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Theoretical study of relaxation at the (110) Ge-GaAs interface
Solid State Communication, Vol. 25, pp. 225—227, 1978.
Pergamun Press.
Printed in Great Britain
THEORETICAL STUDY OF RELAXATION AT THE (110) Ge-GaAs INTERFACE* Warren E. Pickettt and Marvin L. Cohen Department of Physics, University of California and Materials and Molecular Research Division, Lawrence Berkeley Laboratory, Berkeley, California, 94720, USA (Received 27 October 1977 by A. A. Maradudin)
A simple relaxation of the Ge—GaAs (110) interface is studied theoretically using self—consistent psaudopotentials. The relaxation is one in which the distance between the Ge and GaAs atomic planes at the interface is increased by 20~. Band edge discontinuities are found to be insensitive to this type of relaxation in spite of large distortions of the electronic charge density.
The great technological importance of the this IF. The situation for the Ge—GaAs (110) IF properties of the semiconductor—semiconductor is not so clear. Experimentally there is no interface (IF), or heterojunction, has suggestion disruption at evidence this IF; however, 3 recently on an neither is of there any direct of recon— led to detailed theoretical studies’ atomic scale of idealizations of these systems. struction or faceting at the polar (100) and These studies involve electronic structure (111) IFs in Ge—GaAs, which as mentioned above determination using self—consistent pseudopoten— cannot assume the ideal structure. The theore— tials. The input to the calculation consists of tical result for the conduction band edge dis— (1) pseudopotentials which give an adequate des— continuity, ~Ec = 0.40 eV, is not in serious cription of the effective potential (due to conflict with experimental results but is nucleus and core electrons) felt by the valence slightly larger than the “most probable” experi— electrons, and (2) the atomic positions. The mental value of 0.2 eV. The theoretical utility of this pseudopotential approach has results3 for the character of the Ge—Ga and been well documented in bulk and surface Ge—As bonds across the ideal IF are more sug— studies. The relative atomic positions at the gestive of relaxation or reconstruction. The IF IF however are at present unknown, and it is bonds are significantly different in character this situation which will be address in this than the bulk bonds and suggest unbalanced paper, forces on the atoms at the IF. We confine our considerations here to semi— As a first step in studying the sensitivity conductors with similar (or identical) crystal of interfacial properties to variations in lattice types and negligible lattice mismatch, atomic positions we report in this communication Within these restrictions there exists an results for a simple relaxation of the Ge—GaAs obvious “ideal” interfacial structure in which (110) IF and compare with the corresponding one lattice connects simply with the other. results for the ideal IF.2’3 The atomic posi— Calculations so far have been confined to such tions near the IF for the ideal structure are systems, e.g. Ge—GaAs, AlAs—GaAs, GaAs—ZnSe. shown in Fig. 1. The relaxation we consider is The question arises naturally whether the ideal a simple one in which the (110) atomic layers of structure is a realistic one. Baraff et al.1 Ge and GaAs are separated by a distance 20Z have demonstrated quite generally that, for the greater than the (110) layer~in either bulk polar IFs of systems such as Ge—GaAs, the ideal (lattice parameter of 5.658 A is assumed for structure must give rise to a metallic IF (i.e. both bulk materials). Thus the volume of the partially filled bands). Their argument more layer labelled “Interface Layer” in Fig. 1 is specifically implies metallic interfaces for the 2OZ greater in the relaxed geometry than in the ideal polar IFs of systems of the type ANB8—N — ideal structure. A relaxation of this magnitude AMB8—M with M ~ N. Reconstruction of the ideal is presumably greater than one which may IF, and in some cases perhaps a quadrupling of actually occur at this IF. An overestimate of the IF unit cell, is necessary to produce a the relaxation has been made deliberately to semiconducting IF as is found experimentally, allow a clearer identification of the sensitiv— Non—polar IFs with the ideal structure are ity of the various IF properties. not restricted in any way by such general argu— As in previous calculations the super— ments. In particular, calculations on the AlAs— lattice geometry has been chosen. The unit cell GaAs (110) IF agree well with experiment where for each geometry we consider contains five comparison is possible (for the band edge dis— layers of Ge and five layers of GaAs, and thus continuities) and give stable, bulk—like bonds two IFs. The ionic pseudopotentials are those across the IF. There is no indication at used previously.2 The Hamiltonian is construc— present, either experimentally or theoretically, ted in a plane wave representation, then that any reconstruction or relaxation occurs at diagonalized to yield eigenvalues and wave— * Supported in part by the National Science Foundation Grant DMR76—20647. t Supported by an IBM Fellowship. 225
226
THEORETICAL STUDY OF RELAXATION AT THE (110) Ge—GaAs INTERFACE
Vol. 25, No. 4
Relaxed to generate a screening potential,
which is used
to replace the previous screening ispotential the Hamiltonian. This procedure repeated in until convergence of both potential and eigen— values is obtained. functions. The resulting Details charge of the density method, is used Including convergence criteria, are published elsewhere. We first verified that the 5 layer—5 layer unit cell resulted in essentially the same potential and charge density for the ideal
GaAs
Ge
______________________________________________
Ge-GaAs (110) Interface OGa
•Ge
•As
First GaAs
—1
Ideal
-
Layer ,#~
~
~jf
parallel to the interface, for (a) the Fig. 2. The self—consistent potentials, averaged
~ L ~f y
ace First Ge Layer
relaxed geometry and(110) (b)parallel the half ideal geom— etry. interface. unit tions cell The of atomic small is In pictured. each planes arrows case denote one theto ofposi— the the Relaxed Total Ge-GaAs Charge Density Interface
Fig. 1. Atulaic e~~s near the ideal Ge—GaAs (110) interface. Heavy solid lines denote bonds, except for the bonds across the interface which are denoted by heavy bro-ken lines. The chains ABAB and CDCD are the two independent bonding chains perpendicular to the IF.
.6
____ ______
Ge
geometry as the 9 layer—9 layer unit cell for which results have previously been reported. well separated that their mutual interaction is small. ‘Baraff et al.’ have suggested that as few 2—3 atomic eachsufficiently side of the IF This as verifies that layers the IFsonare is sufficient to give a realistic potential and charge density, but our results below indicate that care must be exercised in this respect when relaxation or reconstruction is investigated. The total self—consistent potential averaged parallel to the IF, V(z), is shown in Fig. 2 for both the ideal and relaxed geo— metries. The large peak in V(z) at the IF for the relaxed geometry is a result of the weaken— ing of the attractive ionic potential in this region due to the relaxation. The important quantity to be identified fromV(z) is the posi— tion of the average notential in Ge VGe relative to that of GaAs, VGaA5. The difference ~V = VCe — VGaA5 for the two geometries gives direct— ly the shift in band edge discontinuities due to the relaxation. Using the values AVideal = 0.27 eV for the ideal geometry and AVrelax =
~—~9
Gn
Ga
265
26
~
Fig. 3. A contour plot of the total self—cons— istent valence charge density of Ge—GaAs ia the relaxed geometry. The ABAB (a) and CDCD (b) chains perpemdicular to the interface (see Fig. 1) lie in the planes shown. The average charge density is normalized to unity, with successive contours separated by 0.2 units. The heavy dashed lines denote the stretched interface bonds. 0.13 ±0.10 eV for the relaxed geometry gives a shift in the band edge discontinuities of 0.14 ±0.10 eV. The relatively large uncertain— ty in AVrelax is a result of interaction of
Vol.
25, No. 4
ThEORETICAL STUDY OF RELAXATION AT THE (110) Ge—GaAs INTERFACE
neighboring IFs in the 5 layer—5 layer super— lattice geometry. This indicates that a 5 layer—5 layer cell will be the smallest unit cell useful for studying relaxation at an IF. The shift in the band edges due to this simple relaxation is in the direction such that the conduction band discontinuity at the IF approaches the difference in electron affinities Ax, which would be expected of any reasonable calculation. The experimental value” of Ax is 0.06 eV. Specifically, our results are AEc = 0.40 eV for the ideal (110) IF and AEc = 0.26 ± 0.10 eV for the relaxed geometry. This is to be compared to the “most probable” experimental value of AEc = 0.2 eV mentioned above. Since any relaxation of this simple type which might actually occur would likely be only a fraction of the 20% value chosen for this calculation, we conclude that the band edge discontinuities are insensitive to this type of relaxation. This insensitivity of the band edge disconti— nuities, or equivalently the interfacial potential dipole, makes it impossible to determine whether such a relaxation actually occurs.
As further evidence that this relaxation is unrealistically large the self—consistent charge density in the two atomic planes perpendicular ~ to the IF is presented in Fig. 3. The Ge—Ga bond charge had only a very weak maximum between the two atoms, and the Ge—As bond has no maximum at all with the valence charge density increas— ing monotonically from Ge to As. Such bonds suggest an unstable system. In spite of the gross distortion of these bonds by this “relaxa— tion,” integration of the charge density over the bonding regions suggest that primarily only stretching of each bond has occurred, with no charge transfer between Ge—Ga bond and the Ge—As bond resulting from the bond stretching. As for the ideal geometry we find that the Ge—Ga bond contains 1.89 electrons and the Ge—As bond contains 2.11 electrons. Acknowledgement —— The authors wish to thank Dr. J. C. Phillips for discussions on this topic.
REFERENCES 1. 2. 3. 4.
227
BARAFF, G.A., APPELBAUM, J.A. and HAHANN, D.R., Phys. Rev. Lett. 38, 237 (1977); J. Vac. Sci. Technol. 14, 999 (1977). PIGKETT, W.E., LOUIE, S.G. and COHEN, M.L., Phys. Rev. Lett. 39, 109 (1977). PICKETT, W.E., LOUIE, S.G. and COHEN, M.L., Phys. Rev. B (in press). GOBELI, G.W. and ALLEN, F.G., Phys. Rev. 137, A245 (1965).
Pergamun Press.
Printed in Great Britain
THEORETICAL STUDY OF RELAXATION AT THE (110) Ge-GaAs INTERFACE* Warren E. Pickettt and Marvin L. Cohen Department of Physics, University of California and Materials and Molecular Research Division, Lawrence Berkeley Laboratory, Berkeley, California, 94720, USA (Received 27 October 1977 by A. A. Maradudin)
A simple relaxation of the Ge—GaAs (110) interface is studied theoretically using self—consistent psaudopotentials. The relaxation is one in which the distance between the Ge and GaAs atomic planes at the interface is increased by 20~. Band edge discontinuities are found to be insensitive to this type of relaxation in spite of large distortions of the electronic charge density.
The great technological importance of the this IF. The situation for the Ge—GaAs (110) IF properties of the semiconductor—semiconductor is not so clear. Experimentally there is no interface (IF), or heterojunction, has suggestion disruption at evidence this IF; however, 3 recently on an neither is of there any direct of recon— led to detailed theoretical studies’ atomic scale of idealizations of these systems. struction or faceting at the polar (100) and These studies involve electronic structure (111) IFs in Ge—GaAs, which as mentioned above determination using self—consistent pseudopoten— cannot assume the ideal structure. The theore— tials. The input to the calculation consists of tical result for the conduction band edge dis— (1) pseudopotentials which give an adequate des— continuity, ~Ec = 0.40 eV, is not in serious cription of the effective potential (due to conflict with experimental results but is nucleus and core electrons) felt by the valence slightly larger than the “most probable” experi— electrons, and (2) the atomic positions. The mental value of 0.2 eV. The theoretical utility of this pseudopotential approach has results3 for the character of the Ge—Ga and been well documented in bulk and surface Ge—As bonds across the ideal IF are more sug— studies. The relative atomic positions at the gestive of relaxation or reconstruction. The IF IF however are at present unknown, and it is bonds are significantly different in character this situation which will be address in this than the bulk bonds and suggest unbalanced paper, forces on the atoms at the IF. We confine our considerations here to semi— As a first step in studying the sensitivity conductors with similar (or identical) crystal of interfacial properties to variations in lattice types and negligible lattice mismatch, atomic positions we report in this communication Within these restrictions there exists an results for a simple relaxation of the Ge—GaAs obvious “ideal” interfacial structure in which (110) IF and compare with the corresponding one lattice connects simply with the other. results for the ideal IF.2’3 The atomic posi— Calculations so far have been confined to such tions near the IF for the ideal structure are systems, e.g. Ge—GaAs, AlAs—GaAs, GaAs—ZnSe. shown in Fig. 1. The relaxation we consider is The question arises naturally whether the ideal a simple one in which the (110) atomic layers of structure is a realistic one. Baraff et al.1 Ge and GaAs are separated by a distance 20Z have demonstrated quite generally that, for the greater than the (110) layer~in either bulk polar IFs of systems such as Ge—GaAs, the ideal (lattice parameter of 5.658 A is assumed for structure must give rise to a metallic IF (i.e. both bulk materials). Thus the volume of the partially filled bands). Their argument more layer labelled “Interface Layer” in Fig. 1 is specifically implies metallic interfaces for the 2OZ greater in the relaxed geometry than in the ideal polar IFs of systems of the type ANB8—N — ideal structure. A relaxation of this magnitude AMB8—M with M ~ N. Reconstruction of the ideal is presumably greater than one which may IF, and in some cases perhaps a quadrupling of actually occur at this IF. An overestimate of the IF unit cell, is necessary to produce a the relaxation has been made deliberately to semiconducting IF as is found experimentally, allow a clearer identification of the sensitiv— Non—polar IFs with the ideal structure are ity of the various IF properties. not restricted in any way by such general argu— As in previous calculations the super— ments. In particular, calculations on the AlAs— lattice geometry has been chosen. The unit cell GaAs (110) IF agree well with experiment where for each geometry we consider contains five comparison is possible (for the band edge dis— layers of Ge and five layers of GaAs, and thus continuities) and give stable, bulk—like bonds two IFs. The ionic pseudopotentials are those across the IF. There is no indication at used previously.2 The Hamiltonian is construc— present, either experimentally or theoretically, ted in a plane wave representation, then that any reconstruction or relaxation occurs at diagonalized to yield eigenvalues and wave— * Supported in part by the National Science Foundation Grant DMR76—20647. t Supported by an IBM Fellowship. 225
226
THEORETICAL STUDY OF RELAXATION AT THE (110) Ge—GaAs INTERFACE
Vol. 25, No. 4
Relaxed to generate a screening potential,
which is used
to replace the previous screening ispotential the Hamiltonian. This procedure repeated in until convergence of both potential and eigen— values is obtained. functions. The resulting Details charge of the density method, is used Including convergence criteria, are published elsewhere. We first verified that the 5 layer—5 layer unit cell resulted in essentially the same potential and charge density for the ideal
GaAs
Ge
______________________________________________
Ge-GaAs (110) Interface OGa
•Ge
•As
First GaAs
—1
Ideal
-
Layer ,#~
~
~jf
parallel to the interface, for (a) the Fig. 2. The self—consistent potentials, averaged
~ L ~f y
ace First Ge Layer
relaxed geometry and(110) (b)parallel the half ideal geom— etry. interface. unit tions cell The of atomic small is In pictured. each planes arrows case denote one theto ofposi— the the Relaxed Total Ge-GaAs Charge Density Interface
Fig. 1. Atulaic e~~s near the ideal Ge—GaAs (110) interface. Heavy solid lines denote bonds, except for the bonds across the interface which are denoted by heavy bro-ken lines. The chains ABAB and CDCD are the two independent bonding chains perpendicular to the IF.
.6
____ ______
Ge
geometry as the 9 layer—9 layer unit cell for which results have previously been reported. well separated that their mutual interaction is small. ‘Baraff et al.’ have suggested that as few 2—3 atomic eachsufficiently side of the IF This as verifies that layers the IFsonare is sufficient to give a realistic potential and charge density, but our results below indicate that care must be exercised in this respect when relaxation or reconstruction is investigated. The total self—consistent potential averaged parallel to the IF, V(z), is shown in Fig. 2 for both the ideal and relaxed geo— metries. The large peak in V(z) at the IF for the relaxed geometry is a result of the weaken— ing of the attractive ionic potential in this region due to the relaxation. The important quantity to be identified fromV(z) is the posi— tion of the average notential in Ge VGe relative to that of GaAs, VGaA5. The difference ~V = VCe — VGaA5 for the two geometries gives direct— ly the shift in band edge discontinuities due to the relaxation. Using the values AVideal = 0.27 eV for the ideal geometry and AVrelax =
~—~9
Gn
Ga
265
26
~
Fig. 3. A contour plot of the total self—cons— istent valence charge density of Ge—GaAs ia the relaxed geometry. The ABAB (a) and CDCD (b) chains perpemdicular to the interface (see Fig. 1) lie in the planes shown. The average charge density is normalized to unity, with successive contours separated by 0.2 units. The heavy dashed lines denote the stretched interface bonds. 0.13 ±0.10 eV for the relaxed geometry gives a shift in the band edge discontinuities of 0.14 ±0.10 eV. The relatively large uncertain— ty in AVrelax is a result of interaction of
Vol.
25, No. 4
ThEORETICAL STUDY OF RELAXATION AT THE (110) Ge—GaAs INTERFACE
neighboring IFs in the 5 layer—5 layer super— lattice geometry. This indicates that a 5 layer—5 layer cell will be the smallest unit cell useful for studying relaxation at an IF. The shift in the band edges due to this simple relaxation is in the direction such that the conduction band discontinuity at the IF approaches the difference in electron affinities Ax, which would be expected of any reasonable calculation. The experimental value” of Ax is 0.06 eV. Specifically, our results are AEc = 0.40 eV for the ideal (110) IF and AEc = 0.26 ± 0.10 eV for the relaxed geometry. This is to be compared to the “most probable” experimental value of AEc = 0.2 eV mentioned above. Since any relaxation of this simple type which might actually occur would likely be only a fraction of the 20% value chosen for this calculation, we conclude that the band edge discontinuities are insensitive to this type of relaxation. This insensitivity of the band edge disconti— nuities, or equivalently the interfacial potential dipole, makes it impossible to determine whether such a relaxation actually occurs.
As further evidence that this relaxation is unrealistically large the self—consistent charge density in the two atomic planes perpendicular ~ to the IF is presented in Fig. 3. The Ge—Ga bond charge had only a very weak maximum between the two atoms, and the Ge—As bond has no maximum at all with the valence charge density increas— ing monotonically from Ge to As. Such bonds suggest an unstable system. In spite of the gross distortion of these bonds by this “relaxa— tion,” integration of the charge density over the bonding regions suggest that primarily only stretching of each bond has occurred, with no charge transfer between Ge—Ga bond and the Ge—As bond resulting from the bond stretching. As for the ideal geometry we find that the Ge—Ga bond contains 1.89 electrons and the Ge—As bond contains 2.11 electrons. Acknowledgement —— The authors wish to thank Dr. J. C. Phillips for discussions on this topic.
REFERENCES 1. 2. 3. 4.
227
BARAFF, G.A., APPELBAUM, J.A. and HAHANN, D.R., Phys. Rev. Lett. 38, 237 (1977); J. Vac. Sci. Technol. 14, 999 (1977). PIGKETT, W.E., LOUIE, S.G. and COHEN, M.L., Phys. Rev. Lett. 39, 109 (1977). PICKETT, W.E., LOUIE, S.G. and COHEN, M.L., Phys. Rev. B (in press). GOBELI, G.W. and ALLEN, F.G., Phys. Rev. 137, A245 (1965).