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Computational Solid State Physics 計算物性学特論 5回 5.Band offset at hetero-interfaces and effective mass approximation Energy gaps vs. lattice constants Band alignment at hetero-interfaces B Ec Ec A E Ev A g Ec : conduction band edge EgB A Ev B Ev : valence band edge crystal A crystal B Anderson’s rule for the band alignment (1) χ:electron affinity None of the interface effects are considered. Anderson’s rule for the band alignment (2) Ec A B Ev E ( E ) B B g A Ec : conduction band offset Ev : valence band offset Ec Ev E E B g A g A g Types of band alignment type I type III type II Band bending in a doped hetero-junction (1) Band bending in a doped hetero-junction (2) Effective mass approximation ・Suppose that a perturbation is added to a perfect crystal. ・How is the electronic state? Examples of perturbations an impurity, a quantum well, barrier, superlattice, potential from a patterned gate, space charge potential Effective mass approximation (1) [ H crys V ] (r ) E (r ) H crysnk (r ) nknk (r ) V : external potential nk (r ) : Bloch function dk (r ) m (k )mk (r ) 3 ( 2 ) m assume: conduction band n is minimum at k=0 nk (r ) unk (r )eikr un 0 (r )eikr n 0 (r )eikr ( r ) n 0 ( r ) n ( k ) e ik r dk n 0 ( r ) n ( r ) 3 (2 ) Effective mass approximation(2) [ Hcrys V ] (r ) E (r ) dk H crys (r ) H crys n (k )nk (r ) (2 )3 dk dk ik r n (k ) nknk (r ) n 0 (r ) n (k ) nk e 3 (2 ) (2 )3 nk am k m m H crys (r ) n 0 (r ) am n (k )k e m ikr m dk (2 )3 Effective mass approximation(3) df ( x) ikx ikx e dx ik f ( x ) e dx ikf (k ) dx If f () 0 H crys (r ) n 0 (r ) am (i) m (r ) n 0 (r ) n (i) (r ) m Schroedinger equation for envelope function χ(r) (r ) n0 (r ) n (r ) [ n (i) V (r )] (r ) E (r ) Effective mass approximation (4) ・Schroedinger equation for an envelope function χ(r) 2 2 k n (k ) c 2m * 2 2 [ V (r )] (r ) ( E c ) (r ) 2m * (r ) n0 (r ) n (r ) All the effects of crystal potential are included in εc and effective mass m*. Impurity V (r ) e2 4 0 s r :potential from a donor ion e4 m * m* E c 2 2 2 c Ry m0 s2 8h 0 s Ry=13.6 eV: Rydberg constant Quantum well Quantum corral HEMT 2D-electron confinement in HEMT The sub-band structure at the interface of the GaAs active channel in a HEMT structure. E1 and E2 are the confined levels. The approximate positions of E1 and E2 as well as the shape of the wave functions are indicated in the lower part of the diagram. In the uper part, an approximate form of the potential profile is shown, including contributions of the conduction band offset and of the space charge potential. Superlattice Crystal A Crystal B The Kronig-Penney model, a simple superlattice, showing wells of width w alternating with barriers of thickness b and height V0. The (super)lattice constant is a=b+w. Kronig-Penny model (1) V0 b0 V0b S V ( x) S ( x an) n Schroedinger equation in the effective mass approximation 2 d 2 [ V ( x)] k ( x) E (k ) k ( x) 2 2m dx Bloch condition for superlattice k ( x a) e k ( x) ika k: wave vector of Bloch function in the superlattice Kronog-Penney model (2) Boundary condition at x=0 k (0 ) k (0 ) 0 0 2 d 2 k ( x)dx V ( x) k ( x)dx 0 2 2m 0 dx 0 2 [ k (0 ) k (0 )] V0 k (0) 0 2m (1)continuity of wavefunction (2)connection condition for the 1st derivative of wavefunction (2’) Solution of Schroedinger equation k ( x) sin(k1 x) A cos(k1 x) 2 2 E (k ) k1 2m for 0 x a Kronig-Penney model (3) A e ika (sin k1a A cos k1a) (1) 2 q[1 e ika (cosk1a A sin k1a)] SA 0 2m (2’) Simultaneous equation for E(k) m S sin k1a coska cosk1a 2 k1 2 2 E (k ) k1 2m Kronig-Penney model (4) m P 2S allowed range of cos(ka) Kronig-Penney model (5) Conduction band of crystal A is split into mini-bands with mini-gaps by the Bragg reflection of the superlattice. Problems 5 Calculate the lowest energy level for electrons and light and heavy holes in a GaAs well 6 nm wide sandwiched between layers of Al0.35Ga0.65As. Calculate the photoluminescence energy of the optical transition. Calculate the two-dimensional Schroedinger equation for free electrons confined in a cylindrical well with infinitely high walls for r>a.