72 Phys520.nb
10 Energy Bands
Q: Why are there insulators and conductors? Q: What will happen when an electron moves in a crystal? In Chapter 1, we showed four different levels of approximations. 1. Ignore interactions between electrons, ignore ions, ignore quantum effects and treat electrons as classical particles: i.e. electrons form a (classical) ideal gas 2. Ignore interactions between electrons, ignore ions, keep quantum effects and treat electrons quantum fermion particles: i.e. electrons form an idea Fermi gas. In Ashcroft and Mermin, this is called the “free electron approximation” 3. Ignore interactions between electrons, keep the ions but assumes ions doesn’t move (the adiabatic approximation). Keep quantum effects: In Ashcroft and Mermin, this is called the “independent electron approximation”, but in other literature, this is often called the “free electron approximation”. 4. Keep interactions between electrons, keep the ions but assumes ions doesn’t move (the adiabatic approximation). Keep quantum effects. This is the Landau’s Fermi liquid theory (will be discussed in 540. At the end of the day, the final conclusions of the Landau Fermi liquid theory tells us that approximation #3 works very well). We explored the first two early on. The problem of approximations #1 and #2 lies in the fact that they always predict metallic behavior. But we know that there are insulating materials. Another problem of approximations #1 and #2 predictes that Hall resistivity give us carrier density. For metals with low valence numbers (e.g. 1 valence electron per unit cell), it usually works well. But for materials with more valence electrons, experiments shows a much lower carrier density, and often with a wrong sign (charge carreries in the system carry + charge instead of -). In this chapter, we will take the lattice background into consideration and see what will happen (approximation #3). Turns out that both the two problems mentioned above can be resolved within this approximation (#3).
10.1. Energy Bands and Band Gap
10.1.1. Phenomenon Consider a 1D solid. In the absence of a lattice background, the kinetic energy of one electron can take any positive values ϵ = p2 2 m > 0. Phys520.nb 73
In the presence of a lattice background, the kinetic energy (as a function of the momentum) breaks into pieces. Each piece is known as an energy band. Between two energy bands, there may be a forbidden region, which the energy of an electron can never enter. This forbidden region is called the band gap.
Fig. 1. The dispersion relation for (a) a free electron gas and (b) electrons moving in a crystal.
Fig. 2. The probability density for the ψ+ and ψ- waves.
10.1.2. Why are there two possible energy at k = π /a? Here we consdier the gap that appears at the momentum k = ±π/a. At these two momentum points, there are two linear-independent plane
1 1 waves ⅇⅈ π/a x and ⅇ-ⅈ π/a x. a a We can form two standing waves using these two plane waves
ⅇⅈ π/a x + ⅇ-ⅈ π/a x π ψ+(x) = = 2/a cos x (10.1) 2 a a 74 Phys520.nb
ⅇⅈ π/a x - ⅇ-ⅈ π/a x π ψ-(x) = = 2/a sin x (10.2) a 2 ⅈ The probability density is
2 π cos2 x for + ρ = ψ (x) 2 = a a ( ) ± ± 2 π 10.3 sin2 x for - a a In the absence of lattices, these two standing waves have exactly the same energy
π 2 2 2 ℏ2± 2 2 ℏ k a ℏ π ϵ = = = (10.4) 2 m 2 m 2 m a2 2 2 2 So, we have ϵ+ = ϵ- = ℏ π 2 m a
However, in the presence of the lattice, the ψ+ wave has a lower energy than ψ-, because the electron has a higher probably to appear near an ion for ψ+. This is the reason why at momentum k = π/a we have two possible energies (point A and B in figure 1). ◼ Similar band gaps will arise at k = n π/a for any integer n. π π π π ◼ The momentum region -n < k < -(n - 1) and (n - 1) < k < n is called the nth Brillouin zones (This is the same Brillouin zones as a a a a we learned in the reciprocal lattice). ◼ In side the of these Brillouin zones, the energy is a smooth function and this smooth function is called the nth band. ◼ At each boundary of the Brillouin zones, the energy curve shows a jump and thus an energy gap opens up.
10.1.3. Magnitude of the band gap The size of the gap can be estimated as the following
a a Eg = U(x) ρ+(x) ⅆ x - U(x) ρ-(x) ⅆ x (10.5) 0 0 where U(x) is the potential energy from the ions. Because the lattice is periodic, U(x) is a periodic function U(x + a) = U(x) where a is the lattice constant. In addition, with out loss of generality, we can assume that U(x) is an even function U (x) = -U(x). For such an even periodic function, we can write it as a a Fourier series.
U0 2 π n U(x) = + Un cos x (10.6) 2 n a and
2 a 2 π n Un = U(x) cos x ⅆ x (10.7) a 0 a The energy gap at k = ±π/a is
a 2 a π π 2 a 2 π 2 2 Eg = U(x)[ρ+(x) - ρ-(x)] ⅆ x = U(x)cos x - sin x ⅆ x = U(x) cos x ⅆ x = U1 (10.8) 0 a 0 a a a 0 a The energy gap at k = ±n π/a is
a 2 a π π 2 a 2 π 2 2 Eg = U(x)[ρ+(x) - ρ-(x)] ⅆ x = U(x)cos n x - sin n x ⅆ x = U(x) cos n x ⅆ x = Un (10.9) 0 a 0 a a a 0 a
10.2. Bloch waves
10.2.1. The effect of the lattice potential: the Hamiltonian and the Schrodinger equation The Hamiltonian Phys520.nb 75
p2 ℏ2 2 H = + U(x) = - ∂x +U(x) (10.10) 2 m 2 m where U(x) is a periodic potential, U (x) = U(x + a). The Schrodinger equation is
ℏ2 2 - ∂x ψ(x) + U(x) ψ(x) = ϵ ψ(x) (10.11) 2 m This equation is in general hard (or impossible) to solve analytically. However, F. Bloch managed to prove a very important theorem, which states that the solution to this equation must take the following form:
ⅈ k x ψk(x) = uk(x) ⅇ (10.12)
Here, u(x) is a periodic function uk(x + a) = uk(x), which has the same period as the lattice.
→ → → → → ◼ This conclusion is true in any dimensions. For example, in 3D, we have ψr = u→r exp ⅈ k · r where ur is a 3D periodic function k → → → → u→r = u→ r + T where T is any lattice vector. k k This type of waves are known as the Bloch wave. They have some nice properties ◼ The plane waves are a special type of Bloch waves with the function u(x) = constant. ◼ The Bloch wave is NOT an eigenstate of the momentum operator. In other words, it doesn’t have a well-defined momentum (unless → → u(x) = constant). However, the parameter k in the Bloch wave behaves very similarly to the momentum. We call ℏ k the lattice momentum. ◼ Lattice momentum is not the momentum, but it is conserved in a lattice.
10.2.2. The proof of the Bloch theorem Define the translation operator
Ta ψ(r) = ψ(r + a) (10.13) where a is the lattice constant. This operator shift our quantum system in real space by the lattice constant a.
-1 Under translation by a, a quantum operator O changes as O → Ta O Ta . Therefore, for the Hamiltonian
-1 Ta H(x) Ta = H(x + a) (10.14) Because
ℏ2 2 H(x) = - ∂x +U(x) (10.15) 2 m and
ℏ2 ℏ2 2 2 H(x + a) = - ∂x +U(x + a) = - ∂x +U(x) (10.16) 2 m 2 m we know immediately that H (x) = H(x + a). And thus
-1 Ta H(x) Ta = H(x + a) = H(x) (10.17) In other words,
Ta H(x) = H(x) Ta (10.18) so
[Ta, H] = 0 (10.19)
If two operators commute, we know that we can find common eigenstates of these two operators. For Ta and H, this means that we can find a set of (complete orthonormal) basis ψk(x), such that ⅈ k a H ψk(x) = ϵ ψk(x) and Ta ψk(x) = ⅇ ψk(x) (10.20)
Here, we used the fact that H is an Hermitian operator, so that its eigenvalues ϵ must be real. For Ta, because it is a unitary operator, its eigenvalue must be a complex with absolute value 1. 76 Phys520.nb
Define
-ⅈ k x uk(x) = ⅇ ψk(x) (10.21)
It is easy to prove that uk(x) is a periodic function uk(x) = uk(x + a).
-ⅈ k (x+a) -ⅈ k (x+a) -ⅈ k (x+a) ⅈ k a -ⅈ k x uk(x + a) = ⅇ ψk(x + a) = ⅇ Ta ψk(x) = ⅇ ⅇ ψk(x) = ⅇ ψk(x) = uk(x) (10.22) Therefore,
ⅈ k x ψk(x) = uk(x) ⅇ (10.23)
where uk(x) is a periodic function.
10.3. Properties of Bloch waves and crystal momentum Eigen wave functions for a single particle moving in a periodic potential: Bloch waves
ⅈ k x ( ) ψnk(x) = un k(x) ⅇ 10.24
Here k is known as the crystal momentum. The corresponding eigenenergy is ϵn,k, which satisfies
H ψn,k(x) = ϵn,k ψn,k(x) (10.25)
10.3.1. value of k? ◼ The crystal momenta are only well-defined modulo G, where G is a reciprocal vector, i.e., k and k + G are equivallent. ◼ Therefore, we can limit the crystal momentum to the first Brillouin zone without loss of information (e.g. for a 1D system, -π/a ≤ k < π/a). Proof : Suppose we have a Bloch wave with crystal momentum k
ⅈ k x ψn k(x) = un k(x) ⅇ (10.26) Define k ' = k + G
ⅈ k x ⅈ k'-G x -ⅈ G x ⅈ k' x ( ) ψn k(x) = un k(x) ⅇ = un k'-G(x) ⅇ = un k'-G(x) ⅇ ⅇ 10.27 -ⅈ G x Define un k'(x) = un, k'-G(x) ⅇ , -ⅈ G x ⅈ k' x ⅈ k' x ( ) ψn k(x) = un k'-G(x) ⅇ ⅇ = un k'(x) ⅇ 10.28
It is easy to check that un,k'(x) = un,k'(x + T) where T is any lattice vector.
-ⅈ G (x+T) -ⅈ G x un,k'(x + T) = un, k'-G(x + T) ⅇ = un, k'-G(x) ⅇ = un,k'(x) (10.29)
ⅈ k' x Therefore, ψn k(x) = un k'(x) ⅇ is also a Bloch wave function with crystal momentum k ' ⅈ k' x ψn k(x) = un k'(x) ⅇ = ψn,k'(x) (10.30) Conclusion: as long as k and k ' differs by a reciprocal lattice vector G, we can write a Bloch wavefunction with crystal momentum k as a Bloch wavefunction with crystal momentum k '. In other words, crystal momentum is only well defined modulo G
The most important consequence is that this resuls implies that the dispersion relation ϵn,k is a periodci function ϵn,k = ϵn,k+G. A Brillouin zone is one periodicty. As a results, we can choose to draw the dispersion in three different ways as shown below. Phys520.nb 77
Fig. 3. Dispersion relation ϵn(k) shown in extended zone, reduced zone and periodic zone.
10.3.2. Reduced Brillouin zone Since the crystal momentum is only well defined modulo G, we can limit the momentum to the first Brillouin zone (e.g. for a 1D system, -π/a ≤ k < π/a) without loss of information.
We can plot the eigen-energy ϵn,k as a function of k in the first Brillouin zone. This way of plotting ϵn,k is known as the reduced Brillouin zone (See figure b above) The band index n is (from bottom to top) n = 1, 2, 3, …
10.3.3. Periodic Brillouin zone
There is another way of plotting ϵn,k. We can start from the reduced Brillouin zone and then we plot ϵn,k as a periodic function of k, repeating the same curve in the second, third, ... Brillouin zone (See figure c above)
10.3.4. Folding, Reduced Brillouin zone and extended Brillouin zone for free particles without lattices We know that plane waves is a special case of Bloch waves (where the periodic potential is V = 0). Therefore, we can present the dispersion of a free particle ϵ = ℏ2 k2 2 m in the same way (in the reduced BZ and periodic BZ). The can be achieve using the following procedures: start from the dispersion ϵ(k) = ℏ2 k2 2 m, then move the curve in the regions (2 n - 1) π/a < k < 2 n π/a to -π/a < k < 0 and move the curve in the regions 2 n π/a < k < (2 n + 1) π/a to 0 < k < π/a. So we get the figure below.