices [I], layered materials with a periodICIty larger than the atomic lattice. Due to the rapid development of molecular-beam epitaxy, it was possible to realise this concept in the closely lattice matched GaAs/AlyGal_,·As system during the 1970s. The quantum-mechanical effects In this system depend on the relative positions of the conduction and valence-band edges in the host materials. At the GaAs/AlrGal_rAs heterojunctlOn, the band gap of GaAs lies completely within the band gap of At·Gal_, As, which leads to confinement of both electrons and holes In the GaAs layers. Another band-edge line-up is proVIded by the Inl_yGayAs/GaSbl_yAsv system 12J with alloy compositIOns chosen so that.ll =0.9l8.r+0.082 for latl1ce match. As opposed to GaAs/At Gal_,·As. there is a possibility to get the conduction-band edge of Inl_rGaxAs lower In energy than the valence-band edge of GaSb1_yAs y, thus creating a ,y,tem where the two host band gaps do not overlap. It was predIcted [2] that such a band hne-up would gIVe nse to new and interesting properties. One of the most im portant effects is that electrons and holes no longer w!ll be confined in the ,arne matenal; a spatial separation of carriers will occur. As
0749-6036/94/050077 +06 S08 00/0
a result. this superlattice's band gap can be made smaller than that of any of the host matenals. A semiconductorsemimetal transition of the superlattice [31 occurs for layer thicknesses exceeding 100 A. in spite of the semIconductor character of the host materials. During the 1980s, the technological Interest in low bandgap materials increased. Since the smallest bulk band gap achievable in III-V compounds is 0.17 eV for InSb, corresponding to a wavelength of 7 11m for interband absorption. other materials had to be found in order to satisfy the need for long-wavelength IR detection. The most common material to date used in applications is the alloy Hg l _, Cd,.Te. Smith et al. [4] suggested that a HgTe/CdTe superlattice would have Improved properties, much owing to the higher effectIve mass in a superlatllce and to the fact that layer thicknesses rather than alloy composition would determine the cut-off wavelength. However, it was found that growth and processing of these II- VI materials are more dIfficult than for III-V compounds. As a III-V alternative Osbourn [5] sugge,ted InAs o ;Sb o ,JInAsl_ySby. Here, the strain from the latl1ce mIsmatch causes the superlattice band gap to shrink. Thi, superlattice is of type II and needs ratherthick layers to reach the deSlfed wavelength regIOn. As a consequence of thIS and the separation of carriers, absorption cannot be as
I(} 1994 AcademiC Press Limited
78
Superlatttces and Microstructures, Vol 76, No 7, 7994
high as m HgTe/CdTe superlattlces. Since the InAs/GaSb superlattIce suffers from the same difficulty 161. the GaSb mu,t be alloyed wtth InSb in order to raise the heavy-hole ,ubband [71 and thus mcrea,e the cut-off way elength Thl, superlatttce has been shown to exhlbtt absorption a, high as Hg 1 _. Cd, Te at the same ttme tt has a much 11lgher effectIve mas" which serves to mhibit unwanted dark current, and Auger recombinatlon rates m device, Furthermore. n can be made wtth more well-known growth and proce"ll1g techmques. giving better uniformtty for devices. For apphcatIon purposes, it is necessary to have a quantitatIve a, well as qualitative understandll1g of growth and matenal a"e,sment, how the different superlattJce parameter, (InAs thickness, Gal_, In, Sb thickness, and Gal_. In, Sb alloy composition) affect the ab,orptIon edge, the effecttve ma"e, and matnx elements. Only some speCific ,uperlattIces have been grown and charactensed expenmentally and theoretIcally. General trends cannot alway, be predicted from these and few examples are not suffiCIent from an applicatIon pOInt of view. In this study, the roles of the respective parameters are ll1vestIgated in a sy,tematlc way. The motivatIOn has been to map the available parameter space 111 search of matenal and layer combmatIon, ,unable for detector apphcatlons.
-/~I'hL --'-Ji [' I, A' .
I, /I =
-f{I'td'-
FI/,
)cP',
}; ['I,),·
~~r;,
E,,,
where /.' , 1:'1/, and I,',,, are the strain-dependent r-point po,Itlon, of the conduction band. light-hole and spin-orbit spltt-off valence band,. rnpecttvely. The coupling between the light-hole and spin-orblt split-off bands depends on a strall1-dependent energy shift M,',. The couphng, }', between the valence and conduction bands was chosen so that CI , = :>'III,,}" =19.2 eV, which IS consistent with the bulk effective ma,~es of electrons and light holes Allowed energy eigenvalues, f, are given by solutIOns to a disper'lOn equation 191
yvhere 'I J, the superlatllce wave number, 1/ and h are layer thlckne"es, Ii = II + h. For conductIOn-band electrom, light hole, and spll1-orba-band holes, k, and II, are given by
:11 L'e'" - ()( (EI/ .. , - (I(E,"., - (I -
The theoretIcal framework used IS a three-band envelopefunctlon model for the conduction band, hght-hole valence band and the spin-orbit split-off band, and an effective-ma" approximation for heavy holes. In short, the envelopefunctlon approximallon rehes on the assumption that the wave functlon ,. can be wntten as a :-dependent 11l1ear combmation of Bloch states _ f ,k~ r~ 'LU'lJr " ()},'n(_) I.(. r) -
"r.::f H'~' +I[~
J
2. Theory
(I)
where: IS the co-ordinate along the growth aXIS, 1/".1 IS the periodic part of the Bloch function correspondll1g to band /I at the r -point, r ~ IS the component of the positIon vector lying in the growth plane, and k-L IS the component of the electron wave vector in the same plane. The Hamiltonian can be denved from Kane's 8 x 8 Hamiltoman [81 includmg the twofold conduction, light-hole, heavy-hole and ,Pll1orbit split-off bands. For reasons of slmphctty, only the band structure for wave vectors along the growth aXIs ha, been calculated. Setting k-L = 0 for a (100) growth plane allows one to decouple the Hamiltonian into two equivalent -' x 3 matnees, repre,entmg the conductton, hght-hole and spll1-orbit ,pht-offbands, and two eqUivalent effectlve-ma'i'i equatIOns for the heavy holes. Includll1g ,mun and dlsregardll1g terms of order ho', one gets the three-band Hamtltoman 191
(2)
.'3( 1,' Al".:,
k, =
1
I',
(E
/;.,' 1
-
I
[~
~~F~,) - ,I
(I( L';" , _. f)
= 1)2(:>.( 1,.\"., + ~E" - () + EI/" -
(I'
(4)
whereas for heavy hole,
f h', = tJ:>,~,/;},.'(L'I'h" -
1
II, -
I)
(5)
-llIhh,
The strall1-dependent band edges have been calculated usll1g ll1terpolatlOn formulae from Ref. 10 modified to yield a correct valence-band offset. The superlattices have been as,umed to have a lattlce constant in the growth plane Imposed by a GaSb ,ubstrate The model gives a sufficiently accurate descnption of electromc states with energies close to the band gap and waye Yectors parallel to the growth axis. The most cntlcal a"umptlon I, that the penodic parts of the Bloch functions are a"umed to be the same In both matenals. However. consldenng the large uncertall1ty in valence-band discontinuity. which has a great Impact on the cut-off wavelength. a more ngorous model would only make calculations less manageable, wtthout guaranteeing better accuracy. It is well known that one-band model, can descnbe the GaAs/AI, Gal_, As system. Compansons have also shown that two-band models for the InAs/Gal _, In, Sb system agree wllh experimental data 1111.
79
SuperiattJces and Microstructures, Vol. 16, No 1, 1994
Energy reV]
0,8 0.6
GaSb
GaSb
OA 0.2
-
I-- 0.0
InAs
-
0.2 '-
~~
L
20
L
40
30
50
60
70
Layer thickness, L [A]
Figure 1. The bulk band-edge energies of InAs and GaSb at 300 K compared to the calculated band structure of InAs/GaSb superlattices with equal thickness, L, of InAs and GaSb layers. Energies are measured relative to the InAs conduction-band edge. The symbols HHI, LHI and EI denote, respeclively, the highest heavy-hole subband. the highest light-hole subband and the lowest conduction subband. 30
3. Results and Discussion
--n.lo
The superlattice band structure is a result of several parameters. In Fig. I is shown the band structure for a series of superlattices with equal thicknesses of InAs and GaSb layers. The resulting subband structure as a function of increasing layer thicknesses is referred to the energy-band edges of the bulk materials. When the barriers get thicker, the Widths of the subbands decrease. With broader wells, the distance between conduction- and valence-band derived subbands decreases, and new subbands are generated with increasing layer thicknesses. By considenng the wave number k, for conduction-band electrons and light holes, one concludes that states with energies in the GaSb band gap are confined to the InAs layers, whereas states with energies in the InAs band gap are confined to the GaSb. Similarly, the heavy holes are confined to the GaSb. Large layer thicknesses result in an inherently low degree of wave-function overlap for interband transitions, which diminishes even further as the individual layers get thicker. For the superlattice periods studied here the optical interband transllion is between heavy holes and the conduclion band. The wavelength of this transition is very sensitive to the energy difference, Ee" = Evl - Ed, i.e. between the top of the valence band of GaSb and the bottom of the conduction band of InAs. In Fig. 2 is shown the wavelength corresponding to this transition, calculated for a temperature of 4 K compared to some experimental results found in Refs 6, 12, 13 and 14. Usually Ec,v is taken to be In the vicimty of 0.15 e V. However, a better fit to expenmental data can be
-..,
25
:::
2: .::::
's:.: "" ;;
Ref
20
.... Rer.
•
15
~
:;:;'" Z C
d.ffcrenle
h
"
Ref. 12
,
. . . .,
....
•
10
o
eV difference
----015 r\
L-_~
10
___- L_ _ _- L____ 30
.til
~
____
511
~_-J
60
70
{."yer th,cklless, L [AJ
Figure 2. Cut-off wavelength for a heavy-hole to conductionband transition for InAs/GaSb superlattices. Experimental data are compared to calculations for 4 K with two different values of the energy difference between the InAs conduction-band edge and the GaSb valence-band edge.
achieved by choosing Ec,v = 0.10 eV. This yields a valenceband discontinuity well within the experimental accuracy of Ref. 16, and the wide range of other experimental data and theoretical calculations, It should be noted that the choice of f.',>", mainly affects superlattices with long cut-off wavelengths, I.e. thick layers, where the fiat-band assumption IS expected to become less accurate. The energy difference F,>, can therefore be considered an effective parameter. Control of the individual layer thicknesses seems to be
80
Superlatttces and Microstructures, Vol 16, No 1, 1994
a)
InAs fixed 11 ML = 33 1.0
I-Ir------.------,-------.------,------,
A
:---:-.
InA'I/GaSh egllal,
12
0.8
s:
--''"," ~ ::
'"
~
0.6 004
0.2
El
t:~
HHl 0.0 ·0.2
r--
LH1
~:J.#
~
20
30
- ,,'Jft:i
-10
.~
-
.,
50
60
GaSh layer thickness
b)
GaSb fixed 11 ML
70
'"::
50
611
70
Layer thickness rAJ
Figure 4. Calculated cut-off wavelength as a function of the variable layer thickness for heavy-hole to conduction-band transillons for five different series of InAs/GaSb superlattices at 300 K: one consisting of equal thicknesses of InAs and GaSb, one with GaSb thickness fixed to II ML (33 A) and three with InAs thickness fixed to 11 ML (33 A), 14 ML (42 A) and 17 ML (51 A), respectively.
= 33 A
0.8
--t"''"",
-10
30
rA.]
1.0
s:
20
0.6 0.4 0.2
~
0.0 -0.2
¥,,;'''r:
20
30
-10
50
InAs layer thickness
60
70
rA.]
Figure 3. a) Calculated band structure at 300 K as a function of GaSb layer thickness for InAs/GaSb superlattices with a fixed InAs layer thickness of 11 monolayers (11 ML = 33 A). b) The same as a function of InAs layer thickness for superlattices with a fixed GaSb layer thickness of II ML (33 A). The figure illustrates how the respective materials contnbute to the positions of the bands. a somewhat neglected area. In Fig. 3 the band structure's dependence on layer thickness (ef. Fig. 1) has been splIt up in its dependence on each of the constituent materials. The InAs layer confines the electrons and serves as a bamer for the holes, whereas the opposite applies for the GaSb layer. As stressed previously, the widening of a well leads to a larger number of subbands, the subbands become narrower and their levels approach the bottom of the well. As indicated, there is a weaker modification of the states confined in the other material. In Fig. 4 the cut-off wavelength is shown as a function of layer thickness for five superlattice series. In all cases an increase In layer thickness leads to an increased cut-off wavelength, but the increase is much more pronounced wllh changed InAs thickness. This is due to the fact that InAs
has a much lower effective mass than GaSb, and since the InAs layers serve as quantum wells for the conduction-band electrons, changing the InAs thickness changes the cut-off wavelength dramatically. This has the effect that the cut-off wavelength for many superlattices is rather insensitive to fluctuations in the GaSb layer thickness. In order to reach the technologically important far-infrared region (.\ > 8 Jim), layers have to be thicker than 50 A. An increased spatial separation between the carriers WIll strongly suppress optical absorption. The absorption coeffiCIent between two energy levels, EI and E 2 , is given by (6)
To Yield the total absorption coefficient, Eq. (6) has to be Integrated over the available k-space for the initial and final states. However, for a heavy-hole to conduction-band transition, the optical matrix element ;\f(E I • E 2 )
=
J~';(r)(e·
pl/'J!r))d3 r
(7)
IS in itself a good figure of merit. Furthermore, In the envelope-function approximation, this can be written as the product of a bulk matrix element and an envelope-function overlap integral,
(8) The latter was calculated for the f-point heavy-hole to conduction-band transition in the superlattices shown In Fig. 4 and the result is presented as a function of cut-off
8
Superlattices and Microstructures, Vol. 16, No 1, 1994 wavelength in Fig. 5. USing FIgs 4 and 5 in connecuon, one can deduce that it is very difficult to make an InAs/GaSb superlattice wtth a good absorption above JO pm. From the inchnation of the curves in Fig. 5, one sees that for moderate matrix elements the wavelength can be extended wtthout losing the absorption by increasing the InAs layer thickness. More important is that by making the GaSb layer thinner than the InAs the optical matrix element can be increased with only a mild drop in cut-off wavelength. The reson for this is that the GaSb acts as a barrier for the electrons, and by making this thinner, we increase the overlap with the hole envelope functions. This conclusion WIll also apply to the InAs/Gal_rInrSb superlattices. A good route to fi nd interesting materials is therefore to choose an InAs/GaSb superlattice with a high optical matrix element thIS way, and since increased In content primarily raises the heavy-hole level in the Gal_rIn".Sb, and the opucal matnx element hardly will be affected, the cut-off wavelength can be raised by alloying GaSb with InSb. In FIg. 6 IS shown how the cut-off wavelength increases with alloy composition. An interesting observation is that superlattices wtth thinner InAs layers have a less emphasised dependence on alloy composition, which can be used to reduce the sensitivtty to alloy fluctuations. One has to bear In mind that it is reasonable only to use alloys with .1' < 0.4 since for larger .r-values the lattice mismatch WIll be reheved through the formation of dislocations, whIch severely hampers the opucal properties. The precise condition for this to occur has yet to be established. It is interesting to compare the InAs/GaSb system wah the InAs/Gal_rInrSb system, where an eqUIvalent cut-off O.X r----r---,.---.....--,-----r----, In '\<; fixed 11 \lL
1
o0
c;aSh hxed 11 ~11: '
25r----,.-----r---.-----r----, 15 ML [n,hi ....., 20
l
.
.:::
Ga
I-x
In Sb thickness 7 ML, \.
!
113
15
~IL
::
~
,." <:l
"
,II MLI 10 W~i[~!
~
-
'" " I.J
17 >IL]
5
o ~-----'-------~----~------~----~ 0.1 0.2 11.3 0.5 o o~ Indium fraction. x
Figure 6. The increase in cut-off wavelength with alloy composition for InAs/Gal_, In, Sb superlattice structures at 300 K wtth 7 ML thick Gal_,InrSb layers and 7,9,11,13 and 15 ML thick InAs layers, respectively.
wavelength can be achieved using considerably thinner layers. Since the heavy-hole band here comes much closer to the conduction band, the Increase in cut-off wavelength with layer thickness is much more sudden. Therefore, the change in cut-off wavelength caused by adding or removing just one monolayer is much larger than for the InAs/GaSb system. Hence, this system is more prone to be affected by fluctuations In layer thicknesses and alloy composition. In conclusIOn, it has been demonstrated how one can find InAs/Gal_, InrSb superlattice materials suitable for farInfrared detector apphcations by choosing an InAs/GaSb superlattice wah a high opucal matrix element and raising the cut-off wavelength by increasing the In amount in the Gal_rIn,.Sb alloy. It has also been established how, for a gIven superlatuce in the interesting region, it is possible to Increase the absorption wah almost unaffected cut-off wavelength by reducing the Gal_rIn,,Sb layer thIckness.
Acknowledgement - Financial support from NFR (SwedIsh Natural Science Research Council) and NUTEK (SwedIsh National Board for Industrial and Technical Development) is gratefully acknowledged.
L-__- - '_ _--'-_ _- ' -_ _- ' -_ _- ' - _ - - - '
SIll
I:!
14
Cut-off wavelength [pm]
Figure 5. Calculated opucal matrix element squared as a function of the cut-off wavelength of an interband transition from heavy holes to the conduction band for dIfferent series of InAs/GaSb superlattices at 300 K: equal thicknesses of InAs and GaSb, fixed InAs thickness of 11 ML (33 A), 14 ML (42 A) and 17 ML (51 A), respecttvely. The cut-off wavelength has a weaker dependence on GaSb layer thickness. The optical matrix element can thus be substanually Increased by simply making the GaSb layers thinner.
REFERENCES [ 1[ L. Esaki and R. Tsu, IBM J. Res. Dev. 14, 61, (1970). [2J G. A. Sai-Halasz, R. Tsu, and L. Esaki, Appl. Phys. Lett. 30, 651, (1977). [3] G. A. Sai-Halasz, L. Esaki, and W. A. Harrison, Phys Rev. B 18, 2812, (1978). 14J D. L. Smith, T. C. McGill, and J. N. Schulman, Appl Phys Lett. 43, 180, (1983). [5J G. C. Osbourn,!. Vac. SCI. Technol. B 2,176, (1984).
82 [11] R. H. Miles, D. H. Chow, J. N. Schulman, and T. C. McGill, App/. Phys. Lett. 57, 801, (1990). rl2J G. A. Sai-Halasz, L. L. Chang, J.-M. Welter, C.-A. Chang, and L. Esaki, Solid State Comm 27, 935, (1978). [13] D. H. Chow, R. H. Miles, 1. N. Schulman, D A. Collins, and T. C. MCGIll, Semicond. Sci. Techn()/ 6, C47, (1991). [14] L. L. Chang, G. A. Sai-Halasz, L. ESakl, and R. L Aggarwal,J. Vac Sci Techno/ 19,589, (1981) [15] G. 1. Gualtieri, G. P. Schwartz, R. G. Nuzzo, R. J. Malik, andJ. F. Walker,]. App/. Phys. 61,5337, (1987).
Superlattlces and Microstructures, Vol 76, No 7,7994 [6J D. K. Arch, G. Wicks, T. Tonaue, and J.-L. Staudenmann,1. App/ Phys. 58, 3933, (1985). [7] D. L. Smith and C. Mailhiot, 1. App/ Phys 62,2545, (1987). [8] E. O. Kane, in Semiconductors and Semimeta/s, edited by R. K. Wlilardson and A. C. Beer (AcademIc Press, New York, 1966), p. 75. r91 1. Y. Marzin, in Heterojunctions and Semiconductor Super/attices, edited by G. Allan (Spnnger Verlag, Berlin, 1985), p. 161. [!o] M. P. C. M. Knjn, Semicnnd. Sci Techn()/ 6, 27, (1991).
77
TAILORING InAs/Gal-.rIn.l'Sb SUPERLATTICES FOR LONG-WAVELENGTH IR APPLICATIONS J. H. Roslund and T. G. Andersson Department of Physics, Chalmers University of Technology and the University of Goteborg, S-412 96 Goteborg, Sweden (Received 22 August 1994)
A systematIc study has been carned out on the design of long-wavelength Infrared detector matenals from InAs/GaSb and InAs/GalnSb superlatllces with absorpl1on energies below 250 meV. i.e .\ > Slim. The influence from layer thicknesses and alloy composition on cut-off wavelength and opllcal matnx element has been analysed. A three-band envelopefunction model Including strain effects was used to calculate conduction-band electron and valence-band light-hole states, while an effective-mass approximation wa~ used to descnbe heavy-hole states. In order to achieve useful absorption coeffiCIents. the period of such superlattices must be less than typically 20 monolayers. Calculations revealed that the absorptIon can be increased Wllh almost unaffected cut-off wavelength by redUCing the GalnSb thIckness. The effects of including a small amount of In In the GaSb in order to reach longer wavelengths were studied. One conclusion is that although it makes longer wavelengths pOSSible. It also makes the cut-off wavelength much more sensitive to monolayer and compositIOn fluctuations.
1. Introduction In 1970, Esaki and Tsu introduced the concept of semiconductor superlattices [I], layered materials with a periodICIty larger than the atomic lattice. Due to the rapid development of molecular-beam epitaxy, it was possible to realise this concept in the closely lattice matched GaAs/AlyGal_,·As system during the 1970s. The quantum-mechanical effects In this system depend on the relative positions of the conduction and valence-band edges in the host materials. At the GaAs/AlrGal_rAs heterojunctlOn, the band gap of GaAs lies completely within the band gap of At·Gal_, As, which leads to confinement of both electrons and holes In the GaAs layers. Another band-edge line-up is proVIded by the Inl_yGayAs/GaSbl_yAsv system 12J with alloy compositIOns chosen so that.ll =0.9l8.r+0.082 for latl1ce match. As opposed to GaAs/At Gal_,·As. there is a possibility to get the conduction-band edge of Inl_rGaxAs lower In energy than the valence-band edge of GaSb1_yAs y, thus creating a ,y,tem where the two host band gaps do not overlap. It was predIcted [2] that such a band hne-up would gIVe nse to new and interesting properties. One of the most im portant effects is that electrons and holes no longer w!ll be confined in the ,arne matenal; a spatial separation of carriers will occur. As
0749-6036/94/050077 +06 S08 00/0
a result. this superlattice's band gap can be made smaller than that of any of the host matenals. A semiconductorsemimetal transition of the superlattice [31 occurs for layer thicknesses exceeding 100 A. in spite of the semIconductor character of the host materials. During the 1980s, the technological Interest in low bandgap materials increased. Since the smallest bulk band gap achievable in III-V compounds is 0.17 eV for InSb, corresponding to a wavelength of 7 11m for interband absorption. other materials had to be found in order to satisfy the need for long-wavelength IR detection. The most common material to date used in applications is the alloy Hg l _, Cd,.Te. Smith et al. [4] suggested that a HgTe/CdTe superlattice would have Improved properties, much owing to the higher effectIve mass in a superlatllce and to the fact that layer thicknesses rather than alloy composition would determine the cut-off wavelength. However, it was found that growth and processing of these II- VI materials are more dIfficult than for III-V compounds. As a III-V alternative Osbourn [5] sugge,ted InAs o ;Sb o ,JInAsl_ySby. Here, the strain from the latl1ce mIsmatch causes the superlattice band gap to shrink. Thi, superlattice is of type II and needs ratherthick layers to reach the deSlfed wavelength regIOn. As a consequence of thIS and the separation of carriers, absorption cannot be as
I(} 1994 AcademiC Press Limited
78
Superlatttces and Microstructures, Vol 76, No 7, 7994
high as m HgTe/CdTe superlattlces. Since the InAs/GaSb superlattIce suffers from the same difficulty 161. the GaSb mu,t be alloyed wtth InSb in order to raise the heavy-hole ,ubband [71 and thus mcrea,e the cut-off way elength Thl, superlatttce has been shown to exhlbtt absorption a, high as Hg 1 _. Cd, Te at the same ttme tt has a much 11lgher effectIve mas" which serves to mhibit unwanted dark current, and Auger recombinatlon rates m device, Furthermore. n can be made wtth more well-known growth and proce"ll1g techmques. giving better uniformtty for devices. For apphcatIon purposes, it is necessary to have a quantitatIve a, well as qualitative understandll1g of growth and matenal a"e,sment, how the different superlattJce parameter, (InAs thickness, Gal_, In, Sb thickness, and Gal_. In, Sb alloy composition) affect the ab,orptIon edge, the effecttve ma"e, and matnx elements. Only some speCific ,uperlattIces have been grown and charactensed expenmentally and theoretIcally. General trends cannot alway, be predicted from these and few examples are not suffiCIent from an applicatIon pOInt of view. In this study, the roles of the respective parameters are ll1vestIgated in a sy,tematlc way. The motivatIOn has been to map the available parameter space 111 search of matenal and layer combmatIon, ,unable for detector apphcatlons.
-/~I'hL --'-Ji [' I, A' .
I, /I =
-f{I'td'-
FI/,
)cP',
}; ['I,),·
~~r;,
E,,,
where /.' , 1:'1/, and I,',,, are the strain-dependent r-point po,Itlon, of the conduction band. light-hole and spin-orbit spltt-off valence band,. rnpecttvely. The coupling between the light-hole and spin-orblt split-off bands depends on a strall1-dependent energy shift M,',. The couphng, }', between the valence and conduction bands was chosen so that CI , = :>'III,,}" =19.2 eV, which IS consistent with the bulk effective ma,~es of electrons and light holes Allowed energy eigenvalues, f, are given by solutIOns to a disper'lOn equation 191
yvhere 'I J, the superlatllce wave number, 1/ and h are layer thlckne"es, Ii = II + h. For conductIOn-band electrom, light hole, and spll1-orba-band holes, k, and II, are given by
:11 L'e'" - ()( (EI/ .. , - (I(E,"., - (I -
The theoretIcal framework used IS a three-band envelopefunctlon model for the conduction band, hght-hole valence band and the spin-orbit split-off band, and an effective-ma" approximation for heavy holes. In short, the envelopefunctlon approximallon rehes on the assumption that the wave functlon ,. can be wntten as a :-dependent 11l1ear combmation of Bloch states _ f ,k~ r~ 'LU'lJr " ()},'n(_) I.(. r) -
"r.::f H'~' +I[~
J
2. Theory
(I)
where: IS the co-ordinate along the growth aXIS, 1/".1 IS the periodic part of the Bloch function correspondll1g to band /I at the r -point, r ~ IS the component of the positIon vector lying in the growth plane, and k-L IS the component of the electron wave vector in the same plane. The Hamiltonian can be denved from Kane's 8 x 8 Hamiltoman [81 includmg the twofold conduction, light-hole, heavy-hole and ,Pll1orbit split-off bands. For reasons of slmphctty, only the band structure for wave vectors along the growth aXIs ha, been calculated. Setting k-L = 0 for a (100) growth plane allows one to decouple the Hamiltonian into two equivalent -' x 3 matnees, repre,entmg the conductton, hght-hole and spll1-orbit ,pht-offbands, and two eqUivalent effectlve-ma'i'i equatIOns for the heavy holes. Includll1g ,mun and dlsregardll1g terms of order ho', one gets the three-band Hamtltoman 191
(2)
.'3( 1,' Al".:,
k, =
1
I',
(E
/;.,' 1
-
I
[~
~~F~,) - ,I
(I( L';" , _. f)
= 1)2(:>.( 1,.\"., + ~E" - () + EI/" -
(I'
(4)
whereas for heavy hole,
f h', = tJ:>,~,/;},.'(L'I'h" -
1
II, -
I)
(5)
-llIhh,
The strall1-dependent band edges have been calculated usll1g ll1terpolatlOn formulae from Ref. 10 modified to yield a correct valence-band offset. The superlattices have been as,umed to have a lattlce constant in the growth plane Imposed by a GaSb ,ubstrate The model gives a sufficiently accurate descnption of electromc states with energies close to the band gap and waye Yectors parallel to the growth axis. The most cntlcal a"umptlon I, that the penodic parts of the Bloch functions are a"umed to be the same In both matenals. However. consldenng the large uncertall1ty in valence-band discontinuity. which has a great Impact on the cut-off wavelength. a more ngorous model would only make calculations less manageable, wtthout guaranteeing better accuracy. It is well known that one-band model, can descnbe the GaAs/AI, Gal_, As system. Compansons have also shown that two-band models for the InAs/Gal _, In, Sb system agree wllh experimental data 1111.
79
SuperiattJces and Microstructures, Vol. 16, No 1, 1994
Energy reV]
0,8 0.6
GaSb
GaSb
OA 0.2
-
I-- 0.0
InAs
-
0.2 '-
~~
L
20
L
40
30
50
60
70
Layer thickness, L [A]
Figure 1. The bulk band-edge energies of InAs and GaSb at 300 K compared to the calculated band structure of InAs/GaSb superlattices with equal thickness, L, of InAs and GaSb layers. Energies are measured relative to the InAs conduction-band edge. The symbols HHI, LHI and EI denote, respeclively, the highest heavy-hole subband. the highest light-hole subband and the lowest conduction subband. 30
3. Results and Discussion
--n.lo
The superlattice band structure is a result of several parameters. In Fig. I is shown the band structure for a series of superlattices with equal thicknesses of InAs and GaSb layers. The resulting subband structure as a function of increasing layer thicknesses is referred to the energy-band edges of the bulk materials. When the barriers get thicker, the Widths of the subbands decrease. With broader wells, the distance between conduction- and valence-band derived subbands decreases, and new subbands are generated with increasing layer thicknesses. By considenng the wave number k, for conduction-band electrons and light holes, one concludes that states with energies in the GaSb band gap are confined to the InAs layers, whereas states with energies in the InAs band gap are confined to the GaSb. Similarly, the heavy holes are confined to the GaSb. Large layer thicknesses result in an inherently low degree of wave-function overlap for interband transitions, which diminishes even further as the individual layers get thicker. For the superlattice periods studied here the optical interband transllion is between heavy holes and the conduclion band. The wavelength of this transition is very sensitive to the energy difference, Ee" = Evl - Ed, i.e. between the top of the valence band of GaSb and the bottom of the conduction band of InAs. In Fig. 2 is shown the wavelength corresponding to this transition, calculated for a temperature of 4 K compared to some experimental results found in Refs 6, 12, 13 and 14. Usually Ec,v is taken to be In the vicimty of 0.15 e V. However, a better fit to expenmental data can be
-..,
25
:::
2: .::::
's:.: "" ;;
Ref
20
.... Rer.
•
15
~
:;:;'" Z C
d.ffcrenle
h
"
Ref. 12
,
. . . .,
....
•
10
o
eV difference
----015 r\
L-_~
10
___- L_ _ _- L____ 30
.til
~
____
511
~_-J
60
70
{."yer th,cklless, L [AJ
Figure 2. Cut-off wavelength for a heavy-hole to conductionband transition for InAs/GaSb superlattices. Experimental data are compared to calculations for 4 K with two different values of the energy difference between the InAs conduction-band edge and the GaSb valence-band edge.
achieved by choosing Ec,v = 0.10 eV. This yields a valenceband discontinuity well within the experimental accuracy of Ref. 16, and the wide range of other experimental data and theoretical calculations, It should be noted that the choice of f.',>", mainly affects superlattices with long cut-off wavelengths, I.e. thick layers, where the fiat-band assumption IS expected to become less accurate. The energy difference F,>, can therefore be considered an effective parameter. Control of the individual layer thicknesses seems to be
80
Superlatttces and Microstructures, Vol 16, No 1, 1994
a)
InAs fixed 11 ML = 33 1.0
I-Ir------.------,-------.------,------,
A
:---:-.
InA'I/GaSh egllal,
12
0.8
s:
--''"," ~ ::
'"
~
0.6 004
0.2
El
t:~
HHl 0.0 ·0.2
r--
LH1
~:J.#
~
20
30
- ,,'Jft:i
-10
.~
-
.,
50
60
GaSh layer thickness
b)
GaSb fixed 11 ML
70
'"::
50
611
70
Layer thickness rAJ
Figure 4. Calculated cut-off wavelength as a function of the variable layer thickness for heavy-hole to conduction-band transillons for five different series of InAs/GaSb superlattices at 300 K: one consisting of equal thicknesses of InAs and GaSb, one with GaSb thickness fixed to II ML (33 A) and three with InAs thickness fixed to 11 ML (33 A), 14 ML (42 A) and 17 ML (51 A), respectively.
= 33 A
0.8
--t"''"",
-10
30
rA.]
1.0
s:
20
0.6 0.4 0.2
~
0.0 -0.2
¥,,;'''r:
20
30
-10
50
InAs layer thickness
60
70
rA.]
Figure 3. a) Calculated band structure at 300 K as a function of GaSb layer thickness for InAs/GaSb superlattices with a fixed InAs layer thickness of 11 monolayers (11 ML = 33 A). b) The same as a function of InAs layer thickness for superlattices with a fixed GaSb layer thickness of II ML (33 A). The figure illustrates how the respective materials contnbute to the positions of the bands. a somewhat neglected area. In Fig. 3 the band structure's dependence on layer thickness (ef. Fig. 1) has been splIt up in its dependence on each of the constituent materials. The InAs layer confines the electrons and serves as a bamer for the holes, whereas the opposite applies for the GaSb layer. As stressed previously, the widening of a well leads to a larger number of subbands, the subbands become narrower and their levels approach the bottom of the well. As indicated, there is a weaker modification of the states confined in the other material. In Fig. 4 the cut-off wavelength is shown as a function of layer thickness for five superlattice series. In all cases an increase In layer thickness leads to an increased cut-off wavelength, but the increase is much more pronounced wllh changed InAs thickness. This is due to the fact that InAs
has a much lower effective mass than GaSb, and since the InAs layers serve as quantum wells for the conduction-band electrons, changing the InAs thickness changes the cut-off wavelength dramatically. This has the effect that the cut-off wavelength for many superlattices is rather insensitive to fluctuations in the GaSb layer thickness. In order to reach the technologically important far-infrared region (.\ > 8 Jim), layers have to be thicker than 50 A. An increased spatial separation between the carriers WIll strongly suppress optical absorption. The absorption coeffiCIent between two energy levels, EI and E 2 , is given by (6)
To Yield the total absorption coefficient, Eq. (6) has to be Integrated over the available k-space for the initial and final states. However, for a heavy-hole to conduction-band transition, the optical matrix element ;\f(E I • E 2 )
=
J~';(r)(e·
pl/'J!r))d3 r
(7)
IS in itself a good figure of merit. Furthermore, In the envelope-function approximation, this can be written as the product of a bulk matrix element and an envelope-function overlap integral,
(8) The latter was calculated for the f-point heavy-hole to conduction-band transition in the superlattices shown In Fig. 4 and the result is presented as a function of cut-off
8
Superlattices and Microstructures, Vol. 16, No 1, 1994 wavelength in Fig. 5. USing FIgs 4 and 5 in connecuon, one can deduce that it is very difficult to make an InAs/GaSb superlattice wtth a good absorption above JO pm. From the inchnation of the curves in Fig. 5, one sees that for moderate matrix elements the wavelength can be extended wtthout losing the absorption by increasing the InAs layer thickness. More important is that by making the GaSb layer thinner than the InAs the optical matrix element can be increased with only a mild drop in cut-off wavelength. The reson for this is that the GaSb acts as a barrier for the electrons, and by making this thinner, we increase the overlap with the hole envelope functions. This conclusion WIll also apply to the InAs/Gal_rInrSb superlattices. A good route to fi nd interesting materials is therefore to choose an InAs/GaSb superlattice with a high optical matrix element thIS way, and since increased In content primarily raises the heavy-hole level in the Gal_rIn".Sb, and the opucal matnx element hardly will be affected, the cut-off wavelength can be raised by alloying GaSb with InSb. In FIg. 6 IS shown how the cut-off wavelength increases with alloy composition. An interesting observation is that superlattices wtth thinner InAs layers have a less emphasised dependence on alloy composition, which can be used to reduce the sensitivtty to alloy fluctuations. One has to bear In mind that it is reasonable only to use alloys with .1' < 0.4 since for larger .r-values the lattice mismatch WIll be reheved through the formation of dislocations, whIch severely hampers the opucal properties. The precise condition for this to occur has yet to be established. It is interesting to compare the InAs/GaSb system wah the InAs/Gal_rInrSb system, where an eqUIvalent cut-off O.X r----r---,.---.....--,-----r----, In '\<; fixed 11 \lL
1
o0
c;aSh hxed 11 ~11: '
25r----,.-----r---.-----r----, 15 ML [n,hi ....., 20
l
.
.:::
Ga
I-x
In Sb thickness 7 ML, \.
!
113
15
~IL
::
~
,." <:l
"
,II MLI 10 W~i[~!
~
-
'" " I.J
17 >IL]
5
o ~-----'-------~----~------~----~ 0.1 0.2 11.3 0.5 o o~ Indium fraction. x
Figure 6. The increase in cut-off wavelength with alloy composition for InAs/Gal_, In, Sb superlattice structures at 300 K wtth 7 ML thick Gal_,InrSb layers and 7,9,11,13 and 15 ML thick InAs layers, respectively.
wavelength can be achieved using considerably thinner layers. Since the heavy-hole band here comes much closer to the conduction band, the Increase in cut-off wavelength with layer thickness is much more sudden. Therefore, the change in cut-off wavelength caused by adding or removing just one monolayer is much larger than for the InAs/GaSb system. Hence, this system is more prone to be affected by fluctuations In layer thicknesses and alloy composition. In conclusIOn, it has been demonstrated how one can find InAs/Gal_, InrSb superlattice materials suitable for farInfrared detector apphcations by choosing an InAs/GaSb superlattice wah a high opucal matrix element and raising the cut-off wavelength by increasing the In amount in the Gal_rIn,.Sb alloy. It has also been established how, for a gIven superlatuce in the interesting region, it is possible to Increase the absorption wah almost unaffected cut-off wavelength by reducing the Gal_rIn,,Sb layer thIckness.
Acknowledgement - Financial support from NFR (SwedIsh Natural Science Research Council) and NUTEK (SwedIsh National Board for Industrial and Technical Development) is gratefully acknowledged.
L-__- - '_ _--'-_ _- ' -_ _- ' -_ _- ' - _ - - - '
SIll
I:!
14
Cut-off wavelength [pm]
Figure 5. Calculated opucal matrix element squared as a function of the cut-off wavelength of an interband transition from heavy holes to the conduction band for dIfferent series of InAs/GaSb superlattices at 300 K: equal thicknesses of InAs and GaSb, fixed InAs thickness of 11 ML (33 A), 14 ML (42 A) and 17 ML (51 A), respecttvely. The cut-off wavelength has a weaker dependence on GaSb layer thickness. The optical matrix element can thus be substanually Increased by simply making the GaSb layers thinner.
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82 [11] R. H. Miles, D. H. Chow, J. N. Schulman, and T. C. McGill, App/. Phys. Lett. 57, 801, (1990). rl2J G. A. Sai-Halasz, L. L. Chang, J.-M. Welter, C.-A. Chang, and L. Esaki, Solid State Comm 27, 935, (1978). [13] D. H. Chow, R. H. Miles, 1. N. Schulman, D A. Collins, and T. C. MCGIll, Semicond. Sci. Techn()/ 6, C47, (1991). [14] L. L. Chang, G. A. Sai-Halasz, L. ESakl, and R. L Aggarwal,J. Vac Sci Techno/ 19,589, (1981) [15] G. 1. Gualtieri, G. P. Schwartz, R. G. Nuzzo, R. J. Malik, andJ. F. Walker,]. App/. Phys. 61,5337, (1987).
Superlattlces and Microstructures, Vol 76, No 7,7994 [6J D. K. Arch, G. Wicks, T. Tonaue, and J.-L. Staudenmann,1. App/ Phys. 58, 3933, (1985). [7] D. L. Smith and C. Mailhiot, 1. App/ Phys 62,2545, (1987). [8] E. O. Kane, in Semiconductors and Semimeta/s, edited by R. K. Wlilardson and A. C. Beer (AcademIc Press, New York, 1966), p. 75. r91 1. Y. Marzin, in Heterojunctions and Semiconductor Super/attices, edited by G. Allan (Spnnger Verlag, Berlin, 1985), p. 161. [!o] M. P. C. M. Knjn, Semicnnd. Sci Techn()/ 6, 27, (1991).