Patterson, Bailey - Solid State Physics Introduction to theory

104 2 Lattice Vibrations and Thermal Properties

The Strain Tensor (εij) (MET, MS)

Consider infinitesimal and uniform strains and let i, j, k be a set of orthogonal axes in the unstrained crystal. Under strain, they will go to a not necessarily orthogonal set i′, j′, k′. We define εij so

i′ = (1 + εxx )i + εxy j + εxz k ,	(2.254a)
j′ = ε yx i + (1 + ε yy ) j + ε yz k ,	(2.254b)
k′ = εzx i + εzy j + (1 + εzz )k .	(2.254c)

Let r represent a point in an unstrained crystal that becomes r′ under uniform infinitesimal strain.

r = xi + yj + zk ,

r′ = xi′ + yj′ + zk′.

Let the displacement of the point be represented by u = r′ – r, so

ux = xεxx + yε yx + zεzx ,

u y = xεxy + yε yy + zεzy ,

uz = xεxz + yε yz + zεzz .

We define the strain components in the following way exx = ∂∂uxx ,

eyy = ∂∂uyy ,

ezz = ∂∂uzz ,

			1		∂u	x		∂uy
e	xy	=					+			,
			2		∂y			∂x

(2.255a)

(2.255b)

(2.256a)

(2.256b)

(2.256c)

(2.257a)

(2.257b)

(2.257c)

(2.257d)

1

∂uy

∂u

e

yz

=

+

z

,

(2.257e)

2

∂z

∂y

ezx

=

1

∂u

z

+

∂u

(2.257f)

2

x .

∂x

∂z

2.3 Three-Dimensional Lattices	105

The diagonal components are the normal strain and the off-diagonal components are the shear strain. Pure rotations have not been considered, and the strain tensor (eij) is symmetric. It is a tensor as it transforms like one. The dilation, or change in volume per unit volume is,

	θ =	δV	= i′ ( j′×k′) = exx + eyy + ezz .	(2.258)
		V

Due to symmetry there are only 6 independent stress, and 6 independent strain components. The six component stresses and strains may be defined by:

σ1 = σ xx ,		(2.259a)
σ2 = σ yy ,		(2.259b)
σ3 = σ zz ,		(2.259c)
σ4 = σ yz	= σ zy ,	(2.259d)
σ5 = σ xz	= σ zx ,	(2.259e)
σ6 = σ xy	= σ yx ,	(2.259f)
ε1 = exx ,		(2.260a)
ε2 = eyy ,		(2.260b)
ε3 = ezz ,		(2.260c)
ε4 = 2eyz = 2ezy ,		(2.260d)
ε5 = 2exz = 2ezx ,		(2.260e)
ε6 = 2exy = 2eyx .		(2.260f)

(The introduction of the 2 in (2.260 d, e, f) is convenient for later purposes).

Hooke’s Law (MET, MS)

The generalized Hooke’s law says stress is proportional to strain or in terms of the six-component representation:

σi = ∑6j=1cijε j , (2.261)

where the cij are the elastic constants of the crystal.

106 2 Lattice Vibrations and Thermal Properties

General Equation of Motion (MET, MS)

It is fairly easy, using Newton’s second law, to derive an expression relating the displacements ui and the stresses σij. Reference can be made to Ghatak and Kothari [2.16, pp 59-62] for details. If σiB denotes body force per unit mass in the direction i and if ρ is the density of the material, the result is

ρ	∂ 2u	i	= ρσiB + ∑ j	∂σij	.	(2.262)
	∂t 2			∂x j

In the absence of external body forces the term σiB, of course, drops out.

Strain Energy (MET, MS)

Equation (2.262) seems rather complicated because there are 36 cij. However, by looking at an expression for the strain energy [2.16, p 63-65] and by using (2.261) it is possible to show

	∂σ	i		∂ 2u
cij =			=	V	,	(2.263)
	∂ε j			∂ε j ∂εi

where uV is the potential energy per unit volume. Thus cij is a symmetric matrix and of the 36 cij, only 21 are independent.

Now consider only cubic crystals. Since the x-, y-, z-axes are equivalent,

c11 = c22 = c33	(2.264a)
and
c44 = c55 = c66 .	(2.264b)

By considering inversion symmetry, we can show all the other off-diagonal elastic constants are zero except for

c12 = c13 = c23 = c21 = c31 = c32 .

Thus there are only three independent elastic constants,22 which can be represented as:

	c		c	c	0	0	0
		11	12	12
	c12		c11	c12	0	0	0
c	c		c	c	0	0	0		(2.265)
	=	12	12	11				.
ij		0	0	0	c44	0	0
		0	0	0	0	c44	0
		0	0	0	0	0	c44

22If one can assume central forces Cauchy proved that c12 = c44, however, this is not a good approximation in real materials.

2.3 Three-Dimensional Lattices	107

Equations of Motion for Cubic Crystals (MET, MS)

From (2.262) (with no external body forces)

ρ

∂2u

i

= ∑ j

∂σij

=

∂σ

xx

+

∂σ xy

+

∂σ

xz

,

∂t 2

∂x j

∂y

∂x

∂x

but

σxx =σ1 = c11ε1 + c12ε2 + c13ε3

=(c11 − c12 )ε1 + c12 (ε1 +ε2 +ε3) ,

σxy = σ6 = c44ε6 ,

σxz = σ5 = c44ε5 .

(2.266)

(2.267a)

(2.267b)

(2.267c)

Using also (2.257), and combining with the above we get an equation for ∂2ux/∂t2. Following a similar procedure we can also get equations for ∂2uy/∂t2 and ∂2uz/∂t2. Seeking solutions of the form

u j

= K jei(k v −ωt)

(2.268)

for j = 1, 2, 3 or x, y, z, we find nontrivial solutions only if

− c

)k

2

(c

44

x

(c12

+ c44 )kxk y

(c12

+ c44 )kxkz

11

+ c

44

k 2 − ρω2

2

(c

+ c

)k

k

(c

+ c

)k

k

= 0 . (2.269)

44

y

x

(c11

− c44 )k y

44

y

z

12

k 2 − ρω2

12

+ c

44

2

− c

)k

(c12 + c44 )kzkx

(c12

+ c44 )kzk y

(c

44

z

11

+ c

44

k 2 − ρω2

Suppose the wave travels along the x direction so ky = kz = 0. We then find the three wave velocities:

v = c11		, v	2	= v	3	= c44	(degenerate) .	(2.270)
1	ρ					ρ

vl is a longitudinal wave and v2, v3 are the two transverse waves. Thus, one way of determining these elastic constants is by measuring appropriate wave velocities. Note that for an isotropic material c11 = c12 + 2c44 so v1 > v2 and v3. The longitudinal sound wave is greater than the transverse sound velocity.

108 2 Lattice Vibrations and Thermal Properties

Problems

2.1Find the normal modes and normal-mode frequencies for a three-atom “lattice” (assume the atoms are of equal mass). Use periodic boundary conditions.

2.2Show when m and m′ are restricted to a range consistent with the first Brillouin zone that

	1	2πi			m′
		∑n exp		(m − m′)n	= δm	,
	N		N

where δmm′ is the Kronecker delta.

2.3Evaluate the specific heat of the linear lattice (given by (2.80)) in the low temperature limit.

2.4Show that Gmn = Gnm, where G is given by (2.100).

2.5This is an essay length problem. It should clarify many points about impurity modes. Solve the five-atom lattice problem shown in Fig. 2.14. Use periodic boundary conditions. To solve this problem define A = β/αand δ= m/M (αand

β are the spring constants) and find the normal modes and eigenfrequencies. For each eigenfrequency, plot mω2/αversus δfor A = 1 and mω2/αversus A for δ = 1. For the first plot: (a) The degeneracy at δ = 1 is split by the presence of the impurity. (b) No frequency is changed by more than the distance to the next unperturbed frequency. This is a general property. (c) The frequencies that are unchanged by changing δcorrespond to modes with a node at the impurity (M).

(d) Identify the mode corresponding to a pure translation of the crystal. (e) Identify the impurity mode(s). (f) Note that as we reduce the mass of M, the frequency of the impurity mode increases. For the second plot: (a) The degeneracy at A = 1 is split by the presence of an impurity. (b) No frequency is changed more than the distance to the next unperturbed frequency. (c) Identify the pure translation mode. (d) Identify the impurity modes. (e) Note that the frequencies of the impurity mode(s) increase with β.

m	m	M	m	m
X1	α	β	β	α
	X2	X3	X4	X5

Fig. 2.14. The five-atom lattice

2.6 Let aq and a†q be the phonon annihilation and creation operators. Show that

[a	q	, a 1 ] = 0	and [a†	, a†	] = 0 .
		q	q	q1

Problems 109

2.7From the phonon annihilation and creation operator commutation relations derive that

aq† nq = nq +1 nq +1 ,

and

aq nq = nq nq −1 .

2.8If a1, a2, and a3 are the primitive translation vectors and if Ωa ≡ a1 (a2×a3), use the method of Jacobians to show that dx dy dz = Ωa dη1 dη2 dη3, where x, y, z are the Cartesian coordinates and η1, η2, and η3 are defined by r = η1a1 +

η2a2 +η3a3.

2.9Show that the bi vectors defined by (2.172) satisfy

Ωab1 = a2 ×a3 , Ωab2 = a3 ×a1, Ωa b3 = a1 ×a2 , where Ωa = a1 (a2 × a3).

2.10If Ωb = b1 (b2 × b3), Ωa = a1 (a2 × a3), the bi are defined by (2.172), and the ai are the primitive translation vectors, show that Ωb = 1/Ωa.

2.11This is a long problem whose results are very important for crystal mathematics. (See (2.178)–(2.184)). Show that

a)	1	∑R exp(iq Rl ) = ∑G		δq,G		,
	N1N2 N3				n
		l	n

where the sum over Rl is a sum over the lattice.

b)	1	∑	q	exp(iq Rl ) = δ R ,0 ,
	N1N2 N3			l

where the sum over q is a sum over one Brillouin zone.

c)In the limit as Vf.p.p. → ∞ (Vf.p.p. means the volume of the parallelepiped representing the actual crystal), one can replace

			∑q f (q)			by	Vf.p.p.	∫ f (q)d3q .
							(2π)3
d)			Ωa		∫	exp(iq R )d3q = δ			R ,0	,
							l
			(2π)3 B.Z.						l
	where the integral is over one Brillouin zone.
e)			1	∫ exp[i(Gl′ −Gl ) r]d 3r = δl′,l ,
		Ωa

where the integral is over a unit cell.

110 2 Lattice Vibrations and Thermal Properties

f)	1				∫ exp[iq (r − r′)]d3q = δ (r − r′) ,
			(2π)3
	where the integral is over all of reciprocal space and δ(r − r′) is the
	Dirac delta function.
g)		1		∫Vf.p.p.→∞ exp[i(q − q′) r]d3r = δ (q − q′) .
		(2π)3

In this problem, the ai are the primitive translation vectors. N1a1, N2a2, and N3a3 are vectors along the edges of the fundamental parallelepiped. Rl defines lattice points in the direct lattice by (2.171). q are vectors in reciprocal space defined by (2.175). The Gl define the lattice points in the reciprocal lattice by (2.173). Ωa = a1 (a2 × a3), and the r are vectors in direct space.

2.12This problem should clarify the discussion of diagonalizing Hq (defined by 2.198). Find the normal mode eigenvalues and eigenvectors associated with

	mi xi = −∑3j=1γ ij x j ,
		k,
m1 = m3 = m, m2		− k,
	= M , and (γij ) =
		0,

A convenient substitution for this purpose is

xi = ui eiωt . mi

2.13 By use of the Debye model, show that

cv T 3	for	T <<θD
and
cv 3k(NK )	for	T >>θD .

− k,	0
2k,	− k .
− k,	k

Here, k = the Boltzmann gas constant, N = the number of unit cells in the fundamental parallelepiped, and K = the number of atoms per unit cell. Show that this result is independent of the Debye model.

2.14The nearest-neighbor one-dimensional lattice vibration problem (compare Sect. 2.2.2) can be exactly solved. For this lattice: (a) Plot the average number (per atom) of phonons (with energies between ω and ω + dω) versus ω for several temperatures. (b) Plot the internal energy per atom versus temperature. (c) Plot the entropy per atom versus temperature. (d) Plot the specific heat per atom versus temperature. [Hint: Try to use convenient dimensionless quantities for both ordinates and abscissa in the plots.]

Problems 111

↑

a1

↓

← a2 →

2.15Find the reciprocal lattice of the two-dimensional square lattice shown above.

2.16Find the reciprocal lattice of the three-dimensional body-centered cubic lattice. Use for primitive lattice vectors

a =

a

(xˆ

+ yˆ − zˆ),

a

2

=

a

(−xˆ

+ yˆ + zˆ),

a

3

=

a

(xˆ

− yˆ + zˆ) .

1

2

2

2

2.17Find the reciprocal lattice of the three-dimensional face-centered cubic lattice. Use as primitive lattice vectors

a =

a

(xˆ

+ yˆ),

a

2

=

a

( yˆ

+ zˆ),

a

3

=

a

( yˆ

+ xˆ) .

1

2

2

2

2.18Sketch the first Brillouin zone in the reciprocal lattice of the fcc lattice. The easiest way to do this is to draw planes that perpendicularly bisect vectors (in reciprocal space) from the origin to other reciprocal lattice points. The volume contained by all planes is the first Brillouin zone. This definition is equivalent to the definition just after (2.176).

2.19Sketch the first Brillouin zone in the reciprocal lattice of the bcc lattice. Problem 2.18 gives a definition of the first Brillouin zone.

2.20Find the dispersion relation for the two-dimensional monatomic square lattice in the harmonic approximation. Assume nearest-neighbor interactions.

2.21Write an exact expression for the heat capacity (at constant area) of the twodimensional square lattice in the nearest-neighbor harmonic approximation. Evaluate this expression in an approximation that is analogous to the Debye approximation, which is used in three dimensions. Find the exact highand low-temperature limits of the specific heat.

2.22Use (2.200) and (2.203), the fact that the polarization vectors satisfy

∑p eq*αpbeqβpb′ = δαβδbb′

(the α and β refer to Cartesian components), and

X 11−q†, p = X q11, p , P−11q†, p = Pq11, p .

112 2 Lattice Vibrations and Thermal Properties

(you should convince yourself that these last two relations are valid) to establish that

X q1, p = −i∑p	eq*, p,b (aq†, p − a−q, p ) .
	2mbωq, p

2.23Show that the specific heat of a lattice at low temperatures goes as the temperature to the power of the dimension of the lattice as in Table 2.5.

2.24Discuss the Einstein theory of specific heat of a crystal in which only one lattice vibrational frequency is considered. Show that this leads to a vanishing of the specific heat at absolute zero, but not as T cubed.

2.25In (2.270) show vl is longitudinal and v2, v3 are transverse.

2.26Derive wave velocities and physically describe the waves that propagate along the [110] directions in a cubic crystal. Use (2.269).

3 Electrons in Periodic Potentials

As we have said, the universe of traditional solid-state physics is defined by the crystalline lattice. The principal actors are the elementary excitations in this lattice. In the previous chapter we discussed one of these, the phonons that are the quanta of lattice vibration. Another is the electron that is perhaps the principal actor in all of solid-state physics. By an electron in a solid we will mean something a little different from a free electron. We will mean a dressed electron or an electron plus certain of its interactions. Thus we will find that it is often convenient to assign an electron in a solid an effective mass.

There is more to discuss on lattice vibrations than was covered in Chap. 2. In particular, we need to analyze anharmonic terms in the potential and see how these terms cause phonon–phonon interactions. This will be done in the next chapter. Electron–phonon interactions are also included in Chap. 4 and before we get there we obviously need to discuss electrons in solids. After making the Born– Oppenheimer approximation (Chap. 2), we still have to deal with a many-electron problem (as well as the behavior of the lattice). A way to reduce the manyelectron problem approximately to an equivalent one-electron problem1 is given by the Hartree and Hartree–Fock methods. The density functional method, which allows at least in principle, the exact evaluation of some ground-state properties is also important. In a certain sense, it can be regarded as an extension of the Hartree–Fock method and it has been much used in recent years.

After justifying the one-electron approximation by discussing the Hartree, Hartree–Fock, and density functional methods, we consider several applications of the elementary quasifree-electron approximation.

We then present the nearly free and tight binding approximations for electrons in a crystalline lattice. After that we discuss various band structure approximations. Finally we discuss some electronic properties of lattice defects. We begin with the variational principle, which is used in several of our developments.

1A much more sophisticated approach than we wish to use is contained in Negele and Orland [3.36]. In general, with the hope that this book may be useful to all who are entering solid-state physics, we have stayed away from most abstract methods of quantum field theory.