半导体器件原理 1.2 Calculating Carrier Concentrations

About density of states, Fermi-Dirac distribution, carrier density calculation, effective density of states, Boltzmann approximation, and water analogy for the bandgap.

• The density of carriers in a solid semiconductor depends on three factors:
  • density of state available in an energy band
  • size of band gap
  • temperature of the operation environment

# Density of States

• State: a space to hold one electron, becomes a hole without an electron.
• Density of states are not uniformly distributed in an energy band.
  • Fewer states closer to the band gap
  • More states further away from the band gap
• $D(E)$: Distribution of states as a function of energy (density of states)
• $E_C$: lowest energy in the conduction band
• $E_V$: highest energy in the valence band
• $E_g=E_C-E_V$: band gap energy
  • Exact value of $E_C$ and $E_V$ does not matter, only the difference matters.
  • $E_g\approx 1.1\mathrm{eV}$ for silicon

# Fermi-Dirac Distribution

• Without energy supplied (absolute zero temperature), all electrons stay at lowest energy states.
• With energy supplied:
  • The distribution of carriers in the states of a band is governed by the Fermi-Dirac distribution function:$$f_e(E)=\frac{1}{1+e^{\frac{E-E_F}{kT}}}$$the probability that an electron state at energy $E$ is occupied by an electron.
  • $E_F$: Fermi energy level, a reference energy level
  • $k$: Boltzmann’s constant
  • $T$: absolute temperature in Kelvin

• Materials with a band gap: $E_F$ lies within the band gap, due to the symmetry of the probability distribution. ($\frac{1}{1+e^x}+\frac{1}{1+e^{-x}}=1$)
• In the valence band: the probability of holes occupying a state$$f_h(E)=1-f_e(E)=\frac{1}{1+e^{\frac{E_F-E}{kT}}}$$

# Calculating Carrier Density

• $n(E)=D(E)\cdot f_e(E)$: density of electrons at energy $E$, usually for conduction band
• $p(E)=D(E)\cdot f_h(E)$: density of holes at energy $E$, usually for valence band
• Total number:$$\begin{array}{rl}n& =\int D(E)\cdot f_e(E)\\ p& =\int D(E)\cdot f_h(E)\end{array}$$$p=n$ because every electron excited to the conduction band leaves a hole in the valence band (electron-hole pair generation).
  • For pure or intrinsic silicon, $n=p=n_i$

# Equivalent (Effective) Density of States

• We do not care about $D(E)$, we only care about the total number of carriers.
• Simplify:
  • Assume energy band is narrow, all states are at $E_C$ or $E_V$.
  • Define an equivalent density of states $N_C$ (total number of states if they are all located at $E_C$) and $N_V$ (similar for valence band).
  • Then:$$\begin{array}{rl}n& =N_C\cdot f_e(E_C)\\ p& =N_V\cdot f_h(E_V)\end{array}$$
  • $$\begin{array}{r}{N}_C=2{\left(\frac{2\pi m_{e}kT}{h^2}\right)}^{3/2}\\ N_V=2{\left(\frac{2\pi m_{h}kT}{h^2}\right)}^{3/2}\end{array}$$
    • $m_e$: effective mass of electrons
    • $m_h$: effective mass of holes
    • $h$: Planck’s constant
• For silicon at room temperature ($T=25{}^{\circ }\mathrm{C}$):$$N_C=2.8\times {10}^{19}{\mathrm{cm}}^{-3},N_V=1.09\times {10}^{19}{\mathrm{cm}}^{-3}$$

# Boltzmann Approximation

• $kT\approx 0.026\mathrm{eV}\text{ at }25{}^{\circ }\mathrm{C}$, $E-E_F\gg kT$
• Thus,$$f_e(E)=\frac{1}{1+e^{\frac{E-E_F}{kT}}}\approx e^{-\frac{E-E_F}{kT}}$$for conduction band when $E_F$ is not too close to the conduction band.
• After the approximation, $f_h(E)=1-f_e(E)$ is no longer valid, and$$f_h(E)\approx e^{-\frac{E_F-E}{kT}}$$
• Finally, the intrinsic electron concentration and the intrinsic hole concentration:$$\begin{array}{r}n=N_C\times f_c(E_C)=N_C\times e^{-\frac{E_C-E_F}{kT}}=n_i\\ p=N_V\times f_h(E_V)=N_V\times e^{-\frac{E_F-E_V}{kT}}=p_i\end{array}$$
• Multiplying the two equations:$$np=n_i^2=N_C{N}_V{e}^{-\frac{E_g}{kT}}$$This shows that the number of carriers increases with temperature, and decreases with band gap.
• Number of carriers of silicon at room temperature:$$n_i=1.45\times {10}^{10}{\mathrm{cm}}^{-3}\sim {10}^{10}{\mathrm{cm}}^{-3}$$

# Water Analogy for the bandgap

• A light, hollow, closed box partially filled with water.
• At the water to air interface, the probability to find a water molecule is $0.5$. This is the Fermi level of the box.
• Uniform external potential -> placing the box in a larger water tank where the water level represents the external potential.
• The box will float, aligning the water level inside the box with the water level outside -> The Fermi level is a reference energy level with respect to the surrounding.
• The band gap is a solid box without water molecules dropped inside the box, it will float in water, and the plane separating the floating part and sinking part is the Fermi level.
• The solid box has cracks, water molecules can jump above through the box through the cracks -> electrons excited from valence band to conduction band.
• External voltage applied -> external water level changes -> Fermi level at the two ends of the semiconductor changes -> current flows.
• Battery only controls the two ends, inside the semiconductor, the Fermi level is subject to the properties of the material.

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