RECENT ADVANCES IN THE THEORY OF PHONONS IN SEMICONDUCTORS

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RECENT ADVANCES IN THE THEORY OF PHONONS IN SEMICONDUCTORS

Roland Martin

To cite this version:

Roland Martin. RECENT ADVANCES IN THE THEORY OF PHONONS IN SEMICONDUCTORS.

Journal de Physique Colloques, 1981, 42 (C6), pp.C6-617-C6-624. �10.1051/jphyscol:19816180�. �jpa-

00221263�

JOURNAL DE PHYSIQUE

Colloque C6, suppl6ment au no12, Tome 42, d6cembre 1981 page C6-6 17

RECENT ADVANCES I N THE THEORY O F PHONONS I N SEMICONDUCTORS

R.M. Martin

Xerox Palo Alto Research Centers, 3333 Coyote Hi22 Road, Pa20 Alto, CA 94304, U. S. A.

Universite' P. e t M. Curie, Dept. Recherches Physiques, 4 place Jussieu, 75230 Paris, France.

Abstract

-

Calculations of structural energies of solids have recently reached a new stage,

in

which it has been shown to be feasible

to

calculate directly crystal structures of minimum energy and distortion energies that determine harmonic phonon eigenmodes and anharmonic terms. In this paper are discussed 1) calculational methods, 2) recent results, which have been shown by several groups to give accurate results for

Si, Ge,

GaAs and Se with no adjustable parameters, 3) the corresponding electronic charge densities, which show graphically the bonding and suggest interpretations for the

origins

of the atomic forces. Comparisons of direct computations of total energy differences with second-order perturbation calculations of harmonic coefficients shows the advantages and disadvantages of each method and how each

can

complement the other.

I. Introduction

The structural properties of solids consist of the equilibrium structure and the dynamics of the motion of the atoms around their equilibrium positions. The subject of this conference is indicative of the division of this field of study into separate areas-both experimentally and theoretically-with the interest here focused upon dynamics. Within this division there are many subdivisions into harmonic phonons, anharmonic interactions, coupling of phonons and photons, etc. However, it has been shown re~entlyl-~ that all of these structure-related properties can be considered theoretically in a unified manner so that equilibrium positions, harmonic and anharmonic forces, effective charges, etc., can be calculated directly from the underlying Hanliltonian of the electron-ion system. The methods that have been devised are particularly well-suited for semiconductors, and complete calculations on Si,2-4 Ge,3s5 GaAs6.' and Sea have shown that very good agreement with experiment (and confident predictions of quantities not known experimentally) can be achieved with ab initio calculations having no adjustable parameters. In this paper

I

will attempt to present a brief review of this work, oriented to emphasize the dynarnical aspects and to provide relations among m a y theoretical papers presented at this c o n f e r e n ~ e . ~ . ~ . l ~ l ~

Struct~lral energies and forces are determined by the total energy of the system of ions and electrons. Within the adiabatic approximation, the electrons respond instantaneously on the time scale of atomic motions and the total energy can be considered a function only of the positions of the ions, which we denote schematically by R. The total energy

can

be ~ r i t t e n " ~ J ~

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19816180

JOURNAL DE PHYSIQUE

Eta(R)= &,(R)

+ E,(R), (1)

where

ql

is the direct ion-ion interaction energy, and

E,,

is the total energy of the electrons moving in the potential field of the ions, including the quantum mechanical kinetic, exchange, and correlation energies. The equilibrium structure is determined by minimization of the total energy, a necessary condition for which is that the force on every atom is zero at its equilibrium position. This is the first step in any ab initio calculation of structural energies.

Distortion energies that determine the stability of the structure, phonon frequencies, etc., are given by EtO,<R) for positions R away from equlibrium. There are two primary ways to proceed at this point: 1) the "direct" method,lJ in which

EL&)

is calculated directly for the distorted crystal and 2) the perturbation method,14 in which the energy is expanded to second order in

AR

to determine harmonic phonon energies. The latter approach, also termed the dielectric function formulation of lattice dynamics,14 has a long history15 and has been successfully applied in free-electron-like metals,15 however, crystals with covalent bonding are much more difficult and explicit evaluation of these formulas has been reported in only a few instance^?^^,^^,^^ The advantage of the perturbation method is that any harmonic phonon at any wavevector q can be treated. The disadvantage is that it requires a heavy machinery and it is limited to harmonic properties. We will first describe the "direct" method, which shows most clearly the unity of the theory of structural properties, and in the later sections we will return to other formulations.

11. The Direct Method

The "direct" or "frozen phonon" method1s2 provides a unified approach for calculation of structural energies. It discards the notion of the orders of perturbation theory and treats the distorted crystal directly as a new clystal with a new structure having symmetry lower than the undistorted crystal, whether the magnitude of the distortion is large or small. Exactly the same methods are used for undistorted and distorted crystals and the total energy is calculated directly as a function of the positions of the ion centers. In a unified manner one can find the equiibrium structure, harmonic and anharmonic energies, and properties of the electrons, such

as

polarizabilities and dipole rn~rnents.~.~ Previously, such direct calculations have been restricted to simple high-symmetry crystal stmctures.19 The extension to distorted structures greatly extends the usefulness of total energy calculations and it provides a very severe test for current theories of exchange and correlation energies in interacting electron systems.

The approach which has proved very successful and feasible for calculation of exchange and correlation energies is the density functional method of Hohenberg, Kohn, and ShamB who proved that the total energy is uniquely related to the ground state charge density n(r) through a functional E,, = F[n]. Even though the exact form of F[n] is unknown, it is extremely useful that the theory is formulated in terms of

n(r)

which is itself a measurable quantity.

For

example, changes in charge density are directly related to the distortion energies, forces are determined rigorously by n(r)

as

shown by the Hellrnan-Feynman the0rem,5.l~.~l and dielectric response functions can be uniquely related to functional deriatives of F[n].2.21 Explicit solutions for n(r) result from the variational requirement that F[n] is minimum for the correct n(r).

In

Ref.

20

it was shown how one could define tractable self-consistent solutions to the variational

equations if the exchange-correlation part of the functional E,, is assumed to be local. Within this local density functional (LDF) approximation the function ex, (n(r)) must be the same

as

for a uniform electron gas exc(n) for which various functions have been proposed and utilized: the well-known n1I3 form with coefficient

-

0.7-0.8,2,5-7 the Wigner interpolation formula4 and others?J9 These forms are very similar

in

the range of n considered here.

The

results given below strongly support the conclusion that the LDF approximation is very good in the semiconductor crystals and that the agreement with experiment is &r than

can

be justified by rigorous anahysis.

The most important advance in the recent work is the development of techniques to solve the self-consistent electronic equations rapidly and accurately enough to calculate the small energy differencies between distorted and undistorted crystals (the non-systematic errors must be less than 0.01% on the total energy). The method to make this feasible in semicondutors is given

in

Ref. 2: the important simplifications are the use of pseudopotentials for the ion cores, the "special points" method22 whereby the sums over the Brillouin Zone can be reduced

to

a few points, Lowdin perturbation theory for higher plane waves, and schemes to quickly reach con~ergence.~~ All these steps can be checked and it has been found possible to achieve essentially arbitrary accuracy in the solutions. Let us note, in particular, that the use of a pseudopotential is just a convenient procedure for eliminating the deep core electrons. It

can

be generated in an ab inizio manner to reproduce very accurately the valance charge density.% The only essential approximation is the assumption that the core is rigid. Even this has been tested by Harmon, Weber, and Hamann%ho have carried out all-electron calculations and have shown that the rigid ionic pseudopotential is very accurate except possibly for very large deformations where effects of non-rigidity of the core were detected.

HI. Results of Direct Calculations

The most systematic and complete investigation of the structural properties of a crystal has been carried out for ~ iusing the direct calculations of ~ - ~

Em,.

The original work of Wendel and Martin2 on the equilibrium and harmonic and anharmonic energies has now been extended in Refs. 3 and 4. Yin, Ihm, and Cohen3 have utilized more accurate potentials and considered many crystal structures and phonons. It is remarkable that the lattice constant, bulk modulus, and phonons at

r, X,

and L, and Gruneissen parameters are all calculated to within

a

few percent of experimental

value^.^

Harmon, Weber, and Hamann4 have derived many of these quantities in all-electron calculations and have in addition calculated the internal strain parameter and the cubic anharmonicity of the

TO@)

phonon. Based upon these results, it is clear that Si is one of the best understood of all crystals from a structural point of view, and that confidence can be placed in other results, e. g. anharmonicty, not known experimentally.

Aside from the interesting results for the energies, particularly to low frequency TA(X) modes and its negative Gruneisen ~ararneter,"~ the calcuiations show directly the role of the electrons in determining the forces. For example, it is shown in Ref. 2 that the ion-ion forces destabilize the TA(X) and the closely-related shear C,,-C, elastic constant. It is the covalent electron contributions that give the shear stability. Furthermore, the low TA(X) frequency results from relaxation of the bonding charge density between certain atoms as is shown in Fig.

C6-620 JOURNAL DE PHYSIQUE

5 of Ref. 2.. This rotational-type relaxation is very similar to the displacement of bond-charges in the adiabatic bond-charge model of Weber,= however, the upper part of figure shows that there is also charge transfer between the bond charges. This is especially important in the bond stretching modes, such as TO

(r),

and is not included in the bond-charge models.25 This is

an

example of the electronic effects that determine the atomic forces and which can be given in ab initio calculations. It may be possible to include such effects in future model calculations, but it is essential to have the ab initio calculation to determine what is the appropriate model.

In the cases of GaAs6s7 and Se8 new features enter the theoretical considerations. GaAs

is an

example of an ionic crystal in which there are long-range electrostatic forces which cannot be included in the types of calculations used for Si and GeA5 It is shown in Ref. 7 how long- ranged Coulomb forces

can

be included in finite cell calculations with a celI size which is feasible (8-12 atoms per cel16s7). The key quantity is the effective charge which may be determined by displacing planes of atoms in the crystal and calculating the dipole moment resulting from the bare ion plus the electronic ~ontributions.~.~ In Fig. 1 are given change in electronic charge density calculationed as described in refs. 6 and 7 for GaAs crystals with displaced (001) planes of atoms, Ga planes in the upper part and As planes in the lower part of the figure. The ionic plus electronic charge contributions define the longitudinal effective charge7 and the calculations give f0.155 for Ga, and -0.148 for As com~ared to the experimental value of

+

0.197, where the sign is unknown experimentally. The close agreement for

Ga

and As taken from two independent calculations shows internal accuracy of the calculations. The power of the direct method illustrated by the fact that exactly the same computer programs and convergence criteria can be used for such apparently diverse problems

as

the effective charge7 in GaAs and the electronic potential for Ge-Gals interfaces2"

l o o t G a DISPLACED

Fig.

1.

The change in charge density per unit displacement of the central plane

of

Etoms, Ga in the upper curve and As

in

the

-

lower. The doub)e arrows indicate the

displacement to the right. The coordinate is

100

-

in the [001] direction and the density is

averaged over (001) planes. The longitudinal effective charge is the sum of the ionic part plus the moment of the electronic density

<

-

shown here. Details

are

given in

Ref.

7.

N 0

-

^t

d

Another problem of general importance encountered in GaAs is that, at points such

as X,

the phonon eigenvectors are not known experimentally. Eigenvectors are just as important

as

eigenfrequencies for a complete description of lattice dynamics, yet they are difficult to measure and generally are unknown except when determined by symmetry. It is possible to determine eigenvectors from cross terms in the total energy and this has been done via direct calcuIations for GaAs at the

X

point6 in the first ab initio calculation of eigenvectors to the knowledge of the author. The results are discussed in detail in Ref. 6 where it is shown that the calculated eigenvectors provide a very stringent test for phenomenological models: of six different models, all of which fit all known frequencies extremely accurately, there is complete disagreement on the eigenvectors and only one (and perhaps a second)

is

even close to the present results.

The unique advantage of the direct method

is

the ability to calculate energies for large displacements of the ions. Let us consider one example, perhaps the most interesting of the results reported so far. In calculations described in Ref. 6, it was found that for several dispIacement patterns in GaAs the energy has the form E,t(R) =

+

Au2

+

Bu4

+

Cu6, with A,C>O, but

B<O.

This can lead to interesting results

as

shown in Fig. 2 for the TA(X) mode

in

GaAs. The harmonic frequency of this mode decreases with pressure (decreasing lattice constant in the figure)

so

that a second minimum developes which eventually has lower energy than the undistorted crystal. Thus the calculations predict a first order phase transition to

a

lower symmetry structure at a pressure where the squared TA frequency is reduced by a factor of about 2, which may occur before the known transition to the metallic NaCl phase.

Fig. 2. Variation of the caIculated total energy of a GaAs crystal with the square of the displacement which is approximately the TA(X) mode. The curves are shown at several pressures and we see the decrease

of

the harmonic part (dashed lines) with decreasing lattice constant a Anharmonicity becomes relatively more importmt until

a

second minimum develops at finite displacement indicating a possible first order phase transition. From Ref. 6.

C6-622 JOURNAL DE PHYSIQUE

In

Se8 the new feature is the low symmetry of the equilibrium structure, so that determining the equilibrium lattice structure

is

obviously the first theoretical problem. There are three independent variables describing the structure a, c/a, and chain radius r. The positions of the atoms for zero forces conjugate to each of these variables determines the structure and the curvature determines, B, C,, and the A

(r)

phonon frequency. Vanderbilt and Joannopoulosa have shown the calculations can be done, with reasonable agreement with experiment, and ibrthermore have proceeded to look for new metastable minima for large displacements corresponding to three-fold and one-fold coordination of atoms. These large changes are thought to be very important

in

the electronic properties of chacogenides and their work

is an

extremely important step going far beyond the phonon approximation to look for new stable or metastable structures.

It is also important to mention that approximate versions of the direct method have been used to predict surface reconstructions on a variety of semiconductor surfaces26, and density hnctional calculations has been reported for a Si surface27 and a Ge-GaAs interface.23 These involve changes in bonding and are very important application of ideas similar to those in Se for prediction of entirely new situations.

Finally, at this conference is presented an entirely new use of the direct method to calculate entire dispersion cur~es.~.'~ The idea is to displace one plane of atoms represented by one atom

in

a large cell (8 or 12 aoms "supercells") and to use the Hellman-Feynman theorem to calculate the forces on

&

the planes of atoms in the vicinity of the displaced plane. It is shown in Ref. 7 that interplanar forces are always short range even in ionic crystals and, provided the cell is large enough, complete, accurate dispersion curves can be calculated. The set of harmonic forces, changes in electronic charge density, many different anharmonic terms, effective charges, etc., can be considered in a unified manner calculated simultaneously from a small number of seif- consistent calculations. Results are given in Refs. 5 and 28 which show that complete dispersion curves, eigenvectors, and dielectric properties such as effective charges and dielectric constant,

can

all be calculated, providing a very complete description of forces in the crystal.

IV.

Perturbation Methods

As was mentioned in the introduction, perturbation methods are very powerful for calculating linear response functions and harmonic forces. Using perturbation theory on the perfect crystal, it is much more straightforward than in the direct method to calculate dielectric fbnctions and harmonic phonons of arbitrary points in the Brillouin Zone; thus this approach will always be preferable for some problems, A comparison of the two approaches is given

in

Ref.

21,

where it

is

shown that ideas from each method can be used to aid calculations in the other method. In particular the usefulness of the density functional can be transferred to the perturbation methodlo21 and the perturbation theory can be used to increase the rapidity

of

convergence

in

the self-consistent cycles required in the direct cal~ulations~~

The most complete perturbation

calculation^^^^^^^^^^^

have been for

Si, as

was the case with the direct method. The only complete dispersion curves have been published by Bertoni, et all7

and by Van Camp, Van Doren, and Devreeseg with more recent results presented at this wnferenceJO The current results do not agree with experiment as well the results of direct calculation^^,^ because they have not been done with accurate pseudopotentials. However, the method has the capability

to

give very complete information on the harmonic lattice dynamics.

Let us note also the calculation by Muramatsu aqd HankeU (using different methods for exchange and correlations) who apply the dielectric function method to look for phonon softening and instability to reconstruction on the Si (111) surface.

One of the powers at the perturbation method is the ability to consider long range forces

and

take the q -, o limits14 much more easily than in the direct method7. If the dielevic function matrix ^E is known, the transverse effective charge e*, for ion i is given by"

e*, = l i m ( q 4 ) _Z, q e-I (q, q

+

G) (ij

+

G ) Vi ( q

+

G), (2)

where V, is the same bare ionic potential potential that determines the electronic states of the undistorted crystal. This has been calculated for many ionic semiconductors in papers at this conference by Restc, and Baldereschiu and by Olego, Vogl, and Cardona,13 who consider volume dependence of the charge. The trends among several crystals can be seen in the results of Ref. 11, however, there are inconsistencies in the use of Vi and that affect the yalues of e*,

V. Summary

In conclusion, recent

calculation^^-^

of energies and forces using electronic Hamiltonians, have brought our understanding of structural properties to a new stage in which the equilibrium structure and lattice dynamics can be treated in a unified manner. Because of the computational simplicity of the density functional method,2O the very difficult problem of electronic exchange and correlation can be handled accurately enough to calculate equilibrium lattice constants and atomic positions, harmonic and anharmonic force constants, ground state electronic properties, such

as

effective charges, etc, The computational technique which has made this possible is the

"direct" method2 and the greatest advantage of this approach is the ability to deal with many different situations

--

perturbative and non-perturbathe-- on the same basis. Extensive results on Si,2-4 Ge,3v5 GeAs,"' and Se8 have firmly established the power of this method. The development of perturbation methods 9-14.17.18 is also very useful and can potentially provide a more detailed description of the harmonic

parts of

the restoring forces. Together these two caIcuIational methods hold the promise of providing very complete structural information on many systems including semiconductor crystals, surfaces, and interfaces.

This work was supported in part by Department of the Navy contract N00014-79-c-0704 issued by the Office of Naval Research. The United States Government has a royalty-free license throught the world in all wpyrightable material contained herein.

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1. D. J. Chadi and R.

M.

Martin, Solid State Commun.

B,

643 (1976).

2.

H.

Wendel and

R. M.

Martin, Phys. Rev. Bl9, 5251 (1979).

C6-624 JOURNAL DE PHYSIQUE

3.

M. T.

Yin and

M. L.

Cohen, Phys. Rev. Lett 45,1004 (1980); J. Ihm, M.

T. Yin,

and

M. L.

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4.

B. N.

Harmon, W. Weber, and D. R. Hamann, this conference and to be published 5.

K.

Kunc and R.

M.

Martin, this conference and to be published

6. K. Kunc and R.

M.

Martin, Phys. Rev. Rapid Commun.,

in

press.

7.

R. M.

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K.

Kunc, Phys. Rev., in press.

8.

D.

Vanderbilt and J.

D.

Joannopoulos, to be published.

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E.

Van Camp, V.

E.

Van Doren, and J.

T.

Detreese, Phys. Rev. Lett. 42, 1224 (1979).

10. P.

E.

Van Camp, V.

E.

Van Doren, and J.

T.

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11.

R.

Resta and A. Baldereschi, to be published and

this

conference.

12. A. Muramatsu and W. Hanke, to be published and this conference.

13. D. Olego, M. Cardona, and P. VogI, this conference.

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l88,

1431 (1969); R.

M.

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H.

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M.

L. Cohen; J. Phys.

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K.

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G.

Louie and

M. L.

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L.

Moruzzi, A. R. Williams, and J.

P.

Janak, Phys. Rev.

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Hohenberg

and W.

Kohn,

Phys.

Rev.

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L.

J. Sham, Phys.

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L.

J. Sham and W.

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Phys. Rev.

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R.

M. Martin

and

K.

Kunc, Proc. of CECAM Conf. on Ab

Initio

Calculations of Phonon Spectra, Antwerp, 1981, to be published

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J.

Chadi and

M. L.

Cohen, Phys. Rev.

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5747 (1973);

H.

J. Monkhorst and J.

D.

Pack, Phys. Rev.

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K.

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D. R.

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25. W. Weber, Phys. Rev.

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