def n2D(Vds, Vgs, p, EFs):
ene0 = E0_2d_root(Vds, Vgs, EFs, p)
n2dos = p.ems*const.electron_mass / \
(math.pi*const.hbar**2)*p.temp*const.Boltzmann
beta = const.elementary_charge/(p.temp*const.Boltzmann)
ns = (fdint.fdk(0, beta*(EFs-ene0))+fdint.fdk(0, beta*(EFs-ene0-Vds)))/2*n2dos
return ns
def cylinder_seebeck(cp):
'''seebeck coefficient (V/K)'''
try:
return (constant['k'] /
constant['e']) * (2.0 * fdk(1, cp) / fdk(0, cp) - cp)
except ValueError:
return 0.
def fermi_fdint(nu):
"""
Fermi-Dirac integral of order 1/2 calculated using `fdint` package.
"""
F = fdk(k=0.5, phi=nu)
F *= 2 / np.sqrt(np.pi)
return F
def powerlaw_seebeck(cp, s):
'''
returns the seebeck coeficient (V/K)
Args:
cp: (float/ndarray) reduced chemical potential, unitless
s: (int) energy exponent restricted to integer/half-integer, unitless
Returns: (float/ndarray)
'''
if s == 0: # s=0 requires analytic simplification
return constant['k'] / constant['e'] * ((
(1. + np.exp(-cp)) * fdk(0, cp)) - cp)
else:
return constant['k'] / constant['e'] * ((
(s + 1.) * fdk(s, cp) / s / fdk(s - 1, cp)) - cp)
def calculate_n_j(self, j, Ef):
"""
Calculates electron concentration in the `j`th sub-band [1/m^3]
"""
c = 1. / (np.pi**2 * self.R**2) * np.sqrt(2. * const.k * self.T * self.meG) / const.hbar
eta = (float(Ef) - self.E_lns_CB[j]) / const.k / self.T
n_j = c * fdk(-1 / 2, eta)
self.n_js[j] = n_j
return n_j
def fermi_dirac_integral(k, phi):
if k == 0.5:
gamma = np.sqrt(pi) / 2
elif k == 0:
gamma = 1
elif k == -0.5:
gamma = sqrt(pi)
else:
raise ValueError('k is not the correct value')
fd = (1 / gamma) * fdk(k, phi)
return fd
def calculate_p(self, Ef, T):
"""
Calculates total hole concentration [1/m^3]
"""
c = 1. / (np.pi**2 * self.R**2) * np.sqrt(2. * const.k * T * self.mat.mh_DOS) / const.hbar
p_ = 0.
Eg = self.mat.get_Eg(T)
for E_ln in self.E_lns_VB:
eta = -(float(Ef) + E_ln + Eg) / const.k / T
p_ += c * fdk(-1 / 2, eta)
return p_
def Ids_ballistic2d_0(Vds, Vgs, p, EFs):
ene0 = E0_2d_root(Vds, Vgs, EFs, p)
n2d = p.ems*const.electron_mass / \
(math.pi*const.hbar**2)*p.temp*const.Boltzmann
beta = const.elementary_charge/(p.temp*const.Boltzmann)
ns = (fdint.fdk(0, beta*(EFs-ene0))+fdint.fdk(0, beta*(EFs-ene0-Vds)))/2*n2d
v0 = math.sqrt(2*p.temp*const.Boltzmann /
(math.pi*p.ems*const.electron_mass))
v1 = fdint.fdk(0.5, beta*(EFs-ene0))*(2/math.sqrt(math.pi))
v2 = fdint.fdk(0, beta*(EFs-ene0))
vinj = v0*v1/v2
f1 = 1 - fdint.fdk(0.5, beta*(EFs-ene0-Vds))/fdint.fdk(0.5, beta*(EFs-ene0))
f2 = 1 + fdint.fdk(0, beta*(EFs-ene0-Vds))/fdint.fdk(0, beta*(EFs-ene0))
return const.elementary_charge*ns*vinj*f1/f2
def powerlaw_conductivity(cp, s, sigma_E_0):
'''
returns the electrical conductivity (S/m)
Args:
cp: (float/ndarray) reduced chemical potential, unitless
s: (int) energy exponent restricted to integer/half-integer, unitless
sigma_E_0: (float) powerlaw prefactor, S/m
Returns: (float/ndarray)
'''
if s == 0: # s=0 requires analytic simplification
return sigma_E_0 / (1. + np.exp(-cp))
else:
return sigma_E_0 * s * fdk(s - 1, cp)
def seebeck_spb(eta, Lambda=0.5):
return BOLTZMANN_CONST / ELECTRON_CHARGE * (
(2. + Lambda) * fdk(1. + Lambda, eta) /
((1. + Lambda) * fdk(Lambda, eta)) - eta) * 1e+6
def seebeck_eff_mass_from_carr(eta, n, T, Lambda):
""" eta in kB*T units, n in cm-3, T in K
returns mass in m0 units """
return (2 * np.pi**2 * abs(n) * 10 ** 6 / (fdk(0.5,eta))) ** (2. / 3)\
/ (2 * ELECTRON_MASS * BOLTZMANN_CONST * T / (PLANCK_CONSTANT/2/np.pi) ** 2)
def cylinder_conductivity(cp, T, tau_0, l):
'''electrical conductivity (S/m)'''
return (constant['e']**2. * l / 2. / constant['pi']**2. /
constant['hbar']**2. * constant['k'] * T * tau_0 * fdk(0, cp))
def cylinder_carriers(cp, T, mstar, l):
'''carrier concentration (1/m3)'''
return fdk(0, cp) * (l * mstar / 2. / constant['pi']**2. /
constant['hbar']**2. * constant['k'] * T)
def sphere_conductivity(cp, T, tau_0, mstar):
'''electrical conductivity (S/m)'''
return (tau_0 * fdk(0, cp) * 8. * constant['pi'] * mstar**0.5 *
constant['e']**2. * (2. * constant['k'] * T)**1.5 / 3. /
constant['h']**3.)
def sphere_carriers(cp, T, mstar):
'''carrier concentration (1/m3)'''
return fdk(
0.5, cp) * (4. * constant['pi'] *
(2. * mstar * constant['k'] * T / constant['h']**2.)**1.5)
