半导体器件原理 Summary as of 1.8

Formulas and concepts till 1.8 of the Principle of Semiconductor Devices course. Mainly formulas though.

Some Important Values

For intrinsic silicon at room temperature:

k T \approx 0.026 eV

E_{g} \approx 1.1 eV

n_{i} \approx 1.45 \times 10^{10} {cm}^{- 3} \sim 10^{10} {cm}^{- 3}

V_{ON} \approx 0.7 V

Carrier Concentration

Fermi-Dirac Distribution

\begin{aligned} f_{e} (E) & = \frac{1}{1 + e^{\frac{E - E_{F}}{k T}}} \\ f_{h} (E) & = \frac{1}{1 + e^{\frac{E_{F} - E}{k T}}} \end{aligned}

Boltzmann Approximation

$k T \approx 0.026 eV$ , when $T = 25^{\circ} C$

For electrons, when $E - E_{F} ≫ k T$ :

f_{e} (E) \approx e^{- \frac{E - E_{F}}{k T}}

For holes, when $E_{F} - E ≫ k T$ :

f_{h} (E) \approx e^{- \frac{E_{F} - E}{k T}}

Equivalent DoS

N_{C}, N_{V} \propto T^{\frac{3}{2}}

In intrinsic semiconductor:

n_{i} = N_{C} e^{- \frac{E_{C} - E_{F}}{k T}}

p_{i} = N_{V} e^{- \frac{E_{F} - E_{V}}{k T}}

n_{i} = p_{i}

n_{i}^{2} = n p = N_{C} N_{V} e^{- \frac{E_{g}}{k T}}

Doping

N-type: Donor dopants, $n \approx N_{D}$ , $E_{C} > E_{D}$
- $E_{C} - E_{F} = k T \ln \frac{N_{C}}{N_{D}}$
- $E_{F} - E_{i} = k T \ln \frac{n}{n_{i}}$
P-type: Acceptor dopants, $p \approx N_{A}$ , $E_{A} > E_{V}$
- $E_{F} - E_{V} = k T \ln \frac{N_{V}}{N_{A}}$
- $E_{i} - E_{F} = k T \ln \frac{p}{n_{i}}$

PN Junction Formation

Built-in Potential

\begin{aligned} q V_{bi} & = E_{i p} - E_{i n} \\ = E_{i p} - E_{F} + (E_{F} - E_{i n}) \\ = k T \ln \frac{N_{A} N_{D}}{n_{i}^{2}} \\ \Rightarrow V_{bi} & = \frac{k T}{q} \ln \frac{N_{A} N_{D}}{n_{i}^{2}} \end{aligned}

Depletion Region Width:

N_{A} x_{p} = N_{D} x_{n}

\begin{aligned} E_{max} & = - \frac{q}{ε_{Si}} N_{A} x_{p} \\ = - \frac{q}{ε_{Si}} N_{D} x_{n} \end{aligned}

V_{bi} = \frac{q N_{A} x_{p}^{2}}{2 ε_{Si}} + \frac{q N_{D} x_{n}^{2}}{2 ε_{Si}}

{\begin{cases} x_{p} = \sqrt{\frac{2 ε_{Si} V_{bi}}{q} \cdot \frac{N_{D}}{N_{A} (N_{A} + N_{D})}} \\ x_{n} = \sqrt{\frac{2 ε_{Si} V_{bi}}{q} \cdot \frac{N_{A}}{N_{D} (N_{A} + N_{D})}} \end{cases}

\Rightarrow x_{d} = \sqrt{\frac{2 ε_{Si} V_{bi}}{q} (\frac{1}{N_{A}} + \frac{1}{N_{D}})}

Ideal PN Junction Equation

x_{d} = \sqrt{\frac{2 ε_{Si} (V_{bi} - V_{A})}{q} (\frac{1}{N_{A}} + \frac{1}{N_{D}})}

And the heavily doped side contribution can be neglected.

I_{D} = I_{0} (e^{\frac{q V_{A}}{k T}} - 1)

I_{0} = q (D_{n} \frac{n_{p 0}}{L_{n}} + D_{p} \frac{p_{n 0}}{L_{p}})

Carrier Motions

Minority & Majority Carriers

For minority carriers, the diffusion current dominates
For majority carriers, the drift current dominates

Short Diode

$W_{n} < L_{p}$ and $W_{p} < L_{n}$ , no recombination in the neutral regions.

Minority carrier distribution is a straight line, (diffusion) current is constant across the junction.

J_{n, diff} = q D_{n} \frac{n_{p 0}}{W_{p}} (e^{\frac{q V_{A}}{k T}} - 1)

J_{p, diff} = q D_{p} \frac{p_{n 0}}{W_{n}} (e^{\frac{q V_{A}}{k T}} - 1)

Long Diode

Recombination happen in the neutral regions.

Recombination increases the current.

Real PN Junction Characteristics

Non-ideal Effects

When $V_{A}$ is small, recombination makes current larger than ideal
When $V_{A}$ is large, high-level injection causes accumulation of majority carriers near the depletion region, driving them against the current flow
- Always takes place in the lightly doped side

Turn-on

Depletion region disappears when $V_{A}$ is increased beyond $V_{bi}$ , and the diode becomes a resistor.

Breakdown

Both mechanisms are somehow related to maximum electric field in the depletion region.

\begin{aligned} | E_{max} | & = \sqrt{\frac{2 q V_{bi}}{ε_{Si}} \frac{N_{A} N_{D}}{N_{A} + N_{D}}} \\ (which happens to) & = 2 \frac{V_{bi}}{x_{d}} \end{aligned}

Temperature Effects

Both sides behave more like intrinsic semiconductor
$V_{bi}$ and $V_{ON}$ decreases

PN Junction Switching and Model

Capacitance

Depletion region capacitance (zero bias/reverse bias)
$\begin{aligned} C_{j} & = \frac{ε_{Si}}{W_{d}} \\ = \frac{ε_{Si}}{\sqrt{\frac{2 ε_{Si} (V_{bi} - V_{A})}{q} (\frac{1}{N_{A}} + \frac{1}{N_{D}})}} \\ = \frac{C_{j 0}}{\sqrt{1 - \frac{V_{A}}{V_{bi}}}} \end{aligned}$
Diffusion capacitance (forward bias)
$\begin{aligned} Q_{diff} & = \frac{q}{2} (L_{n} n_{p 0} + L_{p} p_{n 0}) e^{\frac{q V_{A}}{k T}} \\ C_{diff} & = \frac{d Q_{diff}}{d V_{A}} \\ = \frac{q}{k T} \cdot \frac{q}{2} (L_{n} n_{p 0} + L_{p} p_{n 0}) e^{\frac{q V_{A}}{k T}} \\ = \frac{Q_{diff}}{V_{th}} \end{aligned}$

Large Signal Model

A parallel combination of a current source, a depletion capacitance, and a diffusion capacitance, then series with a resistance.

\begin{aligned} I_{D} & = I_{0} (e^{\frac{q V_{A}}{k T}} - 1) & (Current Source) \\ C_{j} & = \frac{C_{j 0}}{\sqrt{1 - \frac{V_{A}}{V_{bi}}}} & (Depletion Capacitance) \\ C_{diff} & = \frac{Q_{diff}}{V_{th}} & (Diffusion Capacitance) \end{aligned}

Small Signal Model

\begin{aligned} g_{d} (V_{b}) & = {\frac{d I_{D}}{d V_{A}} |}_{V_{A} = V_{b}} \\ = \frac{q}{k T} I_{0} e^{\frac{q V_{b}}{k T}} \\ \approx \frac{q}{k T} I_{D} (V_{b}) (if we drop -1 in I_{D}) \\ = \frac{I_{D} (V_{b})}{V_{th}} \end{aligned}

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文章

半导体器件原理 Summary as of 1.8

Some Important Values

Carrier Concentration

Fermi-Dirac Distribution

Boltzmann Approximation

Equivalent DoS

Doping

PN Junction Formation

Built-in Potential

Depletion Region Width:

Ideal PN Junction Equation

Carrier Motions

Minority & Majority Carriers

Short Diode

Long Diode

Real PN Junction Characteristics

Non-ideal Effects

Turn-on

Breakdown

Temperature Effects

PN Junction Switching and Model

Capacitance

Large Signal Model

Small Signal Model

文件历史

博客

笔记

备忘录

生活

文章

半导体器件原理 Summary as of 1.8

# Some Important Values

# Carrier Concentration

# Fermi-Dirac Distribution

# Boltzmann Approximation

# Equivalent DoS

# Doping

# PN Junction Formation

# Built-in Potential

# Depletion Region Width:

# Ideal PN Junction Equation

# Carrier Motions

# Minority & Majority Carriers

# Short Diode

# Long Diode

# Real PN Junction Characteristics

# Non-ideal Effects

# Turn-on

# Breakdown

# Temperature Effects

# PN Junction Switching and Model

# Capacitance

# Large Signal Model

# Small Signal Model

文件历史

Some Important Values

Carrier Concentration

Fermi-Dirac Distribution

Boltzmann Approximation

Equivalent DoS

Doping

PN Junction Formation

Built-in Potential

Depletion Region Width:

Ideal PN Junction Equation

Carrier Motions

Minority & Majority Carriers

Short Diode

Long Diode

Real PN Junction Characteristics

Non-ideal Effects

Turn-on

Breakdown

Temperature Effects

PN Junction Switching and Model

Capacitance

Large Signal Model

Small Signal Model