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More Bandstructure Discussion

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More Bandstructure Discussion

- “Almost free” electron approach to bandstructure.
1 e- Hamiltonian:H = (p)2/(2mo) + V(x); p -iħ(d/dx)

V(x) V(x + a) = Effective potential, period a(lattice repeat distance)

GOAL

- Solve the Schrödinger Equation: Hψ(x) = εψ(x)
Periodic potential V(x)

ψ(x) must have the Bloch form:

ψk(x) = eikx uk(x), with uk(x) = uk(x + a)

- The set of vectors in “k space” of the form G = (nπ/a),
(n = integer) are calledReciprocal Lattice Vectors

- Expand the potential in a Fourier series:
Due to periodicity, only wavevectors for which k = G enter the sum.

V(x) V(x + a) V(x) = ∑GVGeiGx (1)

The VG depend on the functional form of V(x)

V(x) is realV(x)= 2 ∑G>0 VGcos(Gx)

- Expand the wavefunction in a Fourier series ink:
ψ(x) = ∑kCkeikx(2)

Put V(x) from (1) & ψ(x) from (2) into the Schrödinger Equation:

- The Schrödinger Equation: Hψ(x) = εψ(x) or
[-{ħ2/(2mo)}(d2/dx2) + V(x)]ψ(x) = εψ(x)

Insert the Fourier series for both V(x) & ψ(x)

- Manipulation (see BW or Kittel) gets,
For each Fourier component of ψ(x):

(λk - ε)Ck + ∑GVGCk-G = 0 (3)

where λk= (ħ2k2)/(2mo) (the free electron energy)

- Eq. (3) is the k space Schrödinger Equation
A set of coupled, homogeneous, algebraic equations for the Fourier componentsof the wavefunction. Generally, this is intractable: There are an number of Ck !

- The k space Schrödinger Equation is:
(λk - ε)Ck + ∑GVGCk-G = 0 (3)

where λk= (ħ2k2)/(2mo) (the free electron energy)

- Generally, (3) is intractable! # of Ck ! But, in practice, need only a few.
Solution:Determinant of coefficients of theCk is set to0:

That is, it is an determinant!

- Aside:Another Bloch’s Theorem proof:Assume (3) is solved. Then, ψhas the form: ψk(x) = ∑GCk-G ei(k-G)x or
ψk(x) = (∑GCk-Ge-iGx) eikx uk(x)eikx

where uk(x) = ∑G Ck-G e-iGx

It’s easy to show the uk(x) = uk(x + a)

ψk(x) is of the Bloch form!

- The k space Schrödinger Equation:
(λk - ε)Ck + ∑GVGCk-G = 0 (3)

where λk= (ħ2k2)/(2mo) (the free electron energy)

- Eq. (3) is a set of simultaneous, linear, algebraic equations connecting the Ck-Gfor all reciprocal lattice vectors G.
- Note:If VG = 0 for all reciprocal lattice vectors G, then
ε = λk = (ħ2k2)/(2mo)

Free electron energy“bands”.

- The k space Schrödinger Equation is:
(λk - ε)Ck + ∑GVGCk-G = 0 (3)

where λk= (ħ2k2)/(2mo) (the free electron energy)

= Kinetic Energy of the electron in the periodic potential V(x)

- Consider the Special Case:
All VG are small in comparison with the kinetic energy, λk except for

G = (2π/a) & for k at the 1st BZ boundary, k = (π/a)

For k away from the BZ boundary, the energy band is the free electron parabola: ε(k) = λk = (ħ2k2)/(2mo)

For k at the BZ boundary, k = (π/a), Eq. (3) is a

2 2 determinant

- In this special case:As a student exercise (see Kittel), show that, for k at the BZ boundary k = (π/a), the k space Schrödinger Equation becomes 2 algebraic equations:
(λ- ε) C(π/a) + VC(-π/a) = 0

VC(π/a) + (λ- ε)C(-π/a) = 0

where λ= (ħ2π2)/(2a2mo); V = V(2π/a) = V-(2π/a)

- Solutions for the bandsεat the BZ boundary are:
ε = λ V

(from the 2 2 determinant):

Away from the BZ boundary the energy band εis a free electron parabola. At the BZ boundary there is a splitting:

A gap opens up!εG ε+ - ε- = 2V

- Now, lets look at in more detail at knear(but not at!) the BZ boundary to get the k dependence of ε near the BZ boundary: Messy! Student exercise (see Kittel) to show that the
Free Electron Parabola

SPLITS

into 2 bands, with a gap between:

ε(k) = (ħ2π2)/(2a2mo) V

+ ħ2[k- (π/a)2]/(2mo)[1 (ħ2π2 )/(a2moV)]

This also assumes that |V| >> ħ2(π/a)[k- (π/a)]/mo.

For the more general, complicated solution, see Kittel!

Almost Free e-Bandstructure:(Results, from Kittel for the lowest two bands)

ε = (ħ2k2)/(2mo)

V

V

Given an energy band ε(k)(a Schrödinger Equation eigenvalue):

The Electron is a Quantum Mechanical Wave

- From Quantum Mechanics, the energyε(k) & the frequency ω(k) are related by:ε(k) ħω(k)(1)
- Now, from Classical Wave Theory, the wave group velocityv(k) is defined as:v(k) [dω(k)/dk](2)
- Combining (1) & (2) gives: ħv(k) [dε(k)/dk]
- The QM wave (quasi-)momentum is: p ħk

- Now, a simple“Quasi-Classical” Transport Treatment!
- “Mixing up” classical & quantum concepts!

- Assume that the QM electron responds to an EXTERNALforce, FCLASSICALLY(as a particle). That is, assume that
Newton’s 2nd Law is valid: F = (dp/dt)(1)

- Combine this with theQMmomentum p = ħk & get:
F = ħ(dk/dt)(2)

Combine (1) with the classical momentum p = mv:

F = m(dv/dt) (3)

Equate (2) & (3) & also for v in (3) insert the QM group velocity:

v(k) = ħ-1[dε(k)/dk](4)

- So, this “Quasi-classical” treatment gives
F = ħ(dk/dt) = m(d/dt)[v(k)] = m(d/dt)[ħ-1dε(k)/dk](5)

or, using the chain rule of differentiation:

ħ(dk/dt) = mħ-1(dk/dt)(d2ε(k)/dk2) (6)

Note!!(6) can only be true if the e- mass m is given by

m ħ2/[d2 ε(k)/dk2](& NOTmo!) (7)

m EFFECTIVE MASSof e- in the bandε(k)at wavevectork.Notation: m = m* = me

- The Bottom Line is:Under the influence of an external forceF
The e- responds Classically(According to Newton’s 2nd Law)BUTwith a Quantum Mechanical Massm*,notmo!

- m The EFFECTIVE MASSof the e- in band ε(k)at wavevector k
m ħ2/[d2ε(k)/dk2]

- Mathematically,
m [curvature of ε(k)]-1

- This is for 1d. It is easily shown that:
m [curvature of ε(k)]-1

also holds in 3d!!

In that case, the 2nd derivative is taken along specific directions in 3d k space & the effective mass is actually a 2nd rank tensor.

m [curvature of ε(k)]-1

Obviously, we can havem > 0 (positive curvature)

or m < 0 (negative curvature)

- Consider the case of negative curvature:
m < 0 for electrons

For transport & other properties, the charge to mass ratio (q/m) often enters.

For bands with negative curvature, we can either

1. Treat electrons(q = -e) with me < 0

Or 2. Treat holes (q = +e) with mh > 0

Negative me

Positive me

- The linear approximation for L(ε/Vo) does not give accurate effective masses at the BZ edge, k = (π/a).
For k near this value, we must use the exact L(ε/Vo) expression.

- It can be shown (S, Ch. 2) that, in limit of small barriers
(|Vo| << ε), the exact expression for the Krönig-Penney effective mass at the BZ edge is: m = moεG[2(ħ2π 2)/(moa2) εG]-1

with:mo = free electron mass, εG = band gap at the BZ edge.

+ “conduction band”(positive curvature) like:

- “valence band”(negative curvature) like:

The Krönig-Penney model results (near the BZ edge):

m = moεG[2(ħ2π 2)/(moa2) εG]-1

This is obviously too simple for real bands!

- A careful study of this table, finds, for real materials, m εG also!NOTE:In general(m/mo) << 1