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brief analysis of density of states for bulk, quantum well and quantum wire
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Course: Quantum Electronics Arpan Deyasi Quantum Topic: Density of states of Bulk and Quantum Structures Electronics Arpan Deyasi Arpan Deyasi, RCCIIT 6/10/2020 1
What is DoS? Arpan Deyasi E Number of available energy states Quantum per unit energy interval Electronics EC+ dEC dEC per unit dimension EC in real space k EV dEV EV+ dEV 6/10/2020 Arpan Deyasi, RCCIIT 2
What do we mean by ‘dimension’? Arpan Deyasi For ‘bulk’, it is ‘volume’ Quantum For ‘quantum well’, it is ‘area/surface’ Electronics For ‘quantum wire’, it is ‘line/length’ For ‘quantum dot’, it is a ‘point/dot’ 6/10/2020 Arpan Deyasi, RCCIIT 3
Energy band diagram is drawn in E-k plane Arpan Deyasi ‘k’ is wave-vector, not a physical quantity Quantum No of electrons is measured by magnitude of current Electronics So we must know the density of electrons in real space instead of k-space 6/10/2020 Arpan Deyasi, RCCIIT 4
Fermi sphere Arpan Deyasi Quantum Electronics 6/10/2020 Arpan Deyasi, RCCIIT 5
Fermi surface Arpan Deyasi Quantum Electronics 6/10/2020 Arpan Deyasi, RCCIIT 6
DoS for bulk semiconductor Arpan Deyasi Let’s start with Bloch theorem Quantum ( , , ) x y z Electronics Consider a 3D semiconductor with dimensions Lx, Ly, Lz = + + + ( , , ) x L y L z L x y z 6/10/2020 Arpan Deyasi, RCCIIT 7
DoS for bulk semiconductor Arpan Deyasi For validity of wave function Quantum = = = 2 2 2 k L k L n n x x x Electronics y y y k L n z z z (2 ) 3 n n n x y z = k k k Volume in k-space x y z L L L x y z 6/10/2020 Arpan Deyasi, RCCIIT 8
DoS for bulk semiconductor Arpan Deyasi Let Quantum = = = 1 n n n x y z = = = L L L L Electronics x y z 3 (2 ) L = k V Volume of unit cell in k-space 3 6/10/2020 Arpan Deyasi, RCCIIT 9
DoS for bulk semiconductor Arpan Deyasi Volume of Fermi sphere in k-space Quantum 4 3 = 3 V k F Volume of semiconductor in real space Electronics = 3 V L R 6/10/2020 Arpan Deyasi, RCCIIT 10
DoS for bulk semiconductor Arpan Deyasi number of energy states in real space Quantum 1 = N V V F V number of energy states in real space Electronics k 1 1 = N V R F V V k R 6/10/2020 Arpan Deyasi, RCCIIT 11
DoS for bulk semiconductor Arpan Deyasi Quantum 3 4 3 1 L L = 3 N k 3 3 8 Electronics 3 k = = ( ) N N k 2 6 6/10/2020 Arpan Deyasi, RCCIIT 12
DoS for bulk semiconductor Arpan Deyasi Introducing Pauli’s exclusion principle Quantum 3 3 k k = = ( ) 2 N k 2 2 6 3 Electronics 2 k = ( ) N k 2 k 6/10/2020 Arpan Deyasi, RCCIIT 13
DoS for bulk semiconductor Arpan Deyasi From parabolic dispersion relation Quantum 2 2 k = E * 2 m Electronics 2 E k k = * m 2 2 * m E m 2 m E k E = = * * 6/10/2020 Arpan Deyasi, RCCIIT 14
DoS for bulk semiconductor Arpan Deyasi * 1 k E m = Quantum 2 E Electronics N E N k k E = 2 * 1 N E k m = 2 2 E 6/10/2020 Arpan Deyasi, RCCIIT 15
DoS for bulk semiconductor Arpan Deyasi ρ(E) * 2 * m E 1 N E m = Quantum 2 2 2 E Electronics 3/2 * 1 2 N E m = E 2 2 2 E N E E For a particular material 6/10/2020 Arpan Deyasi, RCCIIT 16
DoS for Quantum Well Arpan Deyasi 2 (2 ) L = k A Area in k-space Quantum 2 Area of Fermi circle in k-space Electronics = 2 A k F Area of semiconductor in real space = 2 A L R 6/10/2020 Arpan Deyasi, RCCIIT 17
DoS for Quantum Well Arpan Deyasi number of energy states in real space Quantum 1 A 1 A = N A F k R Electronics 2 1 L L = 2 N k 2 2 4 2 k = = ( ) N N k 4 6/10/2020 Arpan Deyasi, RCCIIT 18
DoS for Quantum Well Arpan Deyasi Introducing Pauli’s exclusion principle Quantum 2 2 k k = = ( ) 2 N k Electronics 4 2 k = ( ) N k k 6/10/2020 Arpan Deyasi, RCCIIT 19
DoS for Quantum Well Arpan Deyasi From parabolic dispersion relation Quantum * 1 k E m = 2 E Electronics N E N k k E = * 1 N E k m = 2 E 6/10/2020 Arpan Deyasi, RCCIIT 20
DoS for Quantum Well Arpan Deyasi * * 2 1 N E m E m = Quantum 2 E Electronics * N E m = 2 DoS is independent of energy? 6/10/2020 Arpan Deyasi, RCCIIT 21
DoS for Quantum Well Arpan Deyasi The result is obtained for a particular sub-band Quantum ρ(E) Considering all the sub-bands Electronics * n N E m = − ( ) E E i 2 = 1 i E E1 E2 E3Ei-1EiEi+1 6/10/2020 Arpan Deyasi, RCCIIT 22
DoS for Quantum Wire Arpan Deyasi 2 = k L Length in k-space Quantum L Length of Fermi line in k-space Electronics = 2 L k F Length of semiconductor in real space = L L R 6/10/2020 Arpan Deyasi, RCCIIT 23
DoS for Quantum Wire Arpan Deyasi number of energy states in real space Quantum 1 L 1 L = N L F k R Electronics 1 L L = 2 N k 2 k = N 6/10/2020 Arpan Deyasi, RCCIIT 24
DoS for Quantum Wire Arpan Deyasi Introducing Pauli’s exclusion principle Quantum 2k = N Electronics 2 = ( ) N k k 6/10/2020 Arpan Deyasi, RCCIIT 25
DoS for Quantum Wire Arpan Deyasi From parabolic dispersion relation Quantum * 1 k E m = 2 E Electronics N E N k k E = * 2 1 N E m = 2 E 6/10/2020 Arpan Deyasi, RCCIIT 26
DoS for Quantum Wire Arpan Deyasi 2 * m 1 N E = Quantum ρ(E) E Electronics E E1E2E3E4E5 6/10/2020 Arpan Deyasi, RCCIIT 27
DoS for Quantum Dot Arpan Deyasi ρ(E) ρ(E) Quantum Electronics E E E1 E2 E3 E4 E1 E2 E3 E4 6/10/2020 Arpan Deyasi, RCCIIT 28