Project Report: Band Structure of GaAs using k.p-theory

Transcription

1 Proect Report: Band Structure of GaAs using k.p-theory Austin Irish Mikael Thorström December 12th Introduction The obective of the proect was to calculate the band structure of both strained and unstrained Gallium arsenide, GaAs, using the standard model. The strained band structure is useful to calculate since the bands will change with the material being sandwiched with different lattice materials. To calculate this the 8-band k p theory will be used, which will give a fairly accurate result for small k-values. 2 Theory Solving the Schrödinger equation for electrons in a crystalline solid reveals the electronic states and energy levels, Eq.1, where p = i h is the momentum operator, m 0 is electron rest mass, V (r) is the periodic position dependent potential in the material, ψ is the electron wave function and E is the energy. [ p 2 ] + V (r) ψ = Eψ (1) This is quite useful for studying the electronic properties of bulk homogeneous semiconductors, however heterostructures and quantum structures complicate solving Eq.1 and approximation methods must be used. k p perturbation theory is one of the methods for solving for the electronic and optical properties of these more complicated crystalline solids. Davies in (7) describes how with Bloch functions and the k p method we can write the wave function as Ψ n,k (r) = e ik r u n,k (r) Using these Bloch functions as solutions to the Schrödinger equation, we get H k u n,k = E n,k u n,k where the Hamiltonian, H, becomes (note the k p operator): 1

2 H k = p2 + hk p + h2 k 2 + V m 0 with components dependent on and independent of k. The Hamiltonian at k=0 reduces to H 0 and we are left with Eq. 2, as described by Cardona and Pollak (6). [ p 2 ] H 0 ψ = + V ψ = Eψ (2) Though technically not necessary, solving the Schrödinger equation only in the Brillouin zone center where k 0, greatly simplifies things. The so called 8-band k p theory does this for eight (four spin up/down degenerate) energy levels, the first conduction band and the first three valence bands: heavy holes, light holes and the split-off band. In matrix notation the Hamiltonian can be represented as [ G ] Γ H = Γ G where G and Γ are 4x4 matrices. Kane (3) first defined these matrices where G(k) = G 1 (k) + G 2 (k) + G so as where E g is the band gap energy and valence band energy E v = E v + 3. Additionally, parameters L, M and N are defined as follows: L = h2 (1 + γ 1 + 4γ 2 ) + P 2 E g M = h2 (1 + γ 1 2γ 2 ) N = 3 h2 m 0 γ 3 + P 2 E g Expanding upon Kane s method, Bahder (5) used work by Pikus and Bir (4) to incorporate the effect of strain into the model. Gershoni (1) describes this and further expands upon their work modeling heterostructures with quantum confinement 2

3 in any number of dimensions. To see the effect of strain on band structure one merely has to add a strain term to matrix G such that G(k) = G 1 (k)+g 2 (k)+g so +G strain where a c [e xx + e yy + e zz ] b e yz ip e x k b e zx ip e y k b e xy ip e z k b e yz + ip e x k le xx + m(e yy + e zz ) ne xy ne xz G strain = b e zx + ip e y k ne xy le yy + m(e xx + e zz ) ne yz b e xy + ip e z k ne xz ne yz le zz + m(e xx + e yy ) Fortunately, G strain greatly simplifies under certain conditions. For the epitaxial InP-GaAs lattice-strained scenario we report on, b = 0, e xx = e yy, e zz = C 12 C 11 e xx and for all e i = 0. Variables a v, b v and d v parameterize the deformation potentials and are defined as follows: a v = 1(l +2m), b 3 v = 1(l m) and d 3 v = 1 3 n. Experimentally, a v = 2.67, b v = 1.7, d v = 4.55, C 11 = 11.81, C 12 = e xx is characterized by the strain due to lattice mismatch, which is between GaAs and InP for our calculation: e xx = e yy = a = and e a zz = Method = a InP a GaAs a GaAs The method used in this proect was to use the 8-band k.p-theory with the help of MATLAB to construct the matrices earlier mentioned. First by assuming to have a one dimensional system, i.e. the k values for y- and z-direction set to 0. The matrices was calculated for each k-value in the x-direction set by a vector, creating a 3 dimensional matrix and the eigenvalues was solved for each k-value giving the energy of each value in all bands. All eigenvalues calculated for each k x was stored in a new matrix which then would be plotted, giving a energy diagram as function of k-value. The parameters used in the constructed matrices are specific for the said material taken from [1] and [2], as earlier mentioned. The same method is used for the two dimensional situation where only k in the z-direction is set to 0. Vectors of k in x- and y-direction was used to calculate the eigenvalues for each k-value giving a three dimensional matrix. When calculating the strained bandstructure of GaAs, which here is strained by putting a layer of InP on top, another G-matrix is added, G strain. Otherwise the calculation is done in the same way as previous. 4 Results Figure 1 shows the plot of the one dimensional band structure comparison of unstrained GaAs and strained by matching to the lattice of InP. Where unstrained is calculated as earlier mentioned and the strained is calculated by using the same 3

4 method but with an extra G-matrix, G strain which adds the effect of lattice mismatch. In this plot we can see eight different bands, including spin up and down for different the sub-bands. The difference in slope/curvature makes it possible to conclude which band represent light and heavy holes, where the light holes have a steeper slope. Figure 1: 1-D comparison of strained and unstrained band structure of GaAs latticematched with InP. Figure 2 shows the plot of the two dimensional band structure, here used 200 steps vector each direction of the k-value and calculated the energy for each combination of k-value. Showing all eight, spin up and down, bands. 4

5 Figure 2: 3D plot of the band structure of unstrained GaAs in two dimensions, showing all eight bands calculated. Figure 3 showing the plot of the four lowest bands of figure 2. Figure 3: 3D plot of the band structure of unstrained GaAs in two dimensions, showing the four lower bands, the split-off and light holes bands. 5

6 5 Conclusion In conclusion, k p theory as developed by Kane and expanded by others is an effective method for calculating electronic bands in crystalline structures, even for complicated materials and environments like heterostructures, quantum devices and strained lattices. With k p theory and Bloch functions any material and complication can be modeled as long as it is periodic (even over small domains) and can be described by a potential. For this proect we show how 8-band k p theory can be used to calculate the first eight electronic bands of GaAs in one or more dimensions. Following the technique of Gershoni, we adapted our model to account for lattice strain by incorporating deformation potentials into our equations. This is nice because it reveals not ust energy bands, but how they shift and split in commonly encountered environments like epitaxial mismatch. The obvious but relatively trivial continuation for this work would be to model materials other than GaAs and InP, to expand dimensionality and to explore different sizes and arrangements of structures. Much of this has been done already in other literature. References [1] D.Gershoni, C. H. Henry and G. A. Baraff, Calculating the Optical Properties of Multidimensional Hetero-structures: Application to the Modeling of Quaternary Quantum Well Lasers [2] E. O. Kane, Energy Band Structure in p-type Germanium and Silicon, General Electric Research Laboratory, Schenectady, New York, [3] E. O. Kane, Energy Band Theory in Handbook on Semiconductors, vol. 1, W. Paul, Ed. Amsterdam, North Holland, 1982, pp [4] G. E. Pikus, G. L. Bir, Effect of Deformation on the Hole Energy Spectrum of Germanium and Silicon, Fiz. Tverd. Tela vol. 1, pp , [5] Thomas B. Bahder, Eight-band k.p Model of Strained Zinc-Blende Crystals, Phys. Rev. vol. B41, pp , [6] Manuel Cardona, Fred H. Pollak, Energy-Band Structure of Germanium and Silicon: The k y Method, Phys. Rev. vol. 142 no. 2, pp , February [7] John H. Davies, The Physics of Low-Dimensional Semiconductors: An Introduction. Cambridge University Press, Cambridge, UK; 1998, pp