346 lines
13 KiB
Fortran
346 lines
13 KiB
Fortran
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!
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!=======================================================================
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!
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MODULE FERMI_VALUES
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!
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! This module computes the Fermi level physical quantities
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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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SUBROUTINE CALC_FERMI(DMN,RS)
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!
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! This subroutine computes Fermi level quantities
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!
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!
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! * DMN : dimension
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! * RS : Wigner-Seitz radius (in units of a_0)
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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 : 27 Jul 2020
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!
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USE REAL_NUMBERS, ONLY : ONE
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USE ENE_CHANGE, ONLY : EV,RYD
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USE CONSTANTS_P1, ONLY : BOHR,H_BAR,M_E,K_B
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USE REAL_NUMBERS, ONLY : TWO,HALF
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USE PI_ETC, ONLY : PI,PI2
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USE UTILITIES_1, ONLY : ALFA
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!
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USE FERMI_SI
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USE FERMI_AU
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!
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IMPLICIT NONE
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!
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CHARACTER (LEN = 2) :: DMN
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!
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REAL (WP), INTENT(IN) :: RS
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REAL (WP) :: KF,EF
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REAL (WP) :: NUM,DEN
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!
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KF = ONE / (ALFA(DMN) * RS) ! a_0 * k_F
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EF = KF * KF * RYD ! Fermi energy in eV
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!
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KF_SI = KF / BOHR ! Fermi momentum in 1/m
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EF_SI = EF * EV ! Fermi energy in J
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VF_SI = H_BAR * KF_SI / M_E ! Fermi velocity in m/s
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TF_SI = EF_SI / K_B ! Fermi temperature in K
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!
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! Computation of n(E_F)
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!
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IF(DMN == '3D') THEN !
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NUM = M_E * KF_SI !
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DEN = H_BAR * H_BAR * PI2 !
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ELSE IF(DMN == '2D') THEN !
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NUM = M_E !
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DEN = H_BAR * H_BAR * PI !
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ELSE IF(DMN == '1D') THEN !
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NUM = TWO * M_E !
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DEN = H_BAR *H_BAR * PI * KF_SI !
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END IF !
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NF_SI = NUM / DEN !
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!
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KF_AU = KF ! Fermi momentum in AU
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EF_AU = HALF * KF * KF ! Fermi energy in AU
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VF_AU = KF ! Fermi velocity in AU
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!
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END SUBROUTINE CALC_FERMI
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!
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!=======================================================================
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!
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END MODULE FERMI_VALUES
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!
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!=======================================================================
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!
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MODULE FERMI_VALUES_M
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!
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! This module computes the Fermi level physical quantities
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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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SUBROUTINE CALC_FERMI_M(DMN,RS)
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!
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! This subroutine computes Fermi level quantities
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! in the presence of an external magnetic field
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!
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!
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! * DMN : dimension
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! * RS : Wigner-Seitz radius (in units of a_0)
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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 : 22 Jun 2020
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!
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USE EXT_FIELDS, ONLY : T,H
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USE REAL_NUMBERS, ONLY : HALF
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USE CONSTANTS_P1, ONLY : BOHR,H_BAR,M_E,K_B
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USE CONSTANTS_P3, ONLY : MU_B
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!
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USE FERMI_SI_M
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USE FERMI_AU_M
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!
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CHARACTER (LEN = 2) :: DMN
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!
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REAL (WP) :: RS
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REAL (WP) :: A
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!
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A=MU_B*H ! --> check B or H
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!
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! Computing k_F in AU
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!
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IF(DMN == '3D') THEN !
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KF_AU_M=KF_M_3D(RS,T,A) !
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ELSE IF(DMN == '2D') THEN !
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KF_AU_M=KF_M_2D(RS,T,A) !
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ELSE IF(DMN == '1D') THEN !
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CONTINUE !
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END IF !
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!
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EF_AU_M=HALF*KF_AU_M*KF_AU_M !
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VF_AU_M=KF_AU_M !
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!
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KF_SI_M=KF_AU_M/BOHR !
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EF_SI_M=HALF*H_BAR*H_BAR*KF_SI_M*KF_SI_M/M_E !
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VF_SI_M=H_BAR*KF_SI_M/M_E !
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TF_SI_M=EF_SI_M/K_B !
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!
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END SUBROUTINE CALC_FERMI_M
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!
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!=======================================================================
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!
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FUNCTION KF_M_3D(RS,T,A)
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!
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! This function computes the temperature-dependent Fermi wave vector
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! in the presence of an external magnetic field for 3D systems
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!
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! References: (7) A. Isihara and D. Y. Kojima, Phys. Rev. B 10,
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! 4925-4931 (1974)
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! Input parameters:
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!
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! * RS : Wigner-Seitz radius (in units of a_0)
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! * T : system temperature in SI
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! * A : sqrt(mu_B * B) : magnitude of the magnetic field
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!
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!
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! Output parameters:
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!
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! * KF_M_3D : Fermi wave number in AU
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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 : 11 Jun 2020
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!
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!
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USE REAL_NUMBERS, ONLY : ONE,FIVE,THIRD,FOURTH,SMALL
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USE CONSTANTS_P1, ONLY : K_B
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USE FERMI_AU, ONLY : KF_AU
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USE PI_ETC, ONLY : PI2
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USE G_FACTORS, ONLY : G_E
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!
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IMPLICIT NONE
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!
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REAL (WP) :: RS,T,A
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REAL (WP) :: KF_M_3D
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REAL (WP) :: BETA,ETA0,LNE0,KFT,KFM,RS2,LRS
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REAL (WP) :: A4
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REAL (WP) :: KG1,KG2,KG3
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!
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A4=A*A*A*A !
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!
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BETA=ONE/(K_B*T) !
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ETA0=BETA*KF_AU*KF_AU !
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LNE0=DLOG(ETA0) !
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!
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RS2=RS*RS !
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LRS=DLOG(RS) !
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!
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KG1=FOURTH*G_E*G_E !
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KG2=KG1 - THIRD !
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KG3=KG1 - FIVE/18.0E0_WP !
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!
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! No magnetic field contribution
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!
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KFT=KF_AU*(ONE - 0.16586E0_WP*RS + RS2*( & !
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0.0084411E0_WP*LRS - 0.027620E0_WP ) - & !
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PI2/(24.0E0_WP*ETA0*ETA0)* ( & !
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ONE - 0.16586E0_WP*RS + RS2*( & !
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0.20808E0_WP + 0.022197E0_WP*LRS - & ! ref. (7) eq. (5.1)
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0.00072406E0_WP*LNE0 - 0.027510E0_WP*LNE0*LNE0 & !
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) & !
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) & !
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) !
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!
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! Magnetic field contribution
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!
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KFM=-A4/(8.0E0_WP*KF_AU*KF_AU*KF_AU) * ( & !
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KG2 + 0.16586E0_WP*RS*KG3 + RS2 * ( & !
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0.054782E0_WP*KG1 - 0.018387E0_WP + & !
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(0.033135E0_WP - 0.072955E0_WP*KG1)*LRS) + & !
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PI2/(24.0E0_WP*ETA0*ETA0)* ( & !
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7.0E0_WP*KG2 + 0.16586E0_WP*(1.6462E0_WP-KG1)*RS + & !
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RS2* ( 1.2783E0_WP - 4.9163E0_WP*KG1 + & ! ref. (7) eq. (5.3)
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(1.25192E0_WP-0.40312E0_WP*KG1)*LRS + & !
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(0.0016895E0_WP-0.0050178E0_WP*KG1)*LNE0 + & !
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(0.064190E0_WP-0.19257E0_WP*KG1)*LNE0*LNE0 & !
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) & !
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) & !
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) !
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!
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IF(A > SMALL) THEN !
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KF_M_3D=KFT+KFM !
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ELSE !
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KF_M_3D=KFT !
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END IF !
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!
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END FUNCTION KF_M_3D
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!
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!=======================================================================
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!
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FUNCTION KF_M_2D(RS,T,A)
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!
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! This function computes the temperature-dependent Fermi wave vector
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! in the presence of an external magnetic field for 3D systems
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!
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! References: (2) A. Isihara and T. Toyoda, Phys. Rev. B 19,
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! 831-845 (1979)
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! (3) A. Isihara and D. Y. Kojima, Phys. Rev. B 19,
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! 846-855 (1979)
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!
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! Input parameters:
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!
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! * RS : Wigner-Seitz radius (in units of a_0)
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! * T : system temperature in SI
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! * A : sqrt(mu_B * B) : magnitude of the magnetic field
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! * FLD : strength of the field
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! FLD = 'WF' weak field --> ref. (2)
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! FLD = 'IF' intermediate field --> ref. (3)
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!
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!
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! Output parameters:
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!
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! * KF_M_2D : Fermi wave number in AU
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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 : 11 Jun 2020
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!
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!
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USE REAL_NUMBERS, ONLY : ZERO,ONE,FOUR,HALF,THIRD,SMALL
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USE CONSTANTS_P1, ONLY : E,K_B
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USE FERMI_AU, ONLY : KF_AU
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USE PI_ETC, ONLY : PI,PI2
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USE G_FACTORS, ONLY : G_E
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USE EXT_FIELDS, ONLY : FLD
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!
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IMPLICIT NONE
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!
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REAL (WP) :: RS,T,A
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REAL (WP) :: KF_M_2D
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REAL (WP) :: BETA,KFT,KFM,RS2,LRS
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REAL (WP) :: S0,ETA0,GAM0
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REAL (WP) :: ALPHA
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REAL (WP) :: A4,LS1,LS2
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REAL (WP) :: G2,XL,X1,X2,X3,SI,I
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!
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INTEGER :: L,LMAX
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!
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LMAX=100 ! max. value of l-sums
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!
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G2=G_E*G_E !
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!
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RS2=RS*RS !
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LRS=DLOG(RS) !
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!
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A4=A*A*A*A !
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ALPHA=A4/(RS2*RS2*KF_AU*KF_AU*KF_AU*KF_AU) !
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!
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BETA=ONE/(K_B*T) !
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ETA0=BETA*KF_AU*KF_AU !
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S0=E*E/KF_AU !
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GAM0=A*A/(KF_AU*KF_AU ) !
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!
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IF(FLD == 'WF') THEN !
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!
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! No magnetic field contribution
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!
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KFT=KF_AU*(ONE - 0.4501E0_WP*RS - 0.1427E0_WP*RS2) ! ref. (2) eq. (7.4)
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!
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! Magnetic field contribution
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!
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KFM=KF_AU*ALPHA*RS2*RS2*( & !
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(1.407E-2_WP*G2 - 3.997E-3_WP)*RS - & !
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4.689E-3_WP*RS*LRS + & !
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(3.372E-3_WP*G2 + 2.287E-2_WP)*RS2 - & !
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6.333E-3_WP*RS*LRS & !
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) !
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!
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IF(A > SMALL) THEN !
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KF_M_2D=KFT+KFM !
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ELSE
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KF_M_2D=KFT !
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END IF !
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!
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ELSE IF(FLD == 'IF') THEN !
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!
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I=0.8149E0_WP ! ref. (3) eq. (5.19)
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!
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! Calculation of l-sums
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!
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SI=-ONE ! init. of sign
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LS1=ZERO !
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LS2=ZERO !
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DO L=1,LMAX !
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XL=DFLOAT(L) !
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X1=XL*PI/GAM0 !
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X2=HALF*G_E*XL*PI !
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X3=XL*PI2/ALPHA !
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SI=-SI ! (-1)^(l+1)
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LS1=LS1+SI*DSIN(X1)*DCOS(X2)/DSINH(X3) !
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LS2=LS2+SI*DSIN(X1)*DCOS(X2)/DSIN(X3) !
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END DO !
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!
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KF_M_2D=KF_AU*(ONE + PI*LS1/ETA0 - & !
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THIRD*I*(S0**(FOUR*THIRD))* & ! ref. (3) eq. (6.1)
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(ONE-PI*LS2/ETA0) & !
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) !
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!
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END IF !
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!
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END FUNCTION KF_M_2D
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!
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END MODULE FERMI_VALUES_M
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