ECE36A Semiconductor Heterostructure Materials Quantum Wells and Superlattices Lecture 9, Nov., 017 Electron envelope wave-function Conduction band quantum wells Quantum well density of states Valence band quantum wells Multiple quantum wells and superlattices Modulation doped heterostructures: Energy levels and D electron gas Ref.: Quantum Semiconductor Structures, C. Weisbuch and B. Vinter, Academic Press, 1991. Principles of the Theory of Solids, J. M. Ziman, end Ed., Cambridge, 197 1
Electron Wave Function In general, to determine the electron wave-function in a solid, we need to start with Schrodinger s equation: Hψ = Eψ 1 H = m P + V ( i ion r i,{ R }) i + 1 i Kinetic energy i Electron-ion interactions i j e 4πε 0 r i r j Electron-electron interactions P i = i i This is complicated to solve, so various levels of approximations are employed, including starting with treating the electron as a single particleà single particle Hamiltonian. H = P m +V r m +V ( r) &ψ r = Eψ ( r ) Potential due to ions in the crystal + approximation for electron-electron interaction plane wave In a periodic crystal V r for all lattice vectors R. =V ( r +R) Bloch s theorem: The wave function can be written in the form: ψ n k,r u n ( k,r,r ) =u k +R n for all lattice vectors R = eik periodic wave.r u n k,r
approximate Allowed energies from characteristic equation Kronig Penny model Nearly free electron characteristic equation Reduced zone scheme 3
Electron Wave Function The nearly free electron model looks at the wave-functions outside the atomic cores where they look very like plane waves; within the atomic cores, they look like atomic orbitals. Both core wave-functions and those arising from overlap of atomic orbitals are solutions for Schrodinger equations à orthogonal à construct orthogonalized plane waves 4
Electron Wave Function The Bloch function in terms of atomic orbitals: ψ n r = 1 N e ik.r φ n r R R atomic orbital To satisfy Schrodinger equation, we need to make a linear combination of atomic orbitals with coefficients to be determined. ψ ( n r) = 1 e ik.r c m φ ( m r R) Very tedious N R,m We suppose that there exists a wave-function a ( n r), that is a solution for Schrodinger equation and satisfy othogonality requirements. ψ ( n r) = 1 e ik.r a N ( n r R) R We can thus write: ψ n r a Wannier functions n r R = f ( R ) n R f n R Unitary transformations of the Bloch functions 5 is the envelope wave-function showing the strength of the wave-function in the neighborhood of R impressed on the local atomic orbitals.
band index Envelope Wave Function ψ n ( r) = f ( R ) a ( n n r R) R Lattice vector envelope wave-function Wannier function for band n The envelope wave-function satisfies the following Schrodinger s equation: i E n Energy band dispersion relation f n r +V ( r ) f ( r ) = Ef ( r ) n n n band-edge profile produced by heterostructure, electric field, etc. For electrons in a band with effective mass, m n, this becomes: 1 4, & f ( m ( r) n r) +V ( r ) f ( r ) = Ef ( r ) n n n where ' A,B B { } = A +BA In a heterostructure, the effective mass becomes a function of position and the boundary conditions for the envelope function are: f ( n r) 1 and f ( m n r) are continuous r 6
Recall: Type I and Type II Quantum Wells Type I and Type II Quantum Wells Type I Quantum well for electrons and holes Type II Quantum well for electrons E c E c E v Quantum well for holes E v Conduction band and valence-band quantum wells can differ substantially in electronic structure due to differences in conduction and valence band structure. 7
Conduction Band Quantum Wells Consider the simplest case of an infinitely deep quantum well: -L/ L/ d m dz f ( z ) = Ef ( z ) L < z < L E n = nπ & m L f n ( z ) = Acos nπz L Asin nπz L (n odd, symmetric) (n even, anti-symmetric) 8
Consider a finite barrier electron quantum well: m B m A Conduction Band Quantum Wells m B V ( z) = E c ( z ) 4 m 1 & d ( z ) & dz f z d dz, 1 m where & ( z ) ' A,B f ( z ) { } = A B & +V z +V z +BA f ( z ) = Ef ( z ) f ( z ) = Ef ( z ) Symmetric wave-functions: f n ( z ) = Boundary conditions: f z = L & is continuous 1 df is continuous m dz L z= Acoskz κ z L/ Be Be κ z+l/ Acos kl = B ka m sin kl A = κb m B L < z < L z > L z < L k m A tan kl = κ m B E = k m A E =V 0 κ m B 9
Anti-symmetric wave-functions: Boundary conditions: Suppose for simplicity that: E = k m =V 0 κ Symmetric Conduction Band Quantum Wells f z = L & 1 df m f n dz L z= ( z ) = is continuous is continuous m A = m B = m m κ = m V 0 Asinkz κ z L/ Be Be κ z+l/ L < z < L z > L z < L Asin kl = B ka m cos kl A = κb m B E = k m A E =V 0 κ m B k m A cot kl = κ m B k = k 0 k k 0 = m V 0 Anti-symmetric k tan kl =κ tan kl >0 tan κl = κ & k = k 0 k & 1 cos kl = ± k k 0 for tan kl >0 kcot kl = κ tan kl <0 cot κl = κ & k = k 0 k & 1 sin kl = ± k k 0 for tan kl <0 10
Conduction Band Quantum Wells Symmetric: cos kl = ± k for tan kl k 0 >0 Anti-symmetric: sin kl = ± k for tan kl k 0 <0 cos kl sin kl k k 0 sign of tan kl + - + - + - confined state in finite potential well confined state in infinite potential well kl There will always be at least one confined state in finite-barrier electron quantum well. Alternating symmetric and anti-symmetric wave-functions. Energy of confined state in finite-barrier QW is lower than that of corresponding infinitebarrier quantum 11 well.
Quantum Well Density of States For an electronic system in D space (e.g. quantum well) of extent in L x, in x,y-direction and confinement in the z-direction, the envelope wave-functions are periodic with periods L x, in x,y-directions. The wave vector components k x, k y must satisfy: k y z y x k x = π L x n x k y = π n y π L x π k x L x The area occupied by one in k-space that is occupied by a D wave-vector (k x,k y ): ( π ) L x L E = E y c + k Assuming a parabolic band-structure in the quantum well, m ( The area in k-space occupied by conduction band-states is: πk ( E ) = π m E E c )' ' fore > E c & The number of distinct values of k for states with energy up to E is then: N states = π ( ( m E E c )/ ) m ( E E c ) L π π x 1 / L x =
Quantum Well Density of States N states = π ( ( m E E c )/ ) m = ( π ) / L x The density of states in the quantum well, g D ( E ), is then given by: electron spin Recall g D ( E ) = dn states g L x de D ( E E c ) π ( E ) = m g π 3D E L x = 1 m π & 3/ ( E E c ) 1/ g D is always finite at the bottom of the D level. All dynamic phenomena remain finite at low kinetic energy and low temperatures such as scattering, optical absorption and gain. 13