APPLICATION AND EXTENSIONS OF THE GREEN'S FUNCTION METHOD

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APPLICATION AND EXTENSIONS OF THE GREEN’S FUNCTION METHOD

B. Segall

To cite this version:

B. Segall. APPLICATION AND EXTENSIONS OF THE GREEN’S FUNCTION METHOD. Journal

de Physique Colloques, 1972, 33 (C3), pp.C3-31-C3-38. �10.1051/jphyscol:1972306�. �jpa-00215040�

JOURNAL DE PHYSIQUE

Colloque C3, suppliment au no 5-6, Tome 33, Mai-Juin 1972, page C3-31

APPLICATION AND EXTENSIONS OF THE GREEN'S FUNCTION METHOD (*)

B. SEGALL

Case Western Reserve University, Cleveland, Ohio, 44106, U. S. A.

RLsum6.

-

L'application conventionnelle de la Methode de la Fonction de Green

(MFG)

au calcul des bandes d'energie du zinc est discutee. Contrairement a l'image qu'on se forme g6n6 ralement du zinc, les bandes d semblent se situer au-dessus du minimum de la bande de conduction.

Ce

r6suItat est confirm6 par des etudes rkentes de IY6rnission de rayons X et de l'6rnission photo- klectrique. Au-dessus des bandes d, I'knergie

E(k)

calculQ resemble

a

celle de I'klectron quasi libre. Les dimensions de la surface de Fermi et le spectre d'absorption a basses knergies pour ces valeurs de

E(k)

correspondent assez bien avec les valeurs expbrimentales. On discute trois dkvelop- pements de la MFG.

Le

premier dkveloppement est I'emploi de la MFG comme sch6ma pour une paramktrisation, les quantites qui peuvent 6tre adaptees sont les dkphasages pour les moments angulaires 1

= 0, 1

et

2.

Les avantages de cette approche sont indiques et l'application de la mkthode pour une &termination semi-empirique des bandes de l'argent est discutke. Le second d6veloppement est I'emploi de la MFG pour determiner des expressions commodes et prhises des paramMres de masse effective pour les Btats aux diffkrents points de symetrie. Les r6sultats pour plusieurs metaux avec des structures f. c. c. et b. c. c. sont indiquks. Finalement, on donne des expressions pour les potentiels de dkformation calculkes par la MFG.

Abstract. - The conventional application of the Green's Function Method (GFM) to the calculation of the energy bands of Zn is discussed. In contrast to the previously accepted picture of Zn, d-bands are found to lie above the conduction band minimum. This result is strongly sup- ported by recent X-ray emission and photoemission studies. Above the d-bands the calculated

E(k)

are nearly-free-electron-like. The Fermi surface dimensions and the low energy absorption

spec-

trum for these

E(k)

are in good accord with the experimental data. Three extensions of the

GFM

are discussed. The first is its use as a parametrization scheme in which the 1

= 0 , l

and

2

phase shifts are the quantities to be adjusted. The advantages of this scheme are noted and its appli- cation to a semiempirical determination of the Ag band structure is discussed. The second is the use of the GFM to determine convenient and accurate expressions for the effective mass parameters for states at various symmetry points. Results for a number of metals with the f. c. c. and b. c.

c.

structure are given. Finally, expressions for the deformation potentials obtained within the

GFM

framework are presented.

Introduction.

-

The Green's function method (GFM) of Korringa, Kohn and Rostoker [I] has proven to be a very powerful tool for studying the electronic structure of crystals. I n Section I of this paper we discuss the results of the conventional application of this method to the hcp metal Zn. A new application and two new extensions of the method are then considered

:

its use as a parametrization scheme (Section 11), as a means for deriving effective mass parameters (Section 111) and for directly deter- mining the deformation potentials (Section IV).

I. Electronic Structure of Zn.

-

Programs for cal- culating the electronic energy bands including all relativistic corrections by the GFM has been set up by G. E. Juras, C . B. Sommers and the author. With these programs we have been investigating some metals which form in the fcc structure, e. g., the noble metals and lead, and some which have the hcp structure like the group I1 metals. Dr. Sommers will discuss some of the work at this meeting. I will restrict myself to

(*)

Supported

in

part

by

the

US Air

Force.

discussing our work on Zn which at present is the most complete of our studies on the hcp metals and which has yielded interesting results.

Zn, which has a 3 d l 0 4 s2 configuration, has been one of the most intensively studied metals both theore- tically and experimentally, and consequently its elec- tronic structure is generally felt t o be very well under- stood. The picture that has emerged from these studies is that the conduction bands of Zn are nearly-free- electron-like. In particular the d-bands are taken t o lie well below the bottom of the conduction band (i. e. E(r,)). I believe it is fair t o say that this picture was based primarily on the fact that the considerable body of data relating to the E@) near the Fermi level, EF, could be well accounted for by a nearly-free- electron model. For example, the last calculation, which was done by Stark and Falicov (SF) 121, used the pseudo-potential approach with the pseudo- potential adjusted to fit de Hass-van Alphen j(dHvA) data. S F did find that they needed a nonlocal pseudo- potential (interestingly enough with a sizable d compo- nent) t o achieve good quantitative agreement. The E(k) they obtained also gave a reasonable account of the

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972306

C3-32 B. SEGALL

low energy optical properties. Such a calculation, however, could not show narrow d-bands. The same is true of the early first principles OPW calculation by Harrison [3]. Thus in both cases the d-bands were excluded at the outset essentially by assumption. The only other investigation that could have shown these bands directly was the admittedly brief and prelimi- nary APW calculations by Mattheiss [4]. He found the d-bands to be about 0.5 Ry below E(T,). We should also note that there were some near UV (ho < 10.8 eV) reflectivity and photoemission studies which were interpreted to support the above picture.

The calculations we are reporting were carried out using a muffin-tin potential constructed from a super- position of atomic Coulomb potentials to which an exchange contribution of the Slater-free-electron form, a 3[(3/8

z)

,(r)]1/3, was added. The E(k) were obtained for three cases

:

the Kohn-Sham exchange (a

=

2/3), the Slater exchange (a

=

1) and the inter- mediate case for a

= 5/6.

Figure 1 gives our results for the intermediate case for all symmetry directions in the Brillouin zone. The salient feature of these results is that, in sharp constrast to the previously accepted picture for Zn, the d-bands lie well within the lowest conduction band. For energies outside the d-band region however, the bands are nearly free-electron-

like and are in general quite similar to those of SF.

The results for the Kohn-Sham and Slater exchanges are qualitatively the same except that the d-bands shift upward by about 1.5 eV for the former and downward by almost 2 eV for the latter. For

ol =

1 they are close to but still above E(T,).

Since this finding conflicts with the accepted picture of Zn, it was necessary to seek all relevant data. There are two recent experimental studies which clearly locate the d-band. The first of these are the X-ray emission and self-absorption studies of Hanzely and Liefeld

[ S ] .

They find a well defined emission peak, clearly associated with the d-bands, which is at 8.4 eV below EF and about 2 eV wide. The other is the study by Nilsson and Lindau [6] of the energy distribution of electrons photo-emitted by high energy (21.2 eV) photons. Nilsson and Lindau also unambigously find a d-electron peak which is about 2 eV wide and which they locate at about 9.5 eV below EF. For comparison, the midpoint of the d-bands in figure 1 is about 8.4 eV below EF and are 2 eV wide.

Regarding the E(k) around EF we note that the Fermi level in figure 1 was located at

by requiring it to lie between the K, and L, levels in

FIG. 1. - E(k) for Zn along symmetry axes.

APPLICATION AND EXTENSIONS OF THE GREEN'S FUNCTION METHOD C3-33

order to be consistent with the existence of the

((

needle

^D

centered at K and the absence of a Fermi surface segment around L. We have also estimated EF utilizing the fact that outside of the d-band region and away from the zone boundaries the bands are parabolic with an effective mass of 0.95. With a small correction for the depression of the lower

TI+

level due to the presence of the d-bands, we find a value of 0.78 Ry in excellent accord with the above value. A comparison of some key Fermi suface dimensions for the E(k) in figure 1 with those determined by dH vA and rf-size effect measurements [7] are given in Table I.

It is seen that the agreement between the calculated and measured values is generally about 1-2 %. A more complete comparison will be made using the relativistic E(k), the calculation of which is now in progress.

The relationship of these results to the low energy (ha < 4 eV) optical reflectivity measurements of Rubloff [8] is also of interest. These measurements yield an optical absorption spectrum with a sharp peak at 0.9 eV present only with E /! c and a broader one present in both polarizations extending from 1.6 eV to 2.1 eV. In a calculation of the absorption associated with the SF E(k), Kasowski [9] found that the low energy peak is due to transitions around the

2

and T directions near E, and that the upper peak

The above noted changes in the d-band location with different exchange contributions also manifest themselves in fairly small but nevertheless important changes in relative level positions near EF. E. g. for a

=

1 the L, drops below the IS, level, while in the a

=

2/3 case the gap at L associated with the 2.1 eV absorption peak increases by about 20 %. It appears then that the a

=

516 choice yields non relativistic E(k) for Zn which are distinctly in better accord with the experimental data discussed above than are the other two.

It is certainly gratifying and also useful to be able to obtain E(k) over a fairly broad range (almost 1 Ry) in good accord with the quasi-particle spectrum that is inferred from available experimental data, as appears to be the case in Zn. However, to press for very close agreement with experiments in a

<<

first principles

^)>

calculation is probably not too realistic or useful at this stage. Here we should use a semiempirical approach.

11. GFM as a Parametrization Scheme. - Some time ago, F. Ham and the author [I] suggested that in solids for which the muffin-tin approximation is applicable the GFM provides a very good basis for a parametrization scheme. One of the reasons for believing this is that the GFM dispersion relation (hw

z

2.0 eV) comes from the roughly parallel bands

in the vicinity of the LS'H direction. He obtained good det ( B,, (E, k) + ^6,,, E~~~ cot

ql

(E) )

⁼

0 (1) agreement for the location of the lower eneryg peak,

but his result for the upper peak was about 0.9 eV too high. We note that the E(k) in figure 1 are in very close agreement with those of SF along the Z and T axes, but are closer together along St than the SF bands and by an amount at L which is almost exactly the discrepancy of 0.9 eV between Kasowski's calcu- lated peak and Rubloff's measurement. Thus, although we have not yet calculated ~,(w), it is clear that the E(k) in figure 1 would yield an absorption spectrum in good accord with the data.

( L

=

1, m) has a convenient form. Here all crystal potential effects are completely incorporated in the phase shifts, g,(E), and these are neatly separated from those terms dependent on the crystal structure.

Secondly, because of the very rapid convergence of the GEM, it generally suffices to include only I

=

0,l and 2 angular momentum components in the trial functions in order to achieve high accuracy in the E(k).

Thus the effect 'of the potential is completely repre- sented by the s, p and d phase shifts. Furthermore the general functional behavior of the q,(E) over

Some dimensions of the Zn Fermi suvface (in A-I)

Third-zone electrons (lens) From r , along TAA From r , along TTK From r, along r Z M Second-zone holes (monster)

From r, along TTK To inside ring Near K point From r, along TZM From K, along KPH

RFSE

(")

-

dHvA (Fitted)

( b )

-

Present calculation

-

(")

STEENHAUT (0. L.) and GOODRICH (R. G.), see reference 7.

(&)

STARK (R. W.) and FALICOV (L. M.), see reference 2, Table 11.

the relevant energy range is far from arbitrary, e. g. it can be ascertained from preliminary calculations for a reasonable potential. Thus it should be feasible to parametrize the y,, or some function of them, in terms of a few disposable parameters which could be determined by empirical data. The advantages of this approach are (1) that the final E Q are determined accurately and (2) that the parameters involved (the y,(E)) have a simple and basic physical signifi- cance in contrast with the quantities (e. g. pseudo- potentials) used in other schemes.

An implementation of such a scheme has been carried out by Cooper, Krieger and the author (CKS) [lo] for the relatively complicated band struc- ture of Ag. The key step is to arrive at functional forms for the E dependence of the y, so that they could be determined over a wide range of E in terms of a few parameters. As a guide we had available four sets of y, from previous first principle calculations (2 for Cu and 2 for Ag). The tan y, for 'one of the Cu and one of the Ag calculations are given in figures 2 and 3 respectively. The general forms of both

0.50

PHASE SHIFT PARAMETRIZATION -.-...-..

F ~ G . 2.

-

Comparison of tan yl' for Cu (for the Chodorow potential) from first principles calculations and from adjusted

parametrized phase shifts.

0.5 FIRST PRINCIPLES -.

CALCULATION

'on 72 --- PHASE SHIFT . . . ..

PARAMETRIZATION

FIG. 3. - Comparison of tan yg' for Ag from a first principles calculation and from adjusted parametrized phase shifts.

of these sets are similar to each other and to the other sets, with the major difference between them being the position of the singularity (a pole) of tan yz. This is the so-called

ct

d-band resonance

⁾⁾

which occurs roughly at the center of the d-bands and which is sometimes represented by tan

q,

%a(E- Ed)-

where Ed is the

^<(

d-resonance

D

energy. However, parts of the d-bands in the noble (and transition) metals fall at low positive energies where the behavior of the tan y, is

tan

y, % ~ ( 2 ' + 1)12

Combining this with the d-band resonance we have the approximate form

a E ~ ' ~

tan y,

^%

-

E

-

Ed

where the E-dependence in the numerator is vital in order to obtain meaningful results for some states.

This approximate form is fairly good except for E > Ed. To achieve a quantitatively good fit to the tan y2(E) over a large energy range (over 1 Ry) it was necessary to introduce additional terms. The final expression we used was

tan y,

=

d2 15" + ^CT(E -

d l ) x

E - dl

where o(x)

=

0 for x < 0 and

=

1 for x > 0.

Corresponding expressions were obtained for the s and p waves. All together ten parameters were intro- duced. The resulting values for the tan yl are shown by the dotted curves in figures 2 and 3 which are seen to differ only slightly from the ones obtained in the ab initio calculations. Except for a few eigenvalues at high energies (1 Ry or more higher than TI), the E(k) corresponding to the fitted y, generally differ from those for the first principles values y, by much less than 0.01 Ry.

With the y, parametrized in a form like (2) there is considerable flexibility in the amount of empirical data that one could input into a calculation. For example, one could make only minor adjustments to a first principle calculation or, if suficient data are available, one could determine a set of q,, and then E(k), almost entirely empirically.

In CKS we applies this approach to Ag. Since there

are ten parameters to be determined, there are the

same number of steps and of data to be used in the

fitting. At each step a condition is set on one or two

of the y , at some energy. Time does not permit me to

detail the sequence of steps that we employed

;

they

are given in CKS [lo]. I will only note the specific

data that was used

:

(1) the 3.97 eV separation of the

top of the d-bands near L from EF, the optical inter-

band threshold [ll], (2) the energy difference

EF - E(L,,)

=

0.3 eV determined from photoemission

studies [12], (3) the d-band width (E(x,) - E(x,)) of

3.9 eV from the photoemission experiments, (4) the

4.3 eV gap between L, (upper) and L,, also from

photoemission work, (5) the energy difference

E(X4) - E(X4

=

5.5 eV from reflectivity measure-

APPLICATION AND EXTENSIONS OF THE GREEN'S FUNCTION METHOD C3-35

ments [ l l ] and (6)

^<(

belly

⁾⁾

radii of the Fermi surface

in the [I001 and [110] directions. The remaining four inputs were obtained from those features of our previous Ag calculations which we believe were likely to be reliable. The resulting semiempirical phase shifts are given in figure 4. With these q,, the band structure shown in figure 5 was calculated.

-

r-

^-r-

-r-7

PARAMETRIZATION SCHEME]

0.50 t o n ? --

0

ton ?I -

FIG. 4.

-

The tan 71' for Ag obtained by the semiempirical phase shift parametrization scheme.

SILVER-SEMMPMCAL PHASE SMFT PARAMETRIZATION SCHEME

io.o,ot (I.WI I I (O.O,OI [ u ~ , a n , o l P,O,M

V."

r

x w L

r

K x

(012KI b

FIG. 5. -The E(k) for Ag calculated from the semiempirical phase shifts shown in figure 4.

A few checks on the reasonableness of these E(k) could be made. First, we estimated E, from the bands and found it to differ from the input value by only 7 x Ry. Second, the computed neck radius turned out to be within a few percent of the measured value. Finally, we found that the calculated gap bet- ween X,, and X I (lower) agrees closely with the tenta- tively identified optical transition [13].

Our calculation has a built-in agreement with key experimental data over a wide energy range. Thus, it is not surprising that our E(k) is in better agreement with the body of available data on Ag than are the several existing first principles calculations

;

but we will not go into detailed comparisons here.

We feel that this work has demonstrated the effec- tiveness of using the GFM as a parametrization scheme and we look forward to its further use and development. It would be useful to extend the scheme to include relativistic corrections. Also, there are probably more efficient ways of parametrizing the

y,.

Again we want to emphasize what we believe is the significant advantage of this approach over other parametrization schemes, namely that the adjustable parameter is a quantity which has a clear cut and basic physical significance. Not only are the phase shifts useful in other applications, but they tell us something about the effective (E and I-dependent) potential in the crystal.

111. Effective Mass Parameters. - The usefulness in both theoretical and semiempirical contexts of means for continuing the En@) in the niehgborhood of a point ko is well known. This is given straightfor- wardly in terms of the diagonal matrix element of the momentum and the inverse effective mass tensor. By the use of k . p perturbation theory the familiar f-sum rule expression is obtained for the latter. Useful though this expression and its generalization to dege- nerate bands has been in solid state theory, it does not generally provide a practical means for accurately evaluating the band parameters since it generally requires knowing an appreciable number of energies and momentum matrix elements.

In 1938 Bardeen [14] obtained a very simple and practical expression for the effective mass within the spherical approximation. His result depends only on the logaritmic derivatives of the s and p radial func- tions evaluated at the Wigner-Seitz sphere and at E(T,) [15]. Unfortunately, the spherical approximation is inapplicable to states other than the s-like TI in nonatomic cubic crystals.

I will briefly discuss work by G. E. Juras and myself [16] on the derivation and evaluation of explicit expressions for band parameters at various symmetry points in the BZ for degenerate as well as non-degene- rate bands for the non-relativistic and relativistic problems. These results which are accurate and very convenient are derived using the GFM. The reasons for expecting that this method would be useful for this purpose are the same as those given in the previous section.

To obtain our results we consider the dispersion relation, eq. (I), for the points

ko

+ 6k,, and we suitably expand it about the solution at ko. To this end we expand the BLL, in a given direction (chosen to exploit symmetry) according to

BLt(E,

ko

+ ^dkn)

⁼

^BL, + 6 k ~ g ( n ) +

For simplicity let us consider the TI state. If, as we

do throughout this paper, we restrict the trial functions

to those with 1

,(

2, only the s component occurs in

the r, function and the determinant consists of one

term, Bo,o(Eo, 0) + E:" cot

yo =

0. For k

#

0, the

only other terms contributing to order ( ~ 3 k ) ~ are those

for I

=

1. Writing

C3-36 B. SEGALL

and taking 6k along any internal symmetry axis we find that

where

and all quantities are evaluated at E

=

E(T,). Once the structure dependent terms are evaluated, this result, which involves only the s and p phase shifts, is as convenient as Bardeen's result. Moreover, this result is more accurate than Bardeen's in that the

anisotropic boundary conditions are properly taken into account. In fact, this result and all others we consider, are exact for the E(k) given by the GFM when the trial function space is restricted to 1 < ^2.

We note that the structure terms, e. g. the BLZ), are computed about as easily as the BLL* themselves. These quantities, or the associated D ~ L , can be calculated once and for all for a given structure and tabulated as was done for the DL, 1171.

Results for other symmetry states are obtained similarly. For example, the longitudinal component of the mass tensor for the L,, (p-like) state for the f. c. c.

lattices is given by

(1) ~ ( 1 )

~ 0 0 ( ~ ~ 2 , 2 0 ) ~ + ~20(~~0',:0),10)~

-

2 B10,20

00,10

- [FOO F20

-

(~00,20)~1 ~ i 2 , l O

(s),

^{= -}

_{[Foe Fzo -} (Boo 20)2] aFlolaE (5) We note that (5), and the similar result for the trans- TABLE I1

verse component, fully incorporates the effects of

d-bands as well as the s and p states. The procedure Band parameters for various symmetry states of various metals

outlined is readily extended to degenerate states. In that case the E,Q in the vicinity of an n-fold degene- rate level are obtained from a secular equation of degree n, det [(Hij - aij(E - k2)]

⁼

0, where the Hv are bilinear in k,, k, and k,. For the 3-fold band in f. c. c. HI,

=

~ k f f ~ ( k z + k:) and HI,

=

Nk, k, (and cyclic permutations). The band parameters L, M and N, which are given by k . p theory by expressions like that in the f-sum rule, are given here by expressions like

The B(,) etc. in this case are evaluated in the [loo] direc- tion and the subscript c denotes the real spherical harmonic with cos 2 q~ dependence on

cp.

Generalization to the relativistic problem is also quite straightforward using the corresponding disper- sion relation. In this case the wave function is expanded in the spin angular momentum functions ~ t : where rc and

,u

are the relativistic quantum numbers for a spherical field. The effective mass for the r6+ ^(corres-

ponding to TI) state o f f . c. c. is readily shown to be

where

@, =

A,,, + E l f 2 cot

Y,

and AK,,,

=

A

,,,,

(with

p =

1/2), are linear combinations of the B,,,,,,, with Clebsch-Gordon coefficients, and the

q,

are the relativistic phase shifts.

Table 11 gives a comparison of the band parameters for a number of different states for several metals obtained directly from expressions of the type discussed above with the same quantities obtained by numeri-

Symmetry m*/m

state by GFM formula

- -

r l , cu ^{.941 5}

r45, CU, L - 1.298

M - .730 8

N 2.799

r12, CUP L - .6475

J - .438 9

XI, Cu along A 4.450

X3, CU - 3.519

X,,CU - - 9.015

X5,Cu - - 2.320

~ , C U - - .I155

L;, Cu along A - .I40 5

to A .248 6

rl, ^{Li 1.293}

rl, ^{~a .970 1}

rl, K 372 6 r 6 + , PI7 .955 5

m*/m by fitting E(k)

- .9414 - 1.298

- ^.731

2.78 .648 - .439 4.45 3.52

- 9.01

- ^2.32

- .12 .I45 .25 1.293

.970 372 5 .955 4

cally fitting the E(k) for the same potential over a suitable range of k. In those cases where the bands can be fitted accurately, the agreement is shown to be excellent. But this is to be expected since the for- mulas are

⁽⁽

exact

^)),

as noted above. In those cases like the bands near L,, and X,, where the curvature rapidly changes sign as one moves away from the zone boundary, it is difficult to obtain accurate parameters from fitting the E(k).

A more graphical illustration of the results is given

in figure 6 where the E(k) for Cu obtained by the

standard application of the GFM are shown for k

APPLICATION AND EXTENSIONS OF THE GREEN'S FUNCTION METHOD C3-37

FIG. 6.

-

Comparison of E(k) for along [I001 direction for Cu (Chodorow potential) calculated by the standard GFM (solid curves) and by quadratic expansions about r a n d X using the

effective mass parameters in Table I1 (dotted curves).

E,,(k) with strain is the last subject we will consider.

This of course gives the deformation potentials which in turn determines the electron-phonon coupling for long wave length acoustic phonons. These important quantities are rather difficult to calculate.

The approach I would suggest is similar to that used for determining the effective masses. Here we expand the dispersion relation for the strained crystal about that for the unstrained crystal. We must note at the outset that since we use the GFM dispersion relation we assume that the muffin-tin form of the potential is valid for the strained as well as the unstrained lattice.

It is possible that this simplifying assumption is not completely satisfactory for certain shear strains which reduce the crystal symmetry. The importance of strain- induced non-muffin tin corrections should be estimated in the future.

For simplicity let us consider a symmetry state (like

r, or L,,) which for the undeformed lattice has only one term in its trial function (for I d 2) so that the determinant in (1) is only of the first order. With a small shear strain, 5, more terms will generally be required and the determinantal equation takes on the form

I . . .

in the [loo] direction as a solid curve. The dotted I

curves are the quadratic expansions about the r a n d X points using the band parameters in Table 11. The substantial range over which there is close agreement for this relatively complicated band structure is to be noted. The smallest range over which there is close agreement understandably occurs for the X,, band because of the above noted rapid change in curvature.

Improved expansions here can be acheived by includ- ing ( ~ 3 k ) ~ terms and other variations of the approach.

From the above we might expect that occupied and low-lying excited states of a

((

simple

D

metal (i. e.

those without d-bands etc.) might be expressible in terms of band parameters, perhaps including the fourth order coefficients. For example, this might be the case for the alkali metals utilizing expansions about only r

and N. Since these expansions give the E(k) directly, the calculation of physical quantities of interest such as the density of states etc. would be greatly simplified.

These expansion parameters can also be very useful in the parametrization of empirical data. This will certainly be the case where small regions of the zone about a symmetry point are involved. The advantage of this approach is the same as that noted in the pre- ceeding section since the adjustable parameters are the

d l

and dy,/dE at the symmetry state energies involved.

IV. Effect of Strain on Electronic Structure

:

Defor- mation Potential.

-

The calculation of the shift in

where only the order of the terms are indicated. The matrix element in the upper left hand corner is the one whose vanishing (indicated by the zero) yields the eigenvalue for 5

=

0. It is easy to see that to order 5

the above leads to A

=

0. By expanding the energy according to E - E,

=

6E

=

Da ta where Da is the deformation potential for a strain of type a, we find

Tg- (1 - 6 3 ~BLL,/~C. + w"' cot v3 :

₍₇₎

Da - ata ( ~ F L I ~ E )

The symbol bf/6ta denotes the change in f per unit strain of type a. The vanishing of the first term in the numerator for isotropic strains results from the fact that the BLLr can be taken independent of scale. The most complicated terms in (7) are the structure terms, 6B/69, which can be calculated once and for all for a given structure. Although we have not yet computed these terms our experience with similar quantities indicates that we could program them without undue difficulty. We also note that the extension to symmetry states involving more than one term in the trial func- tion is straightforward. The results are similar in form to (7) but involve more terms.

While (7) is valid for all strains, it is of most interest

for the shears since the D for isotropic strains can be

obtained fairly conveniently (although not directly as

in (7)) by the conventional GFM as shown by Ham [I81

C3-38 B. SEGALL

and Davis et al. [19]. For the volume preserving shears from the neighboring atoms. This, of course, conside- it appears that the terms involving the change in the r, rably simplifies the determination of the D,.

with strain are probably small in many cases of inte- Dr. Juras and I have recently started work on this rest. They vanish, for example, for the case of cubic problem, but do not have any numerical results to crystal with potential constructed from contributions report at this time.

References

[I] A recent review of the GFM is given in SEGALL (B.) and HAM (F. S.), Methods in Computational

Physics, 1968, 8, 251.

[2] STARK (R. W.) and FALICOV (L. M.), Phys. Rev.

Letters, 1967, 19, 795.

[3] HARRISON (W. A.), Phys. Rev., 1962, 126, 497.

[4] MATTHEISS (L.), Phys. Rev., 1964, 134, A 970.

[5] HANZELY (S.) and LIEFELD (R. T.), National Bureau of Standards. Special Publication (Ed. by L. H.

Bennett), 1971, 323, 319.

[6] NILSSON (P.

0.)

and LINDAU (R. T.), J. Phys. F

: Metal Phys., 1971, 1,

854.

[7] STEENHAUT

(0.

L.) and GOODRICH (R. G.), Phys.

Rev., 1970, B1,4511.

[8] RUBLOFF

(G.), Phys. Rev., 1971, B3, 285.

191 KASOWSKI (R.), Phys. Rev., 1969, 187, 885.

[lo] COOPER (B. R.), KREIGER (E. L.) and SEGALL (B.),

Phys. Rev., 1971, B 4, 1734.

[Ill COOPER (B. R.), EHRENREICH (H.) and PHILIPP (H. R.),

Phys. Rev., 1965, 138, A494.

[12] Most information and references pertaining to photo- emission work in Ag can be found in SMITH (N. V.), Phys. Rev., 1971, 3B, 1862.

[13] EHRENREICH (H.) and PHILIPP (H. R.), Phys. Rev., 1962, 128, 1021.

1141 BARDEEN (J.),

J. Chem. Phys., 1938, 6 , 367.

[15] BROOKS (H.), Suppl. Nuovo Cimento, 1958,

7 ,

165.

[16] SEGALL (B.) and JURAS

(G. E.), Phys. Rev., 1971,

B 4, 3277.

[17] SEGALL (B.) and HAM (F. S.), Document No 7236 AD1, Auxiliary Publication Project, Library of Congress, Washington, DC.

[18] HAM (F. S.), Phys. Rev., 1962, 128, 82.

[19] DAVIS (H. L.), FAULKNER (J. S.) and JOY (H.

W.), Phys. Rev., 1968, 167, 601.