Solid-State Chemistry/Part 3/Semiconductors & Doping

3.1 Semiconductors & Doping

Semiconductors are materials with an electrical conductivity between that of metals and insulators. Their behavior can be precisely controlled through doping, making them the foundation of all modern electronic devices.

Band Theory

When atoms come together in a solid, their discrete energy levels broaden into continuous bands. The two most important bands are the valence band (highest occupied band) and the conduction band (lowest unoccupied band). The energy gap between them is the band gap $E_g$.

- Metals: No band gap (or overlapping bands), $E_g = 0$
- Semiconductors: Small band gap, $E_g \approx 0.1$--$4$ eV
- Insulators: Large band gap, $E_g > 4$ eV

Common band gaps: Si = 1.12 eV, Ge = 0.67 eV, GaAs = 1.42 eV, diamond = 5.5 eV.

Intrinsic Semiconductors

In a pure (intrinsic) semiconductor, electron-hole pairs are created by thermal excitation across the band gap. The intrinsic carrier concentration is:

$$n_i = \sqrt{N_c N_v}\,\exp\!\left(-\frac{E_g}{2k_BT}\right)$$

where $N_c$ and $N_v$ are the effective density of states in the conduction and valence bands respectively. For an intrinsic semiconductor:

$$n = p = n_i, \quad np = n_i^2$$

The mass action law $np = n_i^2$ holds for both intrinsic and extrinsic semiconductors in thermal equilibrium.

Extrinsic Semiconductors: Doping

Doping introduces impurity atoms to control the carrier type and concentration:

n-type (donor doping)

Group V atoms (P, As, Sb) in Si donate an extra electron. Majority carriers: electrons. $n \approx N_d$ when $N_d \gg n_i$.

p-type (acceptor doping)

Group III atoms (B, Al, Ga) in Si accept an electron, creating a hole. Majority carriers: holes. $p \approx N_a$ when $N_a \gg n_i$.

Fermi Level Position

The Fermi level $E_F$ shifts with doping:

$$E_F = E_i + k_BT\ln\!\left(\frac{n}{n_i}\right) \quad \text{(n-type: } E_F > E_i\text{)}$$

$$E_F = E_i - k_BT\ln\!\left(\frac{p}{n_i}\right) \quad \text{(p-type: } E_F < E_i\text{)}$$

Here $E_i$ is the intrinsic Fermi level, located near the middle of the band gap.

Electrical Conductivity

Both electrons and holes contribute to electrical conductivity:

$$\sigma = nq\mu_e + pq\mu_h$$

where $q$ is the elementary charge, $\mu_e$ and $\mu_h$ are the electron and hole mobilities. Typical values for Si at 300 K:$\mu_e \approx 1400$ cm$^2$/Vs, $\mu_h \approx 450$ cm$^2$/Vs.

The p-n Junction

When p-type and n-type semiconductors are brought into contact, a depletion region forms at the junction. The built-in potential is:

$$V_{bi} = \frac{k_BT}{q}\ln\!\left(\frac{N_a N_d}{n_i^2}\right)$$

The depletion width depends on the applied voltage $V$:

$$W = \sqrt{\frac{2\epsilon_s}{q}\left(\frac{1}{N_a} + \frac{1}{N_d}\right)(V_{bi} - V)}$$

Video Lectures

15. Semiconductors

Goodie Bag 5: Electronic Materials

16. Doping

Python: Carrier Concentration vs Temperature

Intrinsic carrier concentration for Si, Ge, and GaAs as a function of temperature, including the temperature dependence of the band gap via the Varshni relation.

Intrinsic Carrier Concentration vs Temperature

Python

Plots ni(T) for Si, Ge, and GaAs using temperature-dependent band gaps

script.py66 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
# ─────────────────────────────────────────────────
# Intrinsic Carrier Concentration vs Temperature
# for Si, Ge, and GaAs
# ─────────────────────────────────────────────────

kB = 8.617e-5  # eV/K (Boltzmann constant)

# Material parameters: (Eg at 0K in eV, alpha eV/K, beta K, Nc300, Nv300)
# Using Varshni: Eg(T) = Eg0 - alpha*T^2/(T+beta)
materials = {
    'Si':   {'Eg0': 1.166, 'alpha': 4.73e-4, 'beta': 636, 'Nc300': 2.8e19, 'Nv300': 1.04e19, 'color': '#22d3ee'},
    'Ge':   {'Eg0': 0.744, 'alpha': 4.77e-4, 'beta': 235, 'Nc300': 1.04e19, 'Nv300': 6.0e18, 'color': '#f59e0b'},
    'GaAs': {'Eg0': 1.519, 'alpha': 5.41e-4, 'beta': 204, 'Nc300': 4.7e17, 'Nv300': 7.0e18, 'color': '#a78bfa'},
}

T = np.linspace(200, 600, 500)  # Temperature range (K)

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 6))
fig.patch.set_facecolor('#0a0a1a')
for ax in (ax1, ax2):
    ax.set_facecolor('#0a0a1a')
    for spine in ax.spines.values():
        spine.set_edgecolor('#334155')

print("=" * 60)
print("Intrinsic Carrier Concentration Calculator")
print("=" * 60)

for name, props in materials.items():
    Eg_T = props['Eg0'] - props['alpha'] * T**2 / (T + props['beta'])
    # Scale Nc, Nv with T^(3/2)
    Nc = props['Nc300'] * (T / 300)**1.5
    Nv = props['Nv300'] * (T / 300)**1.5
    ni = np.sqrt(Nc * Nv) * np.exp(-Eg_T / (2 * kB * T))

ax1.semilogy(T, ni, color=props['color'], linewidth=2, label=name)
    ax2.semilogy(1000 / T, ni, color=props['color'], linewidth=2, label=name)

# Print value at 300 K
    idx300 = np.argmin(np.abs(T - 300))
    print(f"\n{name}:")
    print(f"  Eg(300K) = {Eg_T[idx300]:.3f} eV")
    print(f"  ni(300K) = {ni[idx300]:.3e} cm^-3")

ax1.set_xlabel('Temperature (K)', color='#94a3b8', fontsize=12)
ax1.set_ylabel('Intrinsic carrier concentration (cm$^{-3}$)', color='#94a3b8', fontsize=12)
ax1.set_title('n_i vs Temperature', color='white', fontsize=14)
ax1.legend(fontsize=11, facecolor='#0a0a1a', edgecolor='#334155', labelcolor='white')
ax1.tick_params(colors='#94a3b8')
ax1.grid(True, alpha=0.2)

ax2.set_xlabel('1000/T (K$^{-1}$)', color='#94a3b8', fontsize=12)
ax2.set_ylabel('Intrinsic carrier concentration (cm$^{-3}$)', color='#94a3b8', fontsize=12)
ax2.set_title('n_i vs 1000/T (Arrhenius Plot)', color='white', fontsize=14)
ax2.legend(fontsize=11, facecolor='#0a0a1a', edgecolor='#334155', labelcolor='white')
ax2.tick_params(colors='#94a3b8')
ax2.grid(True, alpha=0.2)

plt.tight_layout()
plt.savefig('carrier_concentration.png', dpi=150, bbox_inches='tight', facecolor='#0a0a1a')
print("\nPlot saved as 'carrier_concentration.png'")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran: Fermi Level & Carrier Concentrations

Computes the Fermi level position and carrier concentrations for n-type and p-type doped silicon at various doping levels and temperatures.

Fermi Level in Doped Silicon

Fortran

Carrier concentrations and Fermi level for n-type and p-type Si

fermi_level.f9084 lines

program fermi_level_doped_si
  implicit none
  integer, parameter :: dp = selected_real_kind(15)
  real(dp), parameter :: kB = 8.617d-5       ! eV/K
  real(dp), parameter :: Eg_Si = 1.12_dp     ! eV at 300K
  real(dp), parameter :: Nc_300 = 2.8d19     ! cm^-3
  real(dp), parameter :: Nv_300 = 1.04d19    ! cm^-3
  real(dp), parameter :: ni_300 = 1.5d10     ! cm^-3

real(dp) :: T, Nd, Na, n, p, Ef, Ei
  real(dp) :: Nc, Nv, ni_T
  integer :: i

! Temperatures and dopant concentrations to test
  real(dp) :: temps(3), n_dope(4), p_dope(4)
  temps = (/ 200.0_dp, 300.0_dp, 400.0_dp /)
  n_dope = (/ 1.0d14, 1.0d16, 1.0d18, 1.0d20 /)
  p_dope = (/ 1.0d14, 1.0d16, 1.0d18, 1.0d20 /)

write(*,'(A)') '=================================================='
  write(*,'(A)') ' Fermi Level & Carrier Concentrations in Doped Si'
  write(*,'(A)') '=================================================='

! n-type doping at 300 K
  T = 300.0_dp
  Nc = Nc_300
  Nv = Nv_300
  Ei = Eg_Si / 2.0_dp + 0.5_dp * kB * T * log(Nv / Nc)

write(*,'(A)') ''
  write(*,'(A,F6.1,A)') ' T = ', T, ' K'
  write(*,'(A,F6.4,A)') ' Ei = ', Ei, ' eV (from valence band)'
  write(*,'(A)') ''
  write(*,'(A)') ' --- n-type (donor doping) ---'
  write(*,'(A)') ' Nd(cm-3)      n(cm-3)       p(cm-3)       Ef-Ei(eV)'
  write(*,'(A)') ' ---------     ---------     ---------     ---------'

do i = 1, 4
    Nd = n_dope(i)
    ! Assuming complete ionization and Nd >> ni
    n = 0.5_dp * (Nd + sqrt(Nd**2 + 4.0_dp * ni_300**2))
    p = ni_300**2 / n
    Ef = Ei + kB * T * log(n / ni_300)
    write(*,'(ES12.3,3X,ES12.3,3X,ES12.3,3X,F8.4)') Nd, n, p, Ef - Ei
  end do

write(*,'(A)') ''
  write(*,'(A)') ' --- p-type (acceptor doping) ---'
  write(*,'(A)') ' Na(cm-3)      p(cm-3)       n(cm-3)       Ei-Ef(eV)'
  write(*,'(A)') ' ---------     ---------     ---------     ---------'

do i = 1, 4
    Na = p_dope(i)
    p = 0.5_dp * (Na + sqrt(Na**2 + 4.0_dp * ni_300**2))
    n = ni_300**2 / p
    Ef = Ei - kB * T * log(p / ni_300)
    write(*,'(ES12.3,3X,ES12.3,3X,ES12.3,3X,F8.4)') Na, p, n, Ei - Ef
  end do

! Temperature dependence for Nd = 1e16
  write(*,'(A)') ''
  write(*,'(A)') ' --- Temperature dependence (Nd = 1e16 cm-3) ---'
  write(*,'(A)') ' T(K)    Nc(cm-3)     ni(cm-3)     n(cm-3)      Ef-Ev(eV)'
  write(*,'(A)') ' ----    --------     --------     --------     ---------'

Nd = 1.0d16
  do i = 1, 3
    T = temps(i)
    Nc = Nc_300 * (T / 300.0_dp)**1.5_dp
    Nv = Nv_300 * (T / 300.0_dp)**1.5_dp
    ni_T = sqrt(Nc * Nv) * exp(-Eg_Si / (2.0_dp * kB * T))
    n = 0.5_dp * (Nd + sqrt(Nd**2 + 4.0_dp * ni_T**2))
    p = ni_T**2 / n
    ! Ef measured from valence band
    Ef = Eg_Si / 2.0_dp + kB * T * log(n / ni_T) &
         + 0.5_dp * kB * T * log(Nv / Nc) + Eg_Si / 2.0_dp
    ! Simpler: Ef from CB -> Ef = Ec - kT*ln(Nc/n), from VB = Eg - kT*ln(Nc/n)
    Ef = Eg_Si - kB * T * log(Nc / n)
    write(*,'(F6.1,2X,ES11.3,2X,ES11.3,2X,ES11.3,2X,F8.4)') T, Nc, ni_T, n, Ef
  end do

write(*,'(A)') ''
  write(*,'(A)') 'Done.'
end program fermi_level_doped_si

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

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