Patterson, Bailey - Solid State Physics Introduction to theory

194 3 Electrons in Periodic Potentials

Interpolation and Pseudopotential Schemes (A)

An energy calculation is practical only at points of high symmetry in the Brillouin zone. This statement is almost true but, of course, as computers become more and more efficient, calculations at a general point in the Brillouin zone become more and more practical. Still, it will be a long time before the calculations are so “dense” in k-space that no (nontrivial) interpolations between calculated values are necessary. Even if such calculations were available, interpolation methods would still be useful for many considerations in which their accuracy was sufficient. The interpolation methods are the LCAO method (already mentioned in the tight binding method section), the pseudopotential method (which is closely related to the OPW method and will be discussed), and the k · p method. Since the first two methods have other uses let us discuss the k · p method.

The k · p Method (A)23 We let the index n label different bands. The solutions of

Hψnk = En (k)ψnk

(3.273)

determine the energy band structure En(k). By Bloch’s theorem, the wave functions can be written as

ψnk = eik rUnk .

Substituting this result into (3.273) and multiplying both sides of the resulting equation by e−ik·r gives

(e−ik r Heik r )Unk = En (k)Unk .	(3.274)
It is possible to define
H ( p + k, r) ≡ e−ik r Heik r .	(3.275)

It is not entirely obvious that such a definition is reasonable; let us check it for a simple example.

If H = p2/2m, then H(p + ћk) = (1/2m)(p2 + 2ћk p + ћ2k2). Also

		1					2
	e−ik r Heik r F =		e−ik r	−		eik r F
		2m			i
							,
	=	1	[ p2 + 2 k p + ( k)2 ]F
		2m

which is the same as [H(p + hk)]F for our example.

23 See Blount [3.6].

3.2 One-Electron Models 195

By a series expansion

∂H

1

∂2H

H ( p + k, r) = H +

∂p

k +

3

(

k

)(

k

j

) . (3.276)

∂p

∂p

2 ∑i, j =1

i

i

j

Note that if H = p2/2m, where p is an operator, then

pH

≡

∂H

=

p

≡ v ,

(3.277)

∂p

m

where v might be called a velocity operator. Further

∂2H

=

1

δil

,

(3.278)

∂p ∂p

m

i

l

so that (3.276) becomes

H ( p +

k, r) H +

k v +

2k

2

.

(3.279)

2m

Then

H ( p + k + k′, r) = H + (k + k′) v +

2

(k

+ k′)2

2m

= H + k v +

2

k 2 + k′

v +

2

k

k′ +

2

k′2

2m

2m

2m

k′

k

2

k′

2

= H ( p + k, r) +

v +

+

.

2m

2m

Defining

v(k) ≡ v +

k / m ,

(3.280)

and

H ′ =

k′ v(k) +

2k′2

,

(3.281)

2m

we see that

H ( p + k +

k′) H ( p +

k, r) + H ′ .

(3.282)

Thus comparing (3.274), (3.275), (3.280), (3.281), and (3.282), we see that if we know Unk, Enk, and v for a k, we can find En,k+k′. for small k′ by perturbation theory. Thus perturbation theory provides a means of interpolating to other energies in the vicinity of Enk.

196 3 Electrons in Periodic Potentials

The Pseudopotential Method (A) The idea of the pseudopotential relates to the simple idea that electron wave functions corresponding to different energies are orthogonal. It is thus perhaps surprising that it has so many ramifications as we will indicate below. Before we give a somewhat detailed exposition of it, let us start with several specific comments that otherwise might be lost in the ensuing details.

1.In one form, the idea of a pseudopotential originated with Enrico Fermi [3.17].

2.The pseudopotential and OPW methods are focused on constructing valence wave functions that are orthogonal to the core wave functions. The pseudopotential method clearly relates to the orthogonalized plane wave method.

3.The pseudopotential as it is often used today was introduced by Phillips and Kleinman [3.40].

4.More general formalisms of the pseudopotential have been given by Cohen and Heine [3.14] and Austin et al [3.3].

5.In the hands of Marvin Cohen it has been used extensively for band-structure calculations of many materials – particularly semiconductors (Cohen [3.11], and also [3.12, 3.13]).

6.W. A. Harrison was another pioneer in relating pseudopotential calculations to the band structure of metals [3.19].

7.The use of the pseudopotential has not died away. Nowadays, e.g., people are using it in conjunction with the density functional method (for an introduction, see, e.g., Marder [3.34, p232ff].

8.Two complications of using the pseudopotential are that it is nonlocal and nonunique. We will show these below, as well as note that it is short range.

9.There are many aspects of the pseudopotential. There is the empirical pseudopotential method (EPM), ab initio calculations, and the pseudopotential can also be considered with other methods for broad discussions of solid-state properties [3.12].

10.As we will show below, the pseudopotential can be used as a way to assess the validity of the nearly free-electron approximation, using the so-called cancellation theorem.

11.Since the pseudopotential, for valence states, is positive it tends to cancel the attractive potential in the core leading to an empty-core method (ECM).

3.2 One-Electron Models 197

12.We will also note that the pseudopotential projects into the space of core wave functions, so its use will not change the valence eigenvalues.

13.Finally, the use of pseudopotentials has grown vastly and we can only give an introduction. For further details, one can start with a monograph like Singh [3.45].

We start with the original Phillips–Kleinman derivation of the pseudopotential because it is particularly transparent.

Using a one-electron picture, we write the Schrödinger equation as

H ψ = E ψ ,

(3.283)

where H is the Hamiltonian of the electron in energy state E with corresponding eigenket |ψ . For core eigenfunctions |c

H c = Ec c .

(3.284)

If |ψ is a valence wave function, we require that it be orthogonal to the core wave functions. Thus for appropriate |φ it can be written

ψ = φ − ∑c′ c′ c′ φ ,

(3.285)

so c|ψ = 0 for all c, c′ the core wave functions. |φ will be a relatively smooth function as the “wiggles” of |ψ in the core region that are necessary to make c|ψ = 0 are included in the second term of (3.285) (This statement is complicated by the nonuniqueness of |φ as we will see below). See also Ziman [3.59, p. 53].

Substituting (3.285) in (3.283) and (3.284) yields, after rearrangement

(H +VR ) φ = E φ	,	(3.286)
where
VR φ = ∑c (E − Ec ) c	c φ .	(3.287)

Note VR has several properties:

a.It is short range since the wave function ψc corresponds to |c and is short range. This follows since if r|r′ = r′|r′ is used to define |r , then ψc(r) = r|c .

b.It is nonlocal since

r′VR φ = ∑c (E − Ec )ψc (r′)∫ψc (r)φ(r)dV ,

or VRφ(r) ≠ f(r)φ(r) but rather the effect of VR on φ involves values of φ(r) for all points in space.

c. The pseudopotential is not unique. This is most easily seen by letting |φ → |φ + δ|φ (provided δ|φ can be expanded in core states). By substitution δ|ψ → 0 but

δVR φ = ∑c (E − Ec ) c δφ c ≠ 0 .

198 3 Electrons in Periodic Potentials

d.Also note that E > Ec, when dealing with valence wave functions so VR > 0 and since V < 0, |V + VR| < |V|. This is an aspect of the cancellation theorem.

e.Note also, by (3.287) that since VR projects |φ into the space of core wave functions it will not affect the valence eigenvalues as we have mentioned and will see in more detail later.

Since H = T + V where T is the kinetic energy operator and V is the potential energy, if we define the total pseudopotential Vp as

Vp = V +VR ,	(3.288)
then (3.286) can be written as
(T + Vp ) φ = E φ .	(3.289)

To derive further properties of the pseudopotential it is useful to develop the formulation of Austin et al. We start with the following five equations:

Hψn = Enψn (n = c or v),

H pφn = (H + VR )φn = Enφn (allowing for several φ),

VRφ = ∑c Fc φ ψc ,

where note Fc is arbitrary so VR is not yet specified.

φc = ∑c′αcc′ψc′ + ∑vαvcψv ,

φv = ∑cαcvψc + ∑v′αvv′ψv′ .

Combining (3.291) with n = c and (3.293), we obtain

(H + VR )(∑c′αcc′ψc′ + ∑vαvcψv ) = En (∑c′αcc′ψc′ + ∑vαvcψv ) .

Using (3.283), we have

∑c′αcc′Ec′ψc′ + ∑vαvc Evψv + ∑c′αcc′VRψc′ + ∑vαvcVRψv

=Ec (∑c′αcc′ψc′ + ∑vαvcψv ).

(3.290)

(3.291)

(3.292)

(3.293)

(3.294)

(3.295)

(3.296)

Using (3.292), this last equation becomes

	∑c′αcc′Ec′ψc′ + ∑vαvc Evψv + ∑c′αcc′∑c	Fc ψc′ ψc	(3.297)
	+ ∑vαvc ∑c Fc ψv ψc = Ec (∑c′αcc′ψc′ + ∑vαvcψv ).

3.2 One-Electron Models 199

This can be recast as

∑c′c′′[(Ec′ − Ec )δcc′′′ + Fc′ ψc′′ ]αcc′′ψc′	(3.298)
+ ∑c′∑vαvc Fc′ ψv ψc′ + ∑vαvc (Ev − Ec )ψv = 0.

Taking the inner product of (3.298) with ψv′ gives

∑vαvc (Ev − Ec )δvv′ = 0 or αvc′(Ev′ − Ec ) = 0 or αvc′ = 0 .

unless there is some sort of strange accidental degeneracy. We shall ignore such degeneracies. This means by (3.293) that

φc = ∑c′αcc′ψc′ .

(3.299)

Equation (3.298) becomes

∑c′c′′[(Ec′ − Ec )δcc′′′ + Fc′ ψc′′ ]αcc′′ψc′ = 0 .

(3.300)

Taking the matrix element of (3.300) with the core state ψc and summing out a resulting Kronecker delta function, we have

∑c′′[(Ec − Ec )δcc′′ + Fc ψc′′ ]αcc′′′ = 0 .

(3.301)

For nontrivial solutions of (3.301), we must have

det[(E	c	− E	c	)δ c′′ + F ψ		c′′	] = 0 .	(3.302)
				c	c

The point to (3.302) is that the “core” eigenvalues E¯c are formally determined. Combining (3.291) with n = v, and using φv from (3.294), we obtain

(H +VR )(∑cαcvψc + ∑v′αvv′ψv′) = Ev (∑cαcvψc + ∑v′αvv′ψv′) .

By (3.283) this becomes

∑cαcv Ecψc + ∑v′αvv′Ev′ψv′ + ∑cαcvVRψc + ∑v′αvv′VRψv′ = Ev (∑cαcvψc + ∑v′αvv′ψv′).

Using (3.292), this becomes

	∑cαcv (Ec − Ev )ψc + ∑v′αvv′(Ev′ − Ev )ψv′	(3.303)
	+ ∑cαcv ∑c Fc ψc ψc′ + ∑v′αvv′∑c Fc ψv′ ψc = 0.
	With a little manipulation we can write (3.303) as
	∑c,c′[(Ec − Ev )δcc′ + Fc ψc′ ]αcv′ψc
	+ ∑cαvv Fc ψv ψc + ∑v′(≠v),cαvv′ Fc ψv′ ψc	(3.304)
	+ (Ev − Ev )αvvψv + ∑v′(≠v) (Ev′ − Ev )αvv′ψv′ = 0.

200 3 Electrons in Periodic Potentials

Taking the inner product of (3.304) with ψv and ψv″, we find
(Ev −			v )αvv = 0 ,		(3.305)
	E
and
(Ev′′ −				v )αvv′′ = 0 .	(3.306)
		E
This implies that Ev = E¯v and
αvv′′ = 0 .

The latter result is really true only in the absence of degeneracy in the set of Ev. Combining with (3.294), we have (if αvv = 1)

φv =ψv + ∑cαcvψc .	(3.307)
Equation (3.304) can now be written
∑c′[(Ec′′ − Ev )δcc′′′ + Fc′′ ψc′ ]αcv′ = − Fc′′ ψv .	(3.308)

With these results we can understand the general pseudopotential theorem as given by Austin at al.:

The pseudo-Hamiltonian HP = H + VR, where VRφ = ∑c Fc|φ ψc, has the same valence eigenvalues Ev as H does. The eigenfunctions are given by (3.299) and

(3.307).

We get a particularly interesting form for the pseudopotential if we choose the arbitrary function to be

F = −Vψ	c	.	(3.309)
c
In this case
VRφ = −∑c ψc V φ ψc ,			(3.310)
and thus the pseudo-Hamiltonian can be written
H pφn = (T +V +VR )φn = Tφn +Vφn − ∑cψc ψc Vφn .			(3.311)
Note that by completeness
Vφn = ∑m amψm
= ∑mψm ψm Vφn		,
= ∑cψc ψc Vφn + ∑vψv ψv Vφn
so
Vφn − ∑cψc ψc Vφn = ∑vψv ψv Vφn .			(3.312)

If the ψc are almost a complete set for Vφn, then the right-hand side of (3.312) is very small and hence

H φ	Tφ	n	.	(3.313)
p n

3.2 One-Electron Models 201

This is another way of looking at the cancellation theorem. Notice this equation is just the free-electron approximation, and, furthermore, Hp has the same eigenvalues as H. Thus we see how the nearly free-electron approximation is partially justified by the pseudopotential.

Physically, the use of a pseudopotential assures us that the valence wave functions are orthogonal to the core wave functions. Using (3.307) and the orthonormality of the core and valence eigenfunction, we can write

ψv = φv − ∑c ψc	ψc φv	(3.314)
≡ (I − ∑c ψc	ψc ) φv .	(3.315)

The operator (I − ∑c|ψc ψc|) simply projects out from |φv all components that are perpendicular to |ψc . We can crudely say that the valence electrons would have to wiggle a lot (and hence raise their energy) to be in the vicinity of the core and also be orthogonal to the core wave function. The valence electron wave functions have to be orthogonal to the core wave functions and so they tend to stay out of the core. This effect can be represented by an effective repulsive pseudopotential that tends to cancel out the attractive core potential when we use the effective equation for calculating volume wave functions.

Since VR can be constructed so as to cause V + VR to be small in the core region, the following simplified form of the pseudopotential VP is sometimes used.

	V	P	(r) = −	Ze	for r > r
				4πε0r	core	(3.316)
	VP (r) = 0				for r ≤ rcore

This is sometimes called the empty-core pseudopotential or empty-core method (ECM).

Cohen [3.12, 3.13], has developed an empirical pseudopotential model (EPM) that has been very effective in relating band-structure calculations to optical properties. He expresses Vp(r) in terms of Fourier components and structure factors (see [3.12, p. 21]). He finds that only a few Fourier components need be used and fitted from experiment to give useful results. If one uses the correct nonlocal version of the pseudopotential, things are more complicated but still doable [3.12, p. 23]. Even screening effects can be incorporated as discussed by Cohen and Heine [3.13].

Note that the pseudopotential can be broken up into different core angular momentum components (where the core wave functions are expressed in atomic form). To see this, write

c = N, L ,

202 3 Electrons in Periodic Potentials

where N is all the quantum number necessary to define c besides L. Thus

VR = ∑c c (E − Ec ) c

= ∑L (∑N N, L (E − EN ,L ) N, L ).

This may help in finding simplified calculations.

For further details see Chelikowsky and Louie [3.10]. This is a Festschrift in honor of Marvin L. Cohen. This volume shows how the calculations of Cohen and his school intertwine with experiment: in many cases explaining experimental results, and in other cases predicting results with consequent experimental verification. We end this discussion of pseudopotentials with a qualitative roundup.

Table 3.4. Band structure and related references

Band-structure calculational	Reference	Comments
techniques

Nearly free electron methods	3.2.3
(NFEM)
Tight binding/LCAO methods	3.2.3
(TBM)
Wigner–Seitz method	[3.57], 3.2.3

Perturbed electron gas of free electrons

Starts from atomic nature of electron states.

First approximate quantitative solution of wave equation in crystal.

Augmented plane wave and	[3.16], [63], 3.2.3	Muffin tin potential with spherical
related methods (APW)		wave functions inside and plane
		wave outside (Slater).
Orthogonalized plane wave	Jones [58] Ch. 6,	Basis functions are plane waves plus
methods (OPW)	[3.58], 3.2.3	core wave functions (Herring).
		Related to pseudopotential.
Empirical pseudopotential	[3.12, 3.20]	Builds in orthogonality to core with
methods (EPM) as well as		a pseudopotential.
Self-consistent and ab initio
pseudopotential methods
Kohn–Korringa–Rostocker or	[3.26]	Related to APW.
KKR Green function methods
Kohn–Sham density	[3.23, 3.25, 3.27,	For calculating ground-state
functional Techniques (for	3.28]	properties.
many-body properties)
k · p Perturbation Theory	[3.5, 3.16, 3.26],	An interpolation scheme.
	3.2.3
G. W. approximation	[3.2]	G is for Green’s function, W for
		Coulomb interaction, Evaluates self-
		energy of quasi-particles.
General reference	[3.1, 3.37]

3.2 One-Electron Models 203

As already mentioned, M. L. Cohen’s early work (in the 1960s) was with the empirical pseudopotential. In brief review, the pseudopotential idea can be traced back to Fermi and is clearly based on the orthogonalized plane wave (OPW) method of Conyers Herring. In the pseudopotential method for a solid, one considers the ion cores as a background in which the valence electrons move. J. C. Phillips and L. Kleinman demonstrated how the requirement of orthogonality of the valence wave function to core atomic functions could be folded into the potential. M. L. Cohen found that the pseudopotentials converged rapidly in Fourier space, and so only a few were needed for practical calculations. These could be fitted from experiment (reflectivity for example), and then the resultant pseudopotential was very useful in determining the optical response – this method was particularly useful for several semiconductors. Band structures, and even electron–phonon interactions were usefully determined in this way. M. L. Cohen and his colleagues have continually expanded the utility of pseudopotentials. One of the earliest extensions was to an angular-momentum-dependent nonlocal pseudopotential, as discussed above. This was adopted early on in order to improve the accuracy, at the cost of more computation. Of course, with modern computers, this is not much of a drawback.

Nowadays, one often uses a pseudopotential-density functional method. One can thus develop ab initio pseudopotentials. The density functional method (in say the local density approximation – LDA) allows one to treat the electron–electron interaction in the core of the atom quite accurately. As we have already shown, the density functional method reduces a many-electron problem to a set of oneelectron equations (the Kohn–Sham equations) in a rational way. Morrel Cohen (another pioneer in the elucidation of pseudopotentials, see Chap. 23 of Chelikowsky and Louie, op cit) has said, with considerable truth, that the Kohn– Sham equations taught us the real meaning of our one-electron calculations. One then uses the pseudopotential to treat the interaction between the valence electrons and the ion core. Again as noted, the pseudopotential allows us to understand why the electron–ion core interaction is apparently so small. This combined pseudopotential-density functional approach has facilitated good predictions of ground-state properties, phonon vibrations, and structural properties such as phase transitions caused by pressure.

There are still problems that need additional attention, such as the correct prediction of bandgaps, but it should not be overlooked that calculations on real materials, not “toy” models are being considered. In a certain sense, M. L. Cohen and his colleagues are developing a “Standard Model of Condensed Matter Physics." The Holy Grail is to feed in only information about the constituents, and from there, at a given temperature and pressure, to predict all solid-state properties. Perhaps at some stage one can even theoretically design materials with desired properties. Along this line, the pseudopotential-density functional method is now being applied to nanostructures such as arrays of quantum dots (nanophysics, quantum dots, etc. are considered in Chap. 12 of Chelikowsky and Louie).

We have now described in some detail the methods of calculating the E(k) relation for electrons in a perfect crystal. Comparisons of actual calculations with