1. The k∙p Method
Brad Malone
Group Meeting
4/24/07

2. First used by Bardeen (1937) and Seitz (1940)
Later extended by Shockley, Dresselhaus, Kittel, and Kane
Uses:
Obtaining analytic expressions for band dispersion and effective masses
Can also be used to get the band structure for the whole zone from zone center energy gaps and optical matrix elements
Bardeen and Seitz used it as a way of calculating effective masses and crystal WFs near high-symmetry points. Analytic expressions for band dispersion and effective masses are obtained around high-symmetry points. Shockley did it for degenerate bands while Dresselhaus and Kane included spin-orbit.

3. With Bloch’s theorem we can express the solutions like so
With Bloch’s theorem we can express the solutions like so. We can get other similar equations for k equal to any other point. Once Un0 and En0 are known we can treat the k terms as perturbations…Only a small number of energy gaps and matrix elements of p (determined experimentally or in some other manner) are used as input.

4. We are going to see how it is used to calculate the effective mass of a nondegenerate band. This is applicable to the conduction band minimum in direct-bandgap semiconductors (w/ the zinc-blende and wurtzite structures). Mass is different from that of a free electron because of coupling between the different electronic states in the bands through the k.p term. Firstly, the difference in energy between the two bands determines how important it is to the effective mass. Bands with lesser energies contribute a positive term to the effective mass, which makes m* smaller than the free electron mass. And so bands with with higher energies increase m* or can even cause it to become negative (like in the top valence bands of the diamond- and zine-blende-type semiconductors). The WFs can only couple if the matrix of p between them is nonzero. Group theory can be very helpful in determining whether or not this is the case. Quick discussion of group theory related to this….. (stuff from Kittel on p. 188 relating to this?)

5. The Matrix Element Theorem-
The matrix element between the operator A and the wave functions Ψ1
and Ψ2 is nonzero only if the direct product of the representations of A
and Ψ1 contains an irreducible representation of Ψ2
Example: The Wigner-Eckart Theorem
T is an irreducible tensor operator or order k (q=-k….k). Their transformation laws are similar as those for the standard angular momentum basis of kets. A vector space is invariant if any vector of that subspace is mapped onto another vector by every rotation (in general, symmetry operation?). If the invariant subspace is impossible to break up into further invariant subspaces, then we say that it is irreducible. The ket above in the Clebsch-Gordan coefficient is in the uncoupled basis.

6. Ge GaAs InP InAs GaSb InSb CdTe
Conduction Band effective mass in III-V and II-VI semiconductors:
-Conduction band has Г1 symmetry; p has Г4 symmetry
These group theory considerations can be used to help determine the conduction band effective mass in many of the III-V and II-VI direct bandgap semiconductors. Gamma 1 is an irreducible representation whose functions are invariant under all symmetry operations of the space group (=A1 in molecular notation). In these semiconductors the lowest conduction band at the zone center has Gamma 1 symmetry. P has Gamma 4 symmetry. So effective mass determined mostly be bands with G4 symmetry. The matrix elements between the nearest conduction band is either zero (as in the case of diamond structures) or much smaller than the matrix elements with the top G4 valence band. So this matrix element dominates the effective mass. Separation is just the direct gap E0. Matrix element between the states is equal to i*P, so p=-ih_grad gives, and P is real. 2P^2/m is about 20eV while Eg is usually less than 2 eV. P^2 is almost constant for most group IV, III-V, and II-VI semiconductors.
Ge GaAs InP InAs GaSb InSb CdTe
Eg (eV)
m*/m (exp)
m*/m (theory)

7. Splitting of a degenerate extremum by spin-orbit

8. We can use the k.p method to get an understanding and to calculate the band dispersion near a degenerate band extremum including spin-orbit interactions. Usually the size of the spin orbit splitting in a semiconductors is comparable to the splitting of the atoms of which it is made. So for SCs with heavier elements, like GaSb, the splitting is on the order of an eV, which is comparable to the bandgap. In lighter SCs, like Si, it’s more like 0.05 eV, which (depending on what you’re doing) may be negligible. We consider the highest energy valence bands at the zone center of semiconductors with the diamond (and zinc-blende) structures. These valence bands have G4 symmetry, and so are p-like. Spin-orbit is a relativistic effect which scales with the atomic number. So for semiconductors with heavier elements like Ge, Ga, As, and Sb, we expect SO coupling to be large and need to include it in the unperturbed Hamiltonian. Rather than use facts about double groups (groups containing symmetry operations on spin wave functions) we make analogy and use similarity of the G4 valence band WFs to atomic p wave functions. The eigenfunctions of the SO interaction are also eigenstates of the total angular momentum J. |X> and such are just the 3 p-like valence band wavefunctions. This gives us 4 j=3/2 states and 2 j=1/2 states which are split by spin orbit.

9. Löwdin’s perturbation method:
Bands that are close to the G4 valence bands that can be coupled by p are the lowest conduction bands with symmetries G1 and G4. The G4 valence bands and the G1 and G4 conduction bands now form a set of 14 unperturbed WFs coupled by the k.p term. LPM can be used to obtain analytic expressions for the valence band dispersions. In it we break up the 14x14 matrix into two parts: WFs of interest and their mutual attractions treated exactly while the remaining treated by perturbation theory. In the present cae we care about the six G4 valence bands and will treat them exactly. Coupling between these states and those of the conduction bands will be treated with an effective matrix element between any two valence band WFs. Then the 14x14 matrix becomes a 6x6, and can be diagonalized numerically without any further approximations. But is we restrict to k values small enough that the matrix elements which couple the J=3/2 and J=1/2 bands are negligible in comparison to spin orbit, the 6x6 becomes a 4x4 and a 2x2. The 2x2 gives the energy of the doubly degenerate j=1/2 band. The 2x2 matrix gives us an effective mass of the j=1/2 split off valence band as shown above. P, as before, is what we get from the matrix elements of the valence band with the G1 conduction band while Q is the result of the matrix element between the valence band states and the G4 conduction band states. E0 is the energy separation between the G1 conduction bands and the j=3/2 valence bands and E0’ is the energy separatoin between the G4 conduction bands and the j=3/2 valence bands. Delta is the spin orbit splitting. By diagonalizing the 4x4 matrix we get the dispersion of the j=3/2 bands. The one with the plus sign corresponds to the light hole and the one with the minus sign to the heavy hole. A,B,C are related to P,Q, and the energy gaps. (A:Terms linear in k are killed by parity or are very small and involve spin-dependent terms which we are not treating here). Hole masses different in different directions. Can also average over k if we want to assume that it’s isotropic. E(k)=E(-k) is true for us even though the crystal may not have that symmetry because our Hamiltonian is invariant under time reversal.