Phonons in bulk crystals

We start with a very brief reminder of the main basic concepts in the theory of lattice dynamics in bulk crystals. This subject is of course abundantly — and well — covered in numerous texts of various kinds [1–6]. Here we shall simply introduce in an organised manner the notation to be followed and the basic elements — often simply the symbols and notation — to be used later.

Since simple systems usually illustrate the features of real systems, we start with a brief summary of linear chains and after that we collect the phonon dispersion and parameters of the semiconductors we shall mostly deal with, i.e. GaAs and AlAs and their ternary compounds.

Linear lattices

Monoatomic linear chain

We begin with the case of a linear lattice with N equidistant atoms of mass M and period a, interacting with an harmonic potential U and a nearest neighbour force constant γ0 and look only at longitudinal vibrations. Then the atomic displacements in the normal mode k are

n=Akei(kna−ωt)

with corresponding eigenvalue

=2γ0M|sin⁡(ka/2)|.

Let us define the collective coordinates Ak

n=1N∑kAkeikna

with inverse relation

k=1N∑kξne−ikna.

The requirement ξn = real then implies that k*=A−k. In these coordinates the phonon Hamiltonian takes the form of N decoupled oscillators. For each oscillator we can introduce the operators ^k and ^k† and then

^=∑kh¯ω(k)(b^k†b^k+12),

b^k,b^k′ ]=0;[ b^k†,b^k′† ]=0;[ b^k,b^k′† ]=δkk′.

If |0,0,…, νk,…, 0,0 > is the state with νk phonons in mode k, then

^k|νk>=νk|νk−1>,b^k†|νk>=νk+1|νk+1>,n^k=b^k†b^k,n^k|νk>=νk|νk>,E0=12∑kh¯ω(k),E=E0+∑kνkh¯ω(k).

Diatomic linear chain

We recall the well-known fact that a linear monoatomic chain with period a can also be formally described as having a period 2a, 3a, etc. The dispersion relation curve ω(k) is then folded in a smaller Brillouin Zone (BZ) and the flatness property ∂ω/∂k|BZ border = 0 no longer holds. If we modify the system in some way so that the fictitious period becomes real, then some gaps appear at the Brillouin Zone border of the initially folded representation and the flatness property reappears. With a view to the eventual treatment of the superlattices it is appropriate to look here in this way at the passage from the monoatomic to the diatomic chain, so we now consider the case of two different atoms (masses Mα, α = 1, 2) in the unit cell interacting with nearest neighbour force constant γ0. For simplicity we assume the atoms to be equidistant so that b = 2a is now the real period. In terms of collective coordinates Ak we now have

n,α=1NMα∑kAkeα(k)eiknb,

α(−k)=eα(k);Ak*=A−k,

where Ak is the amplitude of the wave and eα(k) is the polarisation vector (one-component vector in this 1D case) for α atom.

Then the Lagrangian is given by [1]

=12∑k,αeα2(k)A˜kA˜−k−12∑k,α,βDαβ(k)eα(k)eβ(k)AkA−k

with

11=2γ0/M1;D12=−γ0/M1M2(1+eikb)D21=D12*=D12(−k);D22=2γ0/M2

and the Lagrange equations are

α(k)A¨k+∑βDαβ(k)Ak=0;α=1,2.

These and Äk = − ω2Ak, yield the dispersion relations

s2(k)=γ0[ 1μ+(−1)s1μ2−4M1M2sin⁡2(kb2) ],s=1,2,

where μ is the reduced mass given by

=M1M2M1+M2.

For ka << 1

1(k)≈kb22γ0M1+M2=vsk;ω2(k)≈2γ0μ

where υs is the speed of sound. At the Brillouin Zone border

1(π/a)=2γ0Mmax;ω2(π/a)=2γ0Mmin,

with

max=Max{M1,M2};Mmin=Min{M1,M2},

and the displacements of the atoms in the unit cell are related by

ξn2ξn1)k,s=e2s(k)e1s(k)M2M1,e2s(k)e1s(k)=2γ0M1−ωs2(k)2γ0/M1M2.

In the above equations ω1 represents the acoustical branch, and ω2 the optical branch.

In terms of phonon creation and annihilation operators

^=∑k,sh¯ωs(k)[ b^ks†b^ks+12 ],

^nα(s)=h¯2NMα∑k,seαs(k)ωs(k)(b^ks+b^ks†)eikna.

Qualitatively similar results are obtained when we treat the case of two — different or equal — nonequidistant atoms in the unit cell.

In these systems we have 2N degrees of freedom and 2N second order coupled ordinary differential equations, N being the number of unit cells. The introduction of collective coordinates allows us to solve the problem with a system of 2N/N = 2 algebraic equations.

In the acoustic branch ω tends to zero when k → 0, and then in the long-wavelength limit

n2ξn1≈1.

On the other hand, for the optical branch and ≈0,ω2γ0/μ and

n2ξn1≈−M1M2.

The standard use of the Born-von Karman conditions to count the number of states yields again a total number of modes equal to the number of atoms in the Born-von Karman ring, but now there are two branches in correspondence with the number of atoms in the unit cell and a gap between the two. If we take p cells as a fictitious period then the dispersion relation scheme is folded p times and we loose the flatness property in the neighbourhood of the Brillouin Zone border. Again, if the system is modified so the fictitious period becomes real, then the property is recovered and some gaps open at the Γ and X points of the Brillouin Zone.

General comments on linear lattices

We have examined simply longitudinal modes in linear chains of atoms with only nearest neighbour interactions, which suffices to set the scene and bear out the main qualitative features. For the diatomic chain we have assumed equidistant atoms with only a mass difference. Again, a re-examination of this problem with some changes — e.g. equal or different atoms but with a nearest neighbour spacing a ≠ b/2 — does not change the key qualitative features and if we recover the monoatomic chain by making all atoms and/or interatomic distances equal, then we obtain one branch but artificially folded.

The consideration of p atoms of the same or different classes distributed in the unit cell interacting with an arbitrary number of neighbours implies quantitative changes in the dispersion relation curves and normal modes. But the general picture is still valid, i.e. the states are labelled by the ID wavevector k spanning the first Brillouin Zone, the number of branches (in one dimension) agrees with the number of atoms in the unit cell, the number of states in a branch is determined by the number of unit cells in the crystal (Born-von Karman or periodic boundary condition), there is always an acoustic branch with frequency tending to zero in the long-wavelength limit, and there are p − 1 optical branches in the upper side of the spectrum.

In particular if we consider the linear diatomic chain and we call...