415 lines
13 KiB
Fortran
415 lines
13 KiB
Fortran
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!
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!=======================================================================
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!
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MODULE CONFINEMENT_WF
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!
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USE ACCURACY_REAL
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!
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CONTAINS
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!
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!
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!=======================================================================
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!
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FUNCTION CONFIN_WF(R,TH,SB_I,OM0,N_DEP,N_INV,EPSS)
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!
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! This function computes the confinement wave function
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!
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!
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!
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! Input parameters:
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!
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! * R : radial parameter
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! * TH : polar angle
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! * SB_I : sub-band index (0, 1, 2, ...) --> limited to 10
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! * N_DEP : electron concentration in depletion layer (SI)
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! * N_INV : electron concentration in inversion layer (SI)
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! * EPSS : dielectric constant of the semiconductor
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! * OM0 : frequency of the confinement potential (SI)
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!
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! Intermediate parameters:
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!
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! * CONFIN : type of confinement
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! CONFIN = 'INF_QWW'
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! CONFIN = 'INVLAYE'
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! CONFIN = 'IQWE_LB'
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! CONFIN = 'PC1_QWI'
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! CONFIN = 'PC2_QWI'
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! CONFIN = 'SWC_QWI'
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! * L : length of the quantum well (SI)
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!
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!
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! Output parameters:
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!
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! * CONFIN_WF : wave function
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!
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!
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!
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! Author : D. Sébilleau
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!
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! Last modified : 10 Jun 2020
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!
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!
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USE REAL_NUMBERS, ONLY : ZERO
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USE CONFIN_VAL, ONLY : CONFIN,R0,L
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!
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IMPLICIT NONE
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!
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INTEGER, INTENT(IN) :: SB_I
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!
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REAL (WP), INTENT(IN) :: R,TH,OM0
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REAL (WP), INTENT(IN) :: N_DEP,N_INV,EPSS
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REAL (WP) :: CONFIN_WF
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!
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IF(CONFIN == 'NO-CONF') THEN !
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CONFIN_WF = ZERO !
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ELSE IF(CONFIN == 'INF_QWW') THEN !
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CONFIN_WF = INF_QWW_WF(R,TH,R0,SB_I) !
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ELSE IF(CONFIN == 'INVLAYE') THEN !
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CONFIN_WF = INVLAYE_WF(R,N_DEP,N_INV,EPSS) !
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ELSE IF(CONFIN == 'IQWE_LB') THEN !
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CONFIN_WF = IQWE_LB_WF(R,L) !
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ELSE IF(CONFIN == 'PC1_QWI') THEN !
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CONFIN_WF = PC1_QWI_WF(R,OM0,SB_I) !
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ELSE IF(CONFIN == 'PC2_QWI') THEN !
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CONFIN_WF = PC2_QWI_WF(R,OM0) !
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ELSE IF(CONFIN == 'SWC_QWI') THEN !
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CONFIN_WF = SWC_QWI_WF(R,L) !
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END IF !
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!
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END FUNCTION CONFIN_WF
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!
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!=======================================================================
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!
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FUNCTION INF_QWW_WF(R,TH,R0,SB_I)
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!
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! This function computes the wave function in the two lower subbands
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! of a quantum-well wire.
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!
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! Reference: (1) A. Gold and A. Ghazali, Phys. Rev. B 41, 7626 (1990)
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!
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!
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! Input parameters:
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!
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! * R : radial parameter
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! * TH : polar angle
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! * R0 : radius of the quantum wire (same unit as R)
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! * SB_I : sub-band index
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!
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! Output parameters:
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!
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! * INF_QWW_FF : form factor
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!
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! Author : D. Sébilleau
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!
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! Last modified : 21 Sep 2020
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!
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!
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USE REAL_NUMBERS, ONLY : ONE,THREE
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USE COMPLEX_NUMBERS
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USE PI_ETC, ONLY : PI_INV
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!
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IMPLICIT NONE
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!
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INTEGER :: SB_I
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!
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REAL (WP), INTENT(IN) :: R,TH,R0
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REAL (WP) :: R02,R03,R2,R3
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REAL (WP) :: COEF
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!
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REAL (WP) :: SQRT
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!
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COMPLEX (WP) :: INF_QWW_WF
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!
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COMPLEX (WP) :: CMPLX,EXP
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!
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R02 = R0 * R0 !
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R03 = R02 * R0 !
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R2 = R * R !
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R3 = R2 * R !
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!
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!
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IF(SB_I == 1) THEN !
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IF(R < R0) THEN !
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COEF = SQRT(THREE * PI_INV / R02) !
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INF_QWW_WF = CMPLX(COEF * (ONE - R2 / R02),KIND=WP) !
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ELSE !
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INF_QWW_WF = ZEROC !
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END IF !
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ELSE IF(SB_I == 2) THEN !
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IF(R < R0) THEN !
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COEF = SQRT(12.0E0_WP * PI_INV / R02) !
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INF_QWW_WF = COEF * (R / R0 - R3 / R03) * EXP(IC * TH) !
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ELSE !
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INF_QWW_WF = ZEROC !
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END IF !
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END IF !
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!
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END FUNCTION INF_QWW_WF
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!
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!=======================================================================
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!
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FUNCTION INVLAYE_WF(Z,N_DEP,N_INV,EPSS)
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!
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! This function computes the wave function of the surface inversion layer
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! of a semiconductor in the z direction
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!
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! Reference: (1) M. Jonson, J. Phys. C: Solid State Phys. 9, 3055-3071 (1976)
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!
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!
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! Input parameters:
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!
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! * Z : z coordinate
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! * N_DEP : electron concentration in depletion layer (SI)
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! * N_INV : electron concentration in inversion layer (SI)
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! * EPSS : dielectric constant of the semiconductor
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!
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! Output parameters:
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!
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! * INVLAYE_WF : relaxation time in seconds
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!
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! Author : D. Sébilleau
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!
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! Last modified : 21 Sep 2020
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!
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!
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USE REAL_NUMBERS, ONLY : HALF,THIRD
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USE PI_ETC, ONLY : PI
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USE CONSTANTS_P1, ONLY : H_BAR,E,M_E
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!
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IMPLICIT NONE
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!
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REAL (WP), INTENT(IN) :: Z,N_DEP,N_INV,EPSS
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REAL (WP) :: INVLAYE_WF
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REAL (WP) :: B,B3
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REAL (WP) :: NUM,DEN
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!
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REAL (WP) :: SQRT,EXP
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!
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! Computation of parameter B
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!
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NUM = 48.0E0_WP * PI * E * E * M_E !
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DEN = EPSS * H_BAR * H_BAR !
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!
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B = (NUM * ( N_DEP + 11.0E0_WP * & !
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N_INV / 32.0E0_WP ) / DEN)**THIRD ! ref. 1 eq. (21)
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!
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B3 = B * B * B !
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!
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INVLAYE_WF = SQRT(HALF * B3) * Z * EXP(- HALF * B * Z) ! ref. 1 eq. (19)
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!
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END FUNCTION INVLAYE_WF
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!
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!=======================================================================
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!
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FUNCTION IQWE_LB_WF(Z,L)
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!
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! This function computes the z-axis wave function of the a quantum well
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! with an infinite barrier when only the lower subband is filled
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!
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! Reference: (1) T. Vazifehshenas and T. Salavati-fard,
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! Physica E 41, 1297–1300 (2009)
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!
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!
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! Input parameters:
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!
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! * Z : z value
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! * L : length of the quantum well (SI)
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!
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! Output parameters:
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!
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! * IQWE_LB_WF : form factor
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!
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! Author : D. Sébilleau
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!
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! Last modified : 21 Sep 2020
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!
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!
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USE REAL_NUMBERS, ONLY : TWO
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USE PI_ETC, ONLY : PI
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!
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IMPLICIT NONE
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!
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REAL (WP), INTENT(IN) :: Z,L
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REAL (WP) :: IQWE_LB_WF
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!
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REAL (WP) :: SQRT,SIN
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!
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IQWE_LB_WF = SQRT(TWO / L) * SIN(PI * Z / L) !
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!
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END FUNCTION IQWE_LB_WF
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!
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!=======================================================================
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!
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FUNCTION PC1_QWI_WF(Y,OM0,N)
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!
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! This function computes the wave function of a quantum wire under
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! an harmonic confinement potential of the form 1/2 m omega_0^2 y^2
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! in the y direction
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!
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! Reference: (1) M. Tas, PhD thesis, Middle East Technical University (2004)
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!
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!
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! Input parameters:
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!
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! * Y : parameter in the confinement direction
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! * OM0 : frequency of the confinement potential (SI)
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! * N : sub-band index (0, 1, 2, ...) --> limited to 10
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!
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! Output parameters:
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!
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! * PC1_QWI_WF : form factor
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!
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! Author : D. Sébilleau
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!
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! Last modified : 21 Sep 2020
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!
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!
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USE REAL_NUMBERS, ONLY : ONE,TWO,HALF
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USE PI_ETC, ONLY : PI
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USE CONSTANTS_P1, ONLY : H_BAR,M_E
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USE EXT_FUNCTIONS, ONLY : H_POLYNOMIAL_VALUE
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!
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IMPLICIT NONE
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!
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INTEGER, INTENT(IN) :: N
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!
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REAL (WP), INTENT(IN) :: Y,OM0
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REAL (WP) :: PC1_QWI_WF
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REAL (WP) :: B,YB,ZZ,COEF
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REAL (WP) :: FC(0:10),X(1),P(1,0:10)
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!
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REAL (WP) :: SQRT,EXP
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!
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DATA FC / 1.0E0_WP, 1.0E0_WP, 2.0E0_WP, & !
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6.0E0_WP, 24.0E0_WP, 120.0E0_WP, & ! factorials
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720.0E0_WP, 5040.0E0_WP, 40320.0E0_WP, & !
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362880.0E0_WP, 3628800.0E0_WP / !
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!
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IF(N > 10) THEN !
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WRITE(6,10) !
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STOP !
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END IF !
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!
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! Characteristic length of the harmonic potential
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! (serves as effective diameter of quantum wire)
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!
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B = SQRT(H_BAR / (M_E * OM0)) !
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!
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YB = Y / B !
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!
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ZZ = HALF * YB * YB !
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!
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COEF = SQRT(ONE / (TWO**N * FC(N) * SQRT(PI * B))) !
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!
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! Computing the Hermite polynomial
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!
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X(1) = YB !
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CALL H_POLYNOMIAL_VALUE(1,N,X,P) !
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!
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PC1_QWI_WF = COEF * EXP(- ZZ) * P(1,N) ! ref. 1 eq. (3.67)
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!
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! Format
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!
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10 FORMAT(//,5X,'<<<<< SUB-BAND INDEX TOO LARGE >>>>>',//)
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!
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END FUNCTION PC1_QWI_WF
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!
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!=======================================================================
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!
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FUNCTION PC2_QWI_WF(R,OM0)
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!
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! This function computes the wave function of a quantum wire under
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! an harmonic confinement potential of the form 1/8 m omega_0^2 (x^2+y^2)
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! in the (x,y) direction (lowest sub-band only)
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!
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! Reference: (1) M. Tas, PhD thesis, Middle East Technical University (2004)
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!
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!
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! Input parameters:
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! ->
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! * R : parameter in the confinement directions (projection of r
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! onto the(xy) plane)
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! * OM0 : frequency of the confinement potential (SI)
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!
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! Output parameters:
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!
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! * PC2_QWI_WF : form factor
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!
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! Author : D. Sébilleau
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!
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! Last modified : 21 Sep 2020
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!
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!
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USE REAL_NUMBERS, ONLY : ONE,TWO,FOURTH
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USE PI_ETC, ONLY : PI
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USE CONSTANTS_P1, ONLY : H_BAR,M_E
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!
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IMPLICIT NONE
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!
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REAL (WP), INTENT(IN) :: R,OM0
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REAL (WP) :: PC2_QWI_WF
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REAL (WP) :: B,ZZ,COEF
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!
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REAL (WP) :: SQRT,EXP
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!
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! Characteristic length of the harmonic potential
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! (serves as effective diameter of quantum wire)
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!
|
|||
|
|
B = SQRT(H_BAR / (M_E * OM0)) !
|
|||
|
|
!
|
|||
|
|
ZZ = FOURTH * R * R / (B * B) !
|
|||
|
|
!
|
|||
|
|
COEF =ONE / SQRT(TWO * PI * B * B) !
|
|||
|
|
!
|
|||
|
|
PC2_QWI_WF = COEF * EXP(- ZZ) ! ref. 1 eq. (3.74)
|
|||
|
|
!
|
|||
|
|
END FUNCTION PC2_QWI_WF
|
|||
|
|
!
|
|||
|
|
!=======================================================================
|
|||
|
|
!
|
|||
|
|
FUNCTION SWC_QWI_WF(Y,A)
|
|||
|
|
!
|
|||
|
|
! This function computes the wave function of the a quantum-well wire
|
|||
|
|
! with an infinite barrier when only the lowest subband is filled
|
|||
|
|
!
|
|||
|
|
! The barrier is from -a/2 to a/2
|
|||
|
|
!
|
|||
|
|
!
|
|||
|
|
! Reference: (1) M. Tas, PhD thesis, Middle East Technical University (2004)
|
|||
|
|
!
|
|||
|
|
!
|
|||
|
|
! Input parameters:
|
|||
|
|
!
|
|||
|
|
! * Y : parameter along the y axis
|
|||
|
|
! * A : length of the quantum well (SI)
|
|||
|
|
!
|
|||
|
|
! Output parameters:
|
|||
|
|
!
|
|||
|
|
! * SWC_QWI_WF : form factor
|
|||
|
|
!
|
|||
|
|
! Author : D. Sébilleau
|
|||
|
|
!
|
|||
|
|
! Last modified : 21 Sep 2020
|
|||
|
|
!
|
|||
|
|
!
|
|||
|
|
USE REAL_NUMBERS, ONLY : ZERO,TWO,HALF
|
|||
|
|
USE PI_ETC, ONLY : PI
|
|||
|
|
!
|
|||
|
|
IMPLICIT NONE
|
|||
|
|
!
|
|||
|
|
REAL (WP), INTENT(IN) :: Y,A
|
|||
|
|
REAL (WP) :: SWC_QWI_WF
|
|||
|
|
!
|
|||
|
|
REAL (WP) :: COS
|
|||
|
|
!
|
|||
|
|
IF( (- HALF * A <= Y) .AND. (Y <= HALF * A) ) THEN !
|
|||
|
|
SWC_QWI_WF = TWO * COS(PI * Y / A) / A ! ref. 1 eq. (3.86)
|
|||
|
|
ELSE !
|
|||
|
|
SWC_QWI_WF = ZERO !
|
|||
|
|
END IF !
|
|||
|
|
!
|
|||
|
|
END FUNCTION SWC_QWI_WF
|
|||
|
|
!
|
|||
|
|
END MODULE CONFINEMENT_WF
|