Superlattices and Microstructures, Vol. 9, NO. 3, 199 1 377

RELATION BETWEEN BAND GAP SHRINKAGE AND OVERLAP OF INTERFACE STATES IN POLAR (GaAs),/(Ge& [OOl] SUPERLATTICE

T. Saito and T. lkoma Institute of Industrial Science, University of Tokyo,

7-22-l Roppongi, Minato-ku, Tokyo 106, Japan

(Received 13 August 1990)

We have calculated the band sauctures of (GaAs),,/(Ge& [OOl] superlattices @L’s) with n=lO-1 to understand the origin of the band gap shrinkage in the (GaAs)@q)l [OOl] SL. The calculations are performed by a semiempirical tight-binding method with the sp%* basis. We have studied two kinds of states in the SL’s. The first one corre- sponds to the states at the conduction and valence band edges, which are confined two- dimensionally in the Ge layers. The energy gap between &confined states increases (1.07eV at the X-point for n=2) with decreasing SL ueriod. n, due to the auantum con- finement effect. fie second one corresponds t6the herfade &es consti&ng the two interface bands in the gau. The interface states on the lower and UDW interface bands are located at the Ga-Ge &h As-Ge interfaces, respectively. By de&&sing n, a sudden de- crease in the band gan (Ea=O.l6eV at the R-noin is obtained at n=l. The origin of the band gap shrinkageys’r&aFed to the fact that t6e o&lap of the interface states b&mes so large that they can combine as the band states.

1. Introduction

GaAs/Ge [OOl] heterojunctions and superlattices (SL’s) are studied experimentally1 as a prototype of junctions be- tween polar/nonpolar semiconductors in a technologically important crystalline orientation. Even in an abrupt GaAs/Ge single heterojunction, two kinds of interface are possible; the As-Ge interface in (...GaAsGaw&GeGeGe ...) case and the Ga-Ge interface in (...AsGaAs&/&Ge GeGe...) case. For both kinds of interfaces, two signifi- cant features were predicted by theorists. One is a high density of interface states originated from Ga-Ge and As- Ge bonds which do not exist either in GaAs or in Ge2.3. The other is an electric field in the iunctions deduced fi=om a simple electrostatic consideration4. In (GaAs),,/(Ge& [OOl] X’s, both kinds of interfaces are included in one structure (~~~GaAsGa&&Cie~~~Ge&/~AsGaAs~~~). The above-mentioned two features of GaAs/Ge interfaces are considered to effect the band structurei of (GaAs),/ (Gez),, [OOl] SL’s.

Recently, Ohno has reported that a (GaAs)l/(Ge$l 10011 SL exhibits a metallic character because of disaoDear- &e-of the band gap. However, it is not understooh’how the band gag shrinks with decreasing SL ueriod. n, of (GaAs),/(&&), [OOl] SL’s and how The two features of GaAs/Ge interfaces effect the band gap shrinkage.

In this study, we calculate band structures of (GaAs),J(Ge& [OOl] SL’s with n=lO-1 to understand the origin of the band gap shrinkage in the SL with n=l. We focus our attention on the role of the interface states which is the first of the two features of GaAs/Ge interfaces. The

calculations are performed by a semi-empirical tight- binding method with sp3s* basis without self-consistent procedures. Such methods were successful for obtaining wavefunctions of interface states at polar [OOl] heterojunc- don@. To study the role of electric field which is the set- ond feature, a self-consistent calculation is usually needed7*. We leave a self-consistent calculation as a final goal of our study.

2. Calculations

We employ a nearest-neighbor tight-binding method with a basis of five orbitals (s, pX. pv, p,, s*) per atom. -.- -, -- The tight-binding parameters describing the bandstructures of bulk GaAs and Ge are taken from the values by Voal et al ‘0. Spin-orbit coupling is not include; in-our calculation. For the interatomic matrix elements of the As- Ge and Ga-Ge pairs, we adopt the arithmetic averages of the corresponding matrix elements of bulk GaAs and Ge2.6. The valence band discontinuity between GaAs and Ge, AE, [=E,(Ge)-E,(GaAs)], is assumed to be 0.41eV which is the theoretical value by Harrison*i. The value is close to 0.49eV, the average value of experimental AE,‘s reported by various researcherslz.

3. Band Structure of (GaAs)lol(Gez)lo 10011 Superlattice

The band structure of a (GaAs)Id(Ge$lo [OOl] SL is shown in Fig. 1. The inset shows the Brillouin zone (BZ) of (GaAs),/ (Ge2)” [OOl] SL. The irreducible wedge is

0749-6036/91/030377 +05 $02.00/O 0 1991 Academic Press Limited

378 Superlattices and Microstructures, Vol. 9, No. 3, 1991

GAP ‘I’ I’ iiril

62;...+ X’*

WAVE VECTOR

Fig.1 Band structure of a (GaAs)ld(Ge2)t0 [OOl] super- lattice calculated by the fist neighbor sp%* tight-binding method. The zero of energy corresponds to the valence band maximum of GaAs. The inset shows the Brillouin zone of the (GaAs)n/(Ge$, [OOl] superlattice.

indicated with the labels of the eight symmetry points. The axis from r to X is normal to the projection of the Ga-Ge and As-Ge bonds on the interface planes. The Xz- and Xy- points of fee BZ are folded into the r- and M-points in the SL BZ, respectively. The L-point is folded into the R- and X-points when n is odd and even, respectively.

The valence and conduction bands of the SL consist of many crowded bands formed by the zone folding effect. The minimum energy state of the conduction band edge is the XC state at the X-point. Hence the SL has an indirect band gap.

Additionally, we have found two interface bands, il and i2, lying in the lower and upper regions of the gap, respec- tively. The lower band il is an occupied one which is pulled up from the valence bands, the upper band i2 an empty one pulled down from the conduction bands. In other words, the highest occupied band and the lowest empty band correspond to the lower and upper interface bands, respectively.

The charge densities ]Yk I2 of some of the band edge states (rV, Yc, Xc and Me) and the interface states (r”, ri2, Xi2 and Mi2) are shown in Fig.2 (a) and (b). All of the band edge states are confined two dimensionally in the Ge “well” layers. Chr the other hand, the charge density of the interface state fi* has a peak at the Ga-Ge interface while the charge densities of the Xi2 and p2 states have peaks at the As-Ge interface.

This result for the interface states found in the SL is qualitatively consistent with that obtained by Pollmann and Pantelides in the single heterojunctions. They reported that the interface band appears just above the valence band edge at the Ga-Ge interface and below the conduction band edge at the As-Ge interface. However, the dispersion of the interface band at the As-Ge interface is calculated much more accurately in the present study than in the study of Pollmann and Pantelides, because they used a nearest neighbor sp3 method which does not produce an accurate dispersion of the lowest conduction band.

Superlattices and Microstructures, Vol. 9, No. 3, 7997 379

(b)

GalGe Interface

As!Ge interface

G In’

4s

e tee

Fig.2 Charge densities in a (GaAs)ld(Ge2)10 [OOl] super- lattice. (a) The occupied states; the interface state (r”) and the confined state (I”). (b) The empty states; the interface states (Xi2, ri2 and Mi2) and the confined states (XC, l-c and MC). The panels are stacked up in order of energy. The dashed lines indicate charge densities on the Ga and odd-numbered Ge layers, the solid lines those on the As and even-numbered Ge layers.

4. Change in Band Structure with Decreasing Superlattice Period

Next we study how the band structure changes with de- creasing SL period, n, from 10 to 1. We show the energies of the interface and confined states in (GaAs),,/(Gq), [OOl] SL with n=lO-1 in Fig.3. In fact, with n approaching 1, it is not meaningful very much to distinguish the two kinds of states because the number of the Ga-Ge and As-Ge “interface bonds” approaches that of the Ge-Ge and Ga-As “bulk bonds”. Nevertheless, to see the trend systemati- cally, we formally label the highest occupied state and the lowest empty state at each symmetry point as the interface states, the next highest and next lowest states as the con- fined states, even in case of n=l.

As shown in the figure, the energies of the confined states in the conduction band edge shift up and those in the valence band shift down with decreasing SL period, n, due to the quantum confinement effect. Consequently the energy gap between the confined states increases up to 1.07eV at the X-point for n=2.

The energies of the interface states do not shift very much for nt2. However, at n=l, the energies of Ri2 and

ri2 decrease, the energy of Xi2 increases suddenly. The states labeled “interface states” at n=l are virtually the band edge states because the number of the interface bonds and

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Fig.3 The energies of the interface states and confined states in (GaAs)d (Ge& [OOl] superlattice with n=lO-1.

Superlattlces and Microstructures, Vol. 9, No. 3, 7 99 7

that of the bulk bonds are the same. The energy gap should be defined as the gap between the empty and occupied “interface states” resulting in Eg=O.l6eV at the R-point. The band gap of the (GaAs),/ (Gez), [OOl] SL shrinks suddenly at n=l

5. Origin of Band Gap Shrinkage in (GaAs)t/(Ge2)t [OOl] Superlattice

Figure 4 shows the band structure of a (GaAs)l/(Ge2)t [OOl] SL. The band gap shrinks at the R- and r-points as already shown in Fig.3. The disagreement of the value of the band gap (Eg=O.l6eV) with that reported by Ohno (Eg=OeV) is due to the approximations used in the tight- binding method and the neglect of lattice distortion effect. However, the simple sp%* tight-binding method proves to be useful for predicting the band gap shrinkage.

The conduction band edge at the M- and X,-points do not contribute to the band gap shrinkage. This corresponds to the absence of the interface bands at the same two points in (GaAs)td(Gez)to [OOl] SL. The band gap shrinkage has relation to the presence of the interface bands. To understand the origin of the band gap shrinkage more clearly, we plot the charge density of the Ri2 state at n=6, 2 and 1 in Fig.5. The Ri2 is the conduction band minimum at

WAVE VECTOR

r

Fig.4 Band structure of a (GaAs)t/(Gez)t [OOll super- lattice calculated by the first neighbor sp3s* tight-binding method.

Superlattices and Microstructures, Vol. 9, No. 3, 1991 381

50 Ri2

45. n=6

40.

50

45

40

35

30

25

20

15

10

Fig.5 Charge densities of the empty interface state at the R-point, I@, in (GaAs)n/ (Ge& [OOl] superlattice with n=6,2 and 1. The charge densities on the Ga, As and Ge layers are all indicated by the solid line.

n=l. At n=6 and 2, the Ri2 states at the As-Ge interfaces are still isolated from each other. They overlap and connect each other at n=l . The origin of the band gap shrinkage is related to the fact that the overlap of the interface states be- comes so large that they can combine as the band states.

We have also found band gap shrinkage in Ga/Ge/Asfie [OOl] SLt3 and (GaAs)t/(Ge2)tn [OOll SL (or Ga/As/Ge SL)l4 by the same tight-binding parameters. These results will be reported elsewhere.

6. Conclusions

We have calculated the band structures of (GaAs),,/(Gez), [OOl] SL with n=lO-1 by a semi-empirical tight-binding method with the sp%* basis. We have stud- ied two kinds of states in the SL’s. The first one corre- sponds to the states at the conduction and valence band edges, which are confined two-dimensionally in the Ge lay- ers. The lowest confined state in the conduction band edge for nZ2 is located at the X-point (even n) or R-point (odd n) which corresponds to the fee L-point. The energy gap between the confined states increases (1.07eV at the X- point for n=2) with decreasing SL period, n, due to the quantum confinement effect. The second one corresponds to the interface states constituting the two interface bands in the gap. The interface states on the lower interface band are located at the Ga-Ge interface while the states on the upper interface band located at the As-Ge interface. Bv decreas- ing SL period, n, a sudden decrease in the band gap (Ee=O.l6eV at the R-noint) is obtained at n=l. The origin ofyhe band gap shri;lkage is related to the fact that Fhe overlap of the interface states becomes so large that they can combine as the band states.

i2- =1

I Acknowledgments - We are grateful to Dr. K. Hirakawa for valuable discussions. We also thank Dr. T. J. Thornton for critically reading the manuscript. This work is partly supported by the Grant-in-Aid from the Ministry of Education, Science and Culture, Japan and also by the Industry-University Joint Research Program “Mesoscopic Electronics.”

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