Patterson, Bailey - Solid State Physics Introduction to theory

154 3 Electrons in Periodic Potentials

In the limit as the potential becomes extremely weak, μ → 0, so that kal ≡ ral. Using (3.147), one easily sees that the energies are given by

	E =	2k 2	.	(3.150)
		2m

Equation (3.150) is just what one would expect. It is the free-particle solution.

In the limit as the potential becomes extremely strong, μ → ∞, we can have solutions of (3.148) only if sin ra1 = 0. Thus ral = nπ, where n is an integer, so that the energy is given by

2		2	2
E =	n π				.	(3.151)
		1		2
	2m(a )

Equation (3.151) is expected as these are the “particle-in-a-box” solutions.

It is also interesting to study how the widths of the energy bands vary with the strength of the potential. From (3.148), the edges of the bands of allowed energy occur when P(ral) = ±1. This can certainly occur when ral = nπ. The other values of ral at the band edges are determined in the argument below. At the band edges,

1

μ

1

±

1 = cos ra

+

sin(ra ) .

1

ra

This equation can be recast into the form,

0 =1+

μ

sin(ra1)

.

(3.152)

1

1

ra

1+ cos(ra )

From trigonometric identities

tan ra1

=

sin(ra1)

,

(3.153)

1 + cos(ra1)

2

and

cot ra1

=

sin(ra1)

.

(3.154)

2

1− cos(ra1)

Combining the last three equations gives

0 =1+

μ

tan ra1

or

0 =1−

μ

cot ra1

,

ra1

ra1

2

2

or

tan(ra1 / 2) = −(ra1) / μ,

cot(ra1 / 2) = +(ra1) / μ .

3.2 One-Electron Models 155

Since 1/tan θ = cot θ, these last two equations can be written

cot(ra1 / 2) = −μ /(ra1), tan(ra1 / 2) = +μ /(ra1),

or

(ra1 / 2) cot(ra1 / 2) = −ma1(au) / 2	2 ,	(3.155)
and
(ra1 / 2) tan(ra1 / 2) = +ma1(au) / 2	2 .	(3.156)

Figure 3.6 uses ral = nπ, (3.155), and (3.156) (which determine the upper and lower ends of the energy bands) to illustrate the variation of bandwidth with the strength of the potential.

				1						1
	ra					tan ra
		2							2
ma1au
	2	2
											π	2π	3π	ra1
−	ma1au
	2	2
			1						1
	ra				cot ra
		2						2

Fig. 3.6. Variation of bandwidth with strength of the potential

Note that increasing u decreases the bandwidth of any given band. For a fixed u, the higher r (or the energy) is, the larger is the bandwidth. By careful analysis it can be shown that the bandwidth increases as al decreases. The fact that the bandwidth increases as the lattice spacing decreases has many important consequences as it is valid in the more important three-dimensional case. For example, Fig. 3.7 sketches the variation of the 3s and 3p bonds for solid sodium. Note that at the equilibrium spacing a0, the 3s and 3p bands form one continuous band.

156 3 Electrons in Periodic Potentials

E

					3p
					3s
0.5	1.0	1.5	2.0	2.5	3.0	a/a0

Fig. 3.7. Sketch of variation (with distance between atoms) of bandwidths of Na. Each energy unit represents 2 eV. The equilibrium lattice spacing is a0. Higher bands such as the 4s and 3d are left out

The concept of the effective mass of an electron is very important. A simple example of it can be given within the context of the Kronig–Penney model. Equation (3.148) can be written as

coska1 = P(ra1) .

Let us examine this equation for small k and for r near r0 (= r at k = 0). By a Taylor series expansion for both sides of this equation, we have

1− 12 (ka1)2 =1+ P0′a1(r − r0 ) ,

or

− 1 k 2a1 = r0 2 P0′ r .

Squaring both sides and neglecting terms in k4, we have

r2 = r02 − r0 k 2a′1 .

P0

Defining an effective mass m* as

m = − mP0′ , r0a1

3.2 One-Electron Models 157

we have by (3.147) that

	E =	2r2	= E0 +	2k 2	,	(3.157)
		2m		2m

where E0 = ћ2r02/2m. Except for the definition of mass, this equation is just like an equation for a free particle. Thus for small k we may think of m* as acting as a mass; hence it is called an effective mass. For small k, at any rate, we see that the only effect of the periodic potential is to modify the apparent mass of the particle.

The appearances of allowed energy bands for waves propagating in periodic lattices (as exhibited by the Kronig–Penney model) is a general feature. The physical reasons for this phenomenon are fairly easy to find.

E

a b

c

d

e

f

Fig. 3.8. Wave propagating through periodic potential. E is the kinetic energy of the particle with which there is associated a wave with de Broglie wavelength λ = h/(2mE)1/2 (internal reflections omitted for clarity)

Consider a quantum-mechanical particle moving along with energy E as shown in Fig. 3.8. Associated with the particle is a wave of de Broglie wavelength λ. In regions a–b, c–d, e–f, etc., the potential energy is nonzero. These regions of “hills” in the potential cause the wave to be partially reflected and partially transmitted. After several partial reflections and partial transmissions at a–b, c–d, e–f, etc., it is clear that the situation will be very complex. However, there are two possibilities. The reflections and transmissions may or may not result in destructive interference of the propagating wave. Destructive interference will result in attenuation of the wave. Whether or not we have destructive interference depends clearly on the wavelength of the wave (and of course on the spacings of the “hills” of the potential) and hence on the energy of the particle. Hence we see qualitatively, at any rate, that for some energies the wave will not propagate because of attenuation. This is what we mean by a disallowed band of energy. For other energies, there will be no net attenuation and the wave will propagate. This is what we mean by an allowed band of energy. The Kronig–Penney model calculations were just a way of expressing these qualitative ideas in precise quantum-mechanical form.

158 3 Electrons in Periodic Potentials

3.2.2 The Free-Electron or Quasifree-Electron Approximation (B)

The Kronig–Penney model indicates that for small |ka1| we can take the periodic nature of the solid into account by using an effective mass rather than an actual mass for the electrons. In fact we can always treat independent electrons in a periodic potential in this way so long as we are interested only in a group of electrons that have energy clustered about minima in an E versus k plot (in general this would lead to a tensor effective mass, but let us restrict ourselves to minima such that E k2 + constant near the minima). Let us agree to call the electrons with effective mass quasifree electrons. Perhaps we should also include Landau’s ideas here and say that what we mean by quasifree electrons are Landau quasiparticles with an effective mass enhanced by the periodic potential. We will often use m rather than m*, but will have the idea that m can be replaced by m* where convenient and appropriate. In general, when we actually use a number for the effective mass it is necessary to quote what experiment the effective mass comes from. Only in this way do we know precisely what we are including. There are many interactions beyond that due to the periodic lattice that can influence the effective mass of an electron. Any sort of interaction is liable to change the effective mass (or “renormalize it”). It is now thought that the electron–phonon interaction in metals can be important in determining the effective mass of the electrons.

The quasifree-electron model is most easily arrived at by treating the conduction electrons in a metal by the Hartree approximation. If the positive ion cores are smeared out to give a uniform positive background charge, then the interaction of the ion cores with the electrons exactly cancels the interactions of the electrons with each other (in the Hartree approximation). We are left with just a one-electron, free-electron Schrödinger equation. Of course, we really need additional ideas (such as discussed in Sect. 3.1.4 and in Sect. 4.4 as well as the introduction of Chap. 4) to see why the electrons can be thought of as rather weakly interacting, as seems to be required by the “uncorrelated” nature of the Hartree approximation. Also, if we smear out the positive ion cores, we may then have a hard time justifying the use of an effective mass for the electrons or indeed the use of a periodic potential. At any rate, before we start examining in detail the effect of a three-dimensional lattice on the motion of electrons in a crystal, it is worthwhile to pursue the quasifree-electron picture to see what can be learned. The picture appears to be useful (with some modifications) to describe the motions of electrons in simple monovalent metals. It is also useful for describing the motion of charge carriers in semiconductors. At worst it can be regarded as a useful phenomenological picture.11

Density of States in the Quasifree-Electron Model (B)

Probably the most useful prediction made by the quasifree-electron approximation is a prediction regarding the number of quantum states per unit energy. This

11 See also Kittel C [59, 60].

3.2 One-Electron Models 159

quantity is called the density of states. For a quasifree electron with effective mass m*,

	2
−	2m 2ψ = Eψ .	(3.158)

This equation has the solution (normalized in a volume V)

ψ =		1			exp(ik r) ,			(3.159)
			V
provided that
E =		2		(k 2		+ k 2	+ k 2 ) .	(3.160)
	2m				1	2	3

If periodic boundary conditions are applied on a parallelepiped of sides Niai and volume V, then k is of the form

	n		n			n
k = 2π	1	b +	2		b +	3		b		,	(3.161)
	N
		1	N	2	2	N	3	3
	1

where the ni are integers and the bi are the customary reciprocal lattice vectors that are defined from the ai. (For the case of quasifree electrons, we really do not need the concept of reciprocal lattice, but it is convenient for later purposes to carry it along.) There are thus N1N2N3 k-type states in a volume (2π)3b1 (b2 × b3) of k space. Thus the number of states per unit volume of k space is

N1N2 N3			=	N1N2 N3Ωa	=	V	,	(3.162)
(2π)3b (b ×b )						(2π)3
				(2π)3
1	2	3

where Ω = a1 (a2 × a3). Since the states in k space are uniformly distributed, the number of states per unit volume of real space in d3k is

d3k /(2π)3 .

(3.163)

If E = ћ2k2/2m*, the number of states with energy less than E (with k defined by this equation) is

4π	| k |3	V		Vk3
			=	6π 2 ,
3		(2π)3

where |k| = k, of course. Thus, if N(E) is the number of states in E to E + dE, and N(k) is the number of states in k to k + dk, we have

	d		3		Vk	2
N (E)dE = N (k)dk =		Vk		dk =			dk .
					2π 2
	dk 6π 2

160 3 Electrons in Periodic Potentials

Table 3.2. Dependence of density of states of free electrons D(E) on dimension and energy E.

D(E)

One Dimension

A1 E–1/2

Two Dimensions

A2

Three Dimensions

A3 E1/2

Note that the Ai are constants, and in all

cases the dispersion relation is of the

form Ek = 2k2/(2m*).

But

2

m dE

dE =

kdk, so dk =

2 k ,

m

or

N (E)dE =

V

2m E m

dE ,

2π

2

2

2

or

V

2m

3 2

N (E)dE =

E1 2dE .

(3.164)

2

2

4π

Equation (3.164) is the basic equation for the density of states in the quasifreeelectron approximation. If we include spin, there are two spin states for each k, so (3.164) must be multiplied by 2.

Equation (3.164) is most often used with Fermi–Dirac statistics. The Fermi function f(E) tells us the average number of electrons per state at a given temperature, 0 ≤ f(E) ≤ 1. With Fermi–Dirac statistics, the number of electrons per unit volume with energy between E and E + dE and at temperature T is

K EdE
dn = f (E)K EdE = exp[(E − EF ) / kT ] +1	,	(3.165)
where K = (1/2π2)(2m*/ 2)3/2 and EF is the Fermi energy.
If there are N electrons per unit volume, then EF is determined from
N = ∫∞ K E f (E)dE .		(3.166)
0

3.2 One-Electron Models 161

Once the Fermi energy EF is obtained, the mean energy of an electron gas is determined from

E = ∫∞ Kf (E) E EdE .	(3.167)
0

We shall find (3.166) and (3.167) particularly useful in the next Section where we evaluate the specific heat of an electron gas. We summarize the density of states for free electrons in one, two, and three dimensions in Table 3.2.

Specific Heat of an Electron Gas (B)

This Section and the next one follow the early ground-breaking work of Pauli and Sommerfeld. In this Section all we have to do is to find the Fermi energy from (3.166), perform the indicated integral in (3.167), and then take the temperature derivative. However, to perform these operations exactly is impossible in closed form and so it is useful to develop an approximate way of evaluating the integrals in (3.166) and (3.167). The approximation we will use will be an excellent approximation for metals at all ordinary temperatures.

We first develop a general formula (the Sommerfeld expansion) for the evaluation of integrals of the needed form for “low” temperatures (room temperature qualifies as a very low temperature for the approximation that we will use).

Let f(E) be the Fermi distribution function, and R(E) be a function that vanishes when E vanishes. Define

S = +∫∞ f (E) dR(E) dE
0	dE
= −∫∞ R(E) df (E) dE.
0	dE

(3.168)

(3.169)

At low temperature, f ′(E) has an appreciable value only where E is near the Fermi energy EF. Thus we make a Taylor series expansion of R(E) about the Fermi energy:

R(E) = R(E

F

) + (E − E

F

)R′(E

F

) +

1

(E − E

F

)2 R′′(E

F

) + . (3.170)

2

In (3.170) R″(EF) means

2

.

d

R(E)

dE

2

E =EF

Combining (3.169) and (3.170), we can write

S aR(EF ) + bR′(EF ) + cR′′(EF ) ,

(3.171)

162 3 Electrons in Periodic Potentials

where

			a = −∫∞ f ′(E)dE =1,
			0
			b = −∫∞ (E − EF ) f ′(E)dE = 0,
			0
c = −	1	∫∞ (E − EF )2 f ′(E)dE kT 2		∫∞	x2exdx	=	π 2
							6
2		0	2	−∞ (ex +1)2
Thus we can write
∫∞ f			(E) dR(E) dE = R(EF ) + π 2 (kT )2 R′′(EF ) +
0			dE	6
By (3.166),

(kT )2.

. (3.172)

N = ∫∞ K	d		2	E3 2 f (E)dE	2	KEF3 2	+	π 2	(kT )2 K	1 . (3.173)
	dE 3							6
0				3					2	EF

At absolute zero temperature, the Fermi function f(E) is 1 for 0 ≤ E ≤ EF(0) and zero otherwise. Therefore we can also write

N = ∫	EF (0)		1 2		2	3 2
		KE		dE =		K[EF (0)]	.	(3.174)
					3
0

Equating (3.173) and (3.174), we obtain

[EF (0)]3 2 E3 2 + π 2 (kT )2 .

F 8 EF

Since the second term is a small correction to the first, we can let EF = EF(0) in the second term:

[E		(0)]3 2			π 2	(kT )2				E3 2 .
	F		1	−
					8	[E		(0)]2
							F			F

Again, since the second term is a small correction to the first term, we can use (1 − ε)3/2 ≡ 1 − 3/2ε to obtain

			π	2		kT		2
EF	= EF (0) 1	−	12						.	(3.175)
					EF (0)

For all temperatures that are normally of interest, (3.175) is a good approximation for the variation of the Fermi energy with temperature. We shall need this expression in our calculation of the specific heat.

3.2 One-Electron Models 163

The mean energy E¯ is given by (3.167) or

E = ∫∞ f (E)

d

[ 2 K (E)5 / 2

]dE

2K

EF5 / 2 +

π 2

(kT )2 3K

EF . (3.176)

0

dE

5

5

6

2

Combining (3.176) and (3.175), we obtain

2K

5 / 2

5 / 2 π 2

kT

2

E

+ [EF

[EF (0)]

(0)]

K

.

5

6

EF (0)

The specific heat of the electron gas is then the temperature derivative of E¯:

C	= ∂E	= π 2	k 2K E	F	(0) T .
V	∂T	3

This is commonly written as

CV = γT ,

where

γ =	π 2	k	2	K EF (0) .
	3

There are more convenient forms for γ. From (3.174), K = 32 N[EF (0)]−32 ,

so that

	γ =	π 2	Nk	k	.
		2		EF (0)

The Fermi temperature TF is defined as TF = EF(0)/k so that

γ π 2 Nk . 2 TF

(3.177)

(3.178)

(3.179)

The expansions for E¯ and EF are expansions in powers of kT/EF(0). Clearly our results (such as (3.177)) are valid only when kT << EF(0). But as we already mentioned, this does not limit us to very low temperatures. If 1/40 eV corresponds to 300° K, then EF(0) 1 eV (as for metals) corresponds to approximately 12 000° K. So for temperatures well below 12 000° K, our results are certainly valid.

A similar calculation for the specific heat of a free electron gas using Hartree– Fock theory yields Cv (T/lnT), which is not even qualitatively correct. This shows that Coulomb correlations really do have some importance, and our freeelectron theory does well only because the errors (involved in neglecting both Coulomb corrections and exchange) approximately cancel.