Ridley’s expression

In Ridley’s book1, for a spherical band, the relation between the mobility and the momentum relaxation time is \[ \mu = \frac{e}{m^*} \frac{\left<E_k \tau_m\right>}{\left<E_k\right>} . \] but why?

Then for thermal equilibrium non-degenerate statistics and parabolic bands, \[ \frac{\left<E_k \tau_m\right>}{\left<E_k\right>} = \frac{\int_0^\infty dE_k ~ E_k^{3/2} e^{-E_k / k_B T} \tau_m } {\int_0^\infty dE_k ~ E_k^{3/2} e^{-E_k / k_B T} } \]

a derivation for mobility

Let’s start from the basic definition of the mobility and the Boltzmann transport \[ \mu = \frac{\sigma}{ne} \] where \(\sigma\) is electrical conductivity, and \(n\) is the number of density (concentration) of electrons 2.

this formula is valid when conductivity is due entirely to electrons, if a semiconductor has both electrons and holes, the total conductivity is \(\sigma = e(n\mu_e + p\mu_h)\).

Insert the expression for conductivity from Ashcroft’s book (eq 13.25, p250) 3. \[ \mu = \frac{e}{n} \frac{1}{4 \pi^3} \int d^3k~ v^2(k)~ \tau[\epsilon(k)] \left(- \frac{\partial f}{\partial \epsilon} \right)_{\epsilon=\epsilon(k)} \] where \[ n = \int d\epsilon f(\epsilon). \] This expression is the same as the one in Fischetti’s paper (Eq (9), p2241) 4.

Assuming parabolic band, \[ \epsilon(k) = \frac{\hbar^2 k^2}{2 m^*} \rightarrow k = \frac{\sqrt{2 m^* \epsilon}}{\hbar} \] \[ dk = \frac{\sqrt{m^*}}{\hbar}\frac{1}{\sqrt{2\epsilon}} d\epsilon \]

For non-degenerate semiconductors, \[ f(\epsilon) = \exp\left(- \frac{\epsilon - \mu}{k_B T}\right) \] \[ n = \frac{1}{4 \pi^3}\int d^3k~ f(\epsilon)_{\epsilon=\epsilon_k} \] \[ \frac{\partial f}{\partial \epsilon} = - \frac{1}{k_B T} f(\epsilon) \] \[ I_1 = \int dk~ v^2(k)~ \tau(k) \left(- \frac{\partial f}{\partial \epsilon} \right)_{\epsilon=\epsilon(k)} = 2\pi^2 \int k^2 dk \frac{\hbar^2 k^2}{3 m^{*2}} \tau_k \cdot \frac{1}{k_BT} \cdot \exp\left(- \frac{E-\mu}{k_BT}\right) \] where \(k^2\) and \(2\pi^2\) come from transforming the integral into spherical coordinate, the factor \(3\) comes from the average of x,y,z directions.

\[ \mu = \frac{e}{m^*} \frac{2}{3k_BT} \frac{\int dE~ E^{3/2} \tau(E) \exp(-E/k_BT)}{\int dE~ E^{1/2} \exp(-E/k_BT)} \] where \(3k _BT/2\) represents average kinetic energy of the electrons within the classic non-interacting gas model.

Matthiessen’s rule

When more than one source of scattering is present in the system, it is normally a very good approximation to combine their influences using “Matthiessen’s Rule” (developed from work by Augustus Matthiessen in 1864)

\[ \frac{1}{\mu} = \frac{1}{\mu_\text{LO}} + \frac{1}{\mu_\text{ac}} + \cdots \]

Matthiessen’s rule can also be stated in terms of the scattering time

\[ 1/\tau_m = 1/\tau_{m,ac} + 1/\tau_{m,LO} + \dots \]

a test for GaAs

Example: calculate the mobility as a function of the temperature for GaAs, with parameters from Rode’s paper 5.

source("relaxationTime.R")
source("mobilities.R")
epsinf = 10.92         #        High-frequency dielectric constant
eps0   = 12.9          #        Static dielectric constant
w0 = 8.75e12           #[hz]    frequency
me = 0.0655            #        effective mass
Xi = 7.0               #[ev]    deformation potential
vsL  = 524000          #[cm/s]  Speed of sound 
rho  = 5.360           #[g cm-3]Material density
epsp   = 1.0e0/(1.0e0/epsinf - 1.0e0/eps0)  #effective permittivity
BOLTZ = 8.61734E-5     # [eV K-1]
momentumRelaxationTime <- function(energy,temperature) {
 1.0e0/( 
         1.0e0 / momentumRelaxationTime_LOpolar_isotropic_woscreening_effcharge(permittivity = epsp,
                                                                              freq=w0,
                                                                              eff_mass = me,
                                                                              temperature=temperature,
                                                                              energy = energy)
        +1.0e0 / momentumRelaxationTime_acoustic_isotropic(mass_density = rho,
                                                             sound_velocity = vsL,
                                                             deformation_pot=Xi,
                                                             temperature=temperature,
                                                             eff_mass=me,
                                                             energy=energy)
         )
}
Temperature=seq(50,600,by=10)
mu_gaas = Temperature #just to initialize the vector
i = 0
for (temp in Temperature){
  i = i+1
  mu_gaas[i] = mobility_nondegenerate_parabolic(RelaxationTime=momentumRelaxationTime, 
                                           temperature=temp, 
                                           eff_mass=me, 
                                           energy_step=1e-4,
                                           energy_max = 10.0*BOLTZ*temp)
}

Plot the mobility vs. temperature in semilog scale

plot(Temperature,mu_gaas,type="l",log = "y",xlab = "Temperature (K)",ylab = "Mobility (cm^2/V/sec)")

The mobility at room temperature 300K is

mu_gaas[Temperature==300]
## [1] 13224.52

This value is about 1.5 times Rode’s value (8920, table II).

The mobility at 600K is

mu_gaas[Temperature==600]
## [1] 4552.4

This value is slightly closer to Rode’s value (\(\approx 4000\), Fig.7).

a test for GaSb

Example: calculate the mobility as a function of the temperature for GaSb, still with parameters from Rode’s paper 6.

source("relaxationTime.R")
source("mobilities.R")
epsinf = 13.8          #        High-frequency dielectric constant
eps0   = 15.0          #        Static dielectric constant
w0 = 7.2083e+12        #[hz]    frequency
me = 0.047             #        effective mass
Xi = 6.7               #[ev]    deformation potential
vsL  = 3.35e5          #[cm/s]  Speed of sound 
rho  = 5.66            #[g cm-3]Material density
epsp   = 1.0e0/(1.0e0/epsinf - 1.0e0/eps0)  #effective permittivity
momentumRelaxationTime <- function(energy,temperature) {
 1.0e0/( 
         1.0e0 / momentumRelaxationTime_LOpolar_isotropic_woscreening_effcharge(permittivity = epsp,
                                                                              freq=w0,
                                                                              eff_mass = me,
                                                                              temperature=temperature,
                                                                              energy = energy)
        +1.0e0 / momentumRelaxationTime_acoustic_isotropic(mass_density = rho,
                                                             sound_velocity = vsL,
                                                             deformation_pot=Xi,
                                                             temperature=temperature,
                                                             eff_mass=me,
                                                             energy=energy)
         )
}
Temperature=seq(50,600,by=1)
mu_gasb = Temperature #just to initialize the vector
i = 0
for (temp in Temperature){
  i = i+1
  mu_gasb[i] = mobility_nondegenerate_parabolic(RelaxationTime=momentumRelaxationTime, 
                                           temperature=temp, 
                                           eff_mass=me)
}

Plot the mobility vs. temperature in semilog scale

plot(Temperature,mu_gasb,type="l",xlab = "Temperature (K)",ylab = "Mobility (cm^2/V/sec)",log = "y")

The mobility at room temperature 77K is

mu_gasb[Temperature==77]
## [1] 964152.2

This value is about 1.67 times Rode’s value (\(5.76\times 10^5\), table II).

a test for ZnS

calculate the mobility vs. temperature for ZnS with the parameters from Rode’s paper 7

source("relaxationTime.R")
source("mobilities.R")
epsinf = 5.13          #        High-frequency dielectric constant
eps0   = 8.32          #        Static dielectric constant
w0 = 10.56e12          #[hz]    frequency
me = 0.23              #        effective mass
Xi = 14.5              #[ev]    deformation potential
CL = 12.89             # Elastic constant in 10^10 N/m2
CL = CL*1e4*6.241506363e+18                # 10^10 N/m2  -->  [ev/cm3]
epsp   = 1.0e0/(1.0e0/epsinf - 1.0e0/eps0)  #effective permittivity
momentumRelaxationTime <- function(energy,temperature) {
 1.0e0/( 
         1.0e0 / momentumRelaxationTime_LOpolar_isotropic_woscreening_effcharge(permittivity = epsp,
                                                                              freq=w0,
                                                                              eff_mass = me,
                                                                              temperature=temperature,
                                                                              energy = energy)
        +1.0e0 / momentumRelaxationTime_acoustic_isotropic_2(elastic_const = CL,
                                                             deformation_pot=Xi,
                                                             temperature=temperature,
                                                             eff_mass=me,
                                                             energy=energy)
         )
}
Temperature=seq(50,600,by=1)
mu_zns = Temperature #just to initialize the vector
i = 0
for (temp in Temperature){
  i = i+1
  mu_zns[i] = mobility_nondegenerate_parabolic(RelaxationTime=momentumRelaxationTime, 
                                           temperature=temp, 
                                           eff_mass=me)
}
plot(Temperature,mu_zns,type="l",xlab = "Temperature (K)",ylab = "Mobility (cm^2/V/sec)",log = "y")

The mobility at temperature 100K and 500K is

mu_zns[Temperature==100]
## [1] 6219.847
mu_zns[Temperature==300]
## [1] 396.3957
mu_zns[Temperature==500]
## [1] 151.7914

comparing to 4000 cm2/V/sec at 100K and 100 cm2/V/sec at 500K in Rode’s paper (Fig. 6) 8.

  1. Ridley: Quantum Processes in Semiconductors link↩

  2. Chapter 2: Semiconductor Fundamentals link↩

  3. Solid state physics link↩

  4. Band structure, deformation potentials, and carrier mobility in strained Si, Ge, and SiGe alloys link↩

  5. Electron Mobility in Direct-Gap Polar Semiconductors link↩

  6. Electron Mobility in Direct-Gap Polar Semiconductors link↩

  7. Electron Mobility in II-VI Semiconductors link↩

  8. Electron Mobility in II-VI Semiconductors link↩