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calculation of transmission coefficient and eigenstates using transfer matrix technique for DQWTB structure
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Course: Quantum Electronics Arpan Deyasi Quantum Electronics Calculation of Transmission Coefficient using Transfer Matrix Technique Arpan Deyasi Arpan Deyasi, RCCIIT 9/10/2020 1
Single Quantum Well Arpan Deyasi Quantum Electronics 9/10/2020 Arpan Deyasi, RCCIIT 2
Properties to be evaluated: Arpan Deyasi Electronic Properties Quantum 1. Transmission Coefficient 2. Eigen Energy 3. Density of States Electronics Optical Properties 1. Absorption Coefficient 2. Oscillator Strength 9/10/2020 Arpan Deyasi, RCCIIT 3
Numerical Techniques may be considered for Calculation Arpan Deyasi Transfer Matrix Technique (TMT) Quantum Propagation Matrix Method (PMM) Electronics Perturbation Method WKB Approximation Finite Element Method (FEM) Finite Difference Time Domain Method (FDTD) 9/10/2020 Arpan Deyasi, RCCIIT 4
Q. Which have better accuracy? Arpan Deyasi Quantum Q. Which are faster for calculation? Electronics We have to optimize between them 9/10/2020 Arpan Deyasi, RCCIIT 5
FDTD and FEM are most accurate as per the literatures Arpan Deyasi TMT & PMM are faster which incorporate fast principle Quantum Electronics 9/10/2020 Arpan Deyasi, RCCIIT 6
Arpan Deyasi Today we will start the calculation of Electronic Properties using Transfer Matrix Technique Quantum Electronics We will consider Double Quantum Well structure for our theoretical work 9/10/2020 Arpan Deyasi, RCCIIT 7
Arpan Deyasi DQWTB structure A1 A2 A3 A4 A5 Quantum a a b Electronics Z=a+b Z=0 Z=a Z Z=2a+b B5 B1 B2 B3 B4 I II III IV V 9/10/2020 Arpan Deyasi, RCCIIT 8
Schrödinger Equation for well region Arpan Deyasi for V=0 Quantum * 2 ( ) z E m 2 ( ) z d = w 2 + ( ) ( ) z = 0 z 2 2 2 2 dz Schrödinger Equation for barrier region Electronics for V=V0 ( 2 ) * E V − 2 ( ) z m 2 ( ) z d = 0 b 2 + ( ) ( ) z = 0 z 1 1 2 dz 9/10/2020 Arpan Deyasi, RCCIIT 9
Solution of Schrödinger Equation in different regions Arpan Deyasi = + i z − exp( Quantum ) exp( ) A i z B 1 1 1 1 I = + − exp( ) exp( ) A i z B i z 2 2 2 2 II Electronics = + i z − exp( ) exp( ) A i z B 3 1 3 1 III = + − exp( ) exp( ) A i z B i z 4 2 4 2 IV = + i z − exp( ) exp( ) A i z B 5 1 5 1 V 9/10/2020 Arpan Deyasi, RCCIIT 10
Ben-Daniel Duke Boundary Conditions Arpan Deyasi = Quantum I II Electronics 1 1 d d = I II * * dz dz m m I II A little modification is required in second boundary condition. Why? 9/10/2020 Arpan Deyasi, RCCIIT 11
Both κ1and κ2are functions of m* Arpan Deyasi So to avoid dual effect of m*, we will modify the 2ndcondition as Quantum Electronics d d = I II dz dz 9/10/2020 Arpan Deyasi, RCCIIT 12
at Z = 0 (1stinterface) Arpan Deyasi = Quantum I II + i z − − exp( A ) exp( exp( B ) A = i z B + 1 1 i 1 1 i Electronics exp( ) ) z z 2 2 2 2 + = + A B A B 1 1 2 2 9/10/2020 Arpan Deyasi, RCCIIT 13
at Z = 0 (1stinterface) Arpan Deyasi = ' ' Quantum I II − i z − − exp( exp( A ) exp( exp( B ) i = Electronics A i z i B i − 1 1 1 1 1 1 i ) ) i i z z 2 2 2 2 2 2 − = − A B A B 1 1 1 1 2 2 2 2 9/10/2020 Arpan Deyasi, RCCIIT 14
at Z = 0 (1stinterface) Arpan Deyasi + = + A B A B Quantum 1 1 2 2 − = − A B A B 1 1 1 1 2 2 2 2 Electronics 1 1 − 1 1 − A B A B 1 2 = In matrix notation 1 1 1 2 2 2 9/10/2020 Arpan Deyasi, RCCIIT 15
at Z = 0 (1stinterface) Arpan Deyasi Quantum 1 1 − 1 1 − A B A B 1 2 = 1 1 1 2 2 2 Electronics A B A B 1 2 = M M 1 2 1 2 9/10/2020 Arpan Deyasi, RCCIIT 16
at Z = a (2ndinterface) Arpan Deyasi = Quantum II III + − exp( exp( A ) exp( exp( B ) A = i z B + i z 2 2 i 2 2 i z − Electronics ) ) z 3 1 3 1 + − exp( exp( A ) exp( exp( B ) A = i a B + i a 2 2 i 2 2 i a − ) ) a 3 1 3 1 9/10/2020 Arpan Deyasi, RCCIIT 17
at Z = a (2ndinterface) Arpan Deyasi Quantum = ' ' II III − − − exp( exp( A ) exp( exp( i B ) i = Electronics A i z i B i z 2 i 2 2 i 2 2 2 i z − ) ) z 1 3 1 1 3 1 = − − − exp( exp( A ) exp( exp( B ) A i a B i a 2 2 2 i 2 2 2 i a − ) ) a 1 3 1 1 3 1 9/10/2020 Arpan Deyasi, RCCIIT 18
at Z = a (2ndinterface) Arpan Deyasi + − = + i a − exp( ) exp( ) exp( ) exp( ) A i a B i a A i a B 2 2 Quantum 2 2 3 1 3 1 = − − − exp( exp( A ) exp( exp( B ) A i a B i a 2 2 2 i 2 2 2 i a − ) ) a 1 3 1 1 3 1 Electronics − − exp( exp( = ) a exp( exp( − ) i a i a − A B 2 2 i 2 − ) ) i a In matrix notation 2 2 i a i a − 2 2 2 exp( exp( ) exp( exp( − ) A B i a 3 1 1 i a − ) ) 3 1 1 1 1 9/10/2020 Arpan Deyasi, RCCIIT 19
at Z = a (2ndinterface) Arpan Deyasi − − exp( exp( = Electronics ) a exp( exp( − ) i a i a − A B 2 2 i 2 Quantum − ) ) i a 2 2 i a i a − 2 2 2 exp( exp( ) exp( exp( − ) A B i a 3 1 1 i a − ) ) 3 1 1 1 1 = A B A B 3 2 M M 3 4 3 2 9/10/2020 Arpan Deyasi, RCCIIT 20
at Z = (a+b) (3rdinterface) Arpan Deyasi = III IV Quantum + i z − exp( exp( A ) exp( exp( ) A = i z B + 3 1 i 3 B 1 i − ) ) z z 4 2 4 2 Electronics + + − + exp( exp( A ( ) exp( B ( i )) + A = i a b a b B i a b 3 1 i 3 + 1 − + ( )) exp( ( )) a b 4 2 4 2 9/10/2020 Arpan Deyasi, RCCIIT 21
at Z = (a+b) (3rdinterface) Arpan Deyasi = ' ' III IV Quantum − i z − exp( exp( ) z exp( exp( ) i = A i z i B i − 1 3 A 1 1 3 B 1 i − ) ) i i z Electronics 2 4 2 2 4 2 = + − − − − + exp( exp( ( )) exp( B ( )) + A i a b a b B i a b 1 3 A 1 1 3 1 i + ( )) exp( ( )) i a b 2 4 2 2 4 2 9/10/2020 Arpan Deyasi, RCCIIT 22
at Z = (a+b) (3rdinterface) Arpan Deyasi + + − − a b + exp( exp( A Quantum ( )) + exp( B ( )) + A = i a b a b B + i 3 1 i 3 1 ( )) exp( ( )) i a b 4 2 4 2 = + − − − − + exp( exp( ( )) exp( B ( )) + A i a b a b B i a b 1 3 A 1 1 3 1 i 2 Electronics + ( )) exp( ( )) i a b 4 2 2 4 2 + − − + exp( exp( = ( )) exp( exp( − ( )) A B i a b a b i a b a b In matrix notation 3 1 1 i − − + + + + ( )) ( )) i 3 1 1 1 1 exp( exp( ( )) exp( exp( − ( )) + i a b a b i a b a b A B 2 2 i 4 + ( )) ( )) i 2 2 2 2 4 9/10/2020 Arpan Deyasi, RCCIIT 23
at Z = (a+b) (3rdinterface) Arpan Deyasi + − − + exp( exp( = Electronics ( )) exp( exp( − ( )) A B i a b a b i a b a b 3 1 1 i − − + + + + ( )) ( )) i 1 Quantum 3 1 1 1 exp( exp( ( )) exp( exp( − ( )) + i a b a b i a b a b A B 2 2 i 4 + ( )) ( )) i 2 2 2 2 4 A B A B 3 4 = M M 5 6 3 4 9/10/2020 Arpan Deyasi, RCCIIT 24
at Z = (2a+b) (4thinterface) Arpan Deyasi = IV V Quantum + − exp( exp( A ) exp( exp( B ) A = i z B + i z 4 2 i 4 2 i z − ) ) z 5 1 5 1 Electronics + − − + exp( exp( A (2 )) exp( exp( (2 )) A = i a b a b B + i − a b a b 4 2 i 4 B 2 + + (2 )) (2 )) i 5 1 5 1 9/10/2020 Arpan Deyasi, RCCIIT 25
at Z = (2a+b) (3rdinterface) Arpan Deyasi = ' ' IV V i = Quantum Electronics − − − exp( exp( A ) exp( exp( i B ) A i z i B i z 2 i 4 2 i 2 4 2 i z − ) ) z 1 5 1 1 5 1 = + − − − + exp( exp( (2 )) exp( exp( (2 (2 )) )) A i a b a b B i − a b a b + 2 4 A 2 2 4 B 2 + (2 )) i i 1 5 1 1 5 1 9/10/2020 Arpan Deyasi, RCCIIT 26
at Z = (2a+b) (3rdinterface) Arpan Deyasi ( ) ( ) ) a b + − − a b + exp Quantum (2 ) exp (2 ) A = i B + i − 4 2 i 4 B 2 ( ) ( ) a b + a b + exp (2 ) exp (2 A i 5 1 5 1 = + − − − + exp( exp( (2 )) exp( exp( (2 (2 )) )) A i a b a b B i − a b a b + 2 4 A 2 2 4 B 2 5 Electronics + (2 )) i i 1 1 1 5 1 In matrix notation + − − − + exp( exp( = (2 )) exp( exp( − (2 )) + i a b a b + i a b a b + A B 2 2 i − 4 + (2 )) (2 )) i 2 2 (2 2 exp( exp( − 2 4 exp( exp( )) (2 )) A B i a b a b i a b a b 5 1 1 i + + (2 )) (2 )) i 5 1 1 1 1 9/10/2020 Arpan Deyasi, RCCIIT 27
at Z = (2a+b) (3rdinterface) Arpan Deyasi + − − − + exp( exp( = Electronics (2 )) exp( exp( − (2 )) + i a b a b + i a b a b + A B 2 2 i − 4 Quantum + (2 )) (2 )) i 2 2 (2 2 exp( exp( − 2 4 exp( exp( )) (2 )) A B i a b a b i a b a b 5 1 1 i + + (2 )) (2 )) i 5 1 1 1 1 = A B A B 5 4 M M 7 8 5 4 9/10/2020 Arpan Deyasi, RCCIIT 28
Arpan Deyasi = A B A B 5 4 M M 7 8 Quantum 5 4 = A B A B − Electronics 1 5 4 M M 7 8 5 4 9/10/2020 Arpan Deyasi, RCCIIT 29
Arpan Deyasi A B A B 3 4 = M M 5 6 3 4 Quantum A B A B − 1 3 4 = M M Electronics 5 6 3 4 A B A B − − 1 1 3 5 = M M M M 5 6 7 8 3 5 9/10/2020 Arpan Deyasi, RCCIIT 30
Arpan Deyasi = A B A B 3 2 M M 3 4 3 2 Quantum = A B A B − 1 3 2 M M Electronics 3 4 3 2 = A B A B − − − 1 1 1 5 2 M M M M M M 3 4 5 6 7 8 5 2 9/10/2020 Arpan Deyasi, RCCIIT 31
Arpan Deyasi A B A B 1 2 = M M 1 2 1 2 Quantum A B A B − 1 1 2 = M M Electronics 1 2 1 2 = A B A B − − − − 1 1 1 1 5 1 M M M M M M M M 1 2 3 4 5 6 7 8 5 1 9/10/2020 Arpan Deyasi, RCCIIT 32
Quantum = A B A B Arpan Deyasi − − − − 1 1 1 1 5 1 M M M M M M M M 1 2 3 4 5 6 7 8 5 1 = A B A B 5 1 M Electronics 5 1 A B A B M M M M 5 1 11 12 = 5 1 21 22 9/10/2020 Arpan Deyasi, RCCIIT 33
Arpan Deyasi A B A B M M M M 5 1 11 12 = Quantum 5 1 21 22 = + A M A M B Electronics 1 11 5 12 5 = + B M A M B 1 21 5 22 5 9/10/2020 Arpan Deyasi, RCCIIT 34
Arpan Deyasi A5 A1 M11 M12 Quantum M21 M22 Electronics B5 B1 M12is the transmission coefficient when the wave is traversing from port 2 to port 1 and port 1 is terminated by matched load 9/10/2020 Arpan Deyasi, RCCIIT 35
M12 = 0 for practical device Arpan Deyasi = + A M A M B 1 11 5 12 5 Quantum A A = 1 M 11 Electronics 5 2 1 A A ( ) = = 5 T E * M M 1 11 11 9/10/2020 Arpan Deyasi, RCCIIT 36
Graphical representation of Transmission Coefficient Arpan Deyasi T(E) Quantum Electronics E E1 E2 E3 E0 9/10/2020 Arpan Deyasi, RCCIIT 37