---
title: Transistors and Optoelectronic Devices
module: Semiconductors
moduleNumber: 7
lessonNumber: 5
order: 705
summary: >
  Two junctions in series make a bipolar transistor whose thin base gives current
  gain; a gate over an oxide makes a MOSFET whose inversion channel switches
  digital logic. Run in reverse, a junction converts photons to current. This
  lesson derives the transistor current gain and the MOSFET channel current, then
  treats the LED, the diode laser, and the illuminated solar-cell characteristic.
topics: [Semiconductors]
sources:
  - book: Kittel
    ref: "Ch. 8, Ch. 17–18 — Semiconductor Crystals; Surface and Interface Physics; Nanostructures"
  - book: Ashcroft & Mermin
    ref: "Ch. 29 — Inhomogeneous Semiconductors"
  - book: Hook & Hall
    ref: "Ch. 5 — Semiconductors"
---

The [p-n junction](/condensed-matter/semiconductors/the-pn-junction) is a
one-dimensional device: it rectifies. Combining junctions and gating them makes the
active devices that amplify and switch — the bipolar transistor and the MOSFET —
and running a junction as an emitter or collector of light makes the
optoelectronic devices. All of them are the junction physics of the previous
lessons rearranged in geometry.

## The bipolar junction transistor

A bipolar junction transistor (BJT) is two junctions sharing a thin middle region:
an **npn** device is a heavily doped n **emitter**, a thin lightly doped p
**base**, and an n **collector**. In the useful **forward-active** mode the
emitter-base junction is forward biased and the collector-base junction reverse
biased.

Forward bias on the emitter-base junction injects electrons from the emitter into
the base. The base is made **thin** — its width $W_B$ far smaller than the electron
diffusion length $L_n$ — so injected electrons cross it by diffusion before more
than a small fraction recombine. Reaching the reverse-biased collector junction,
they are swept into the collector by its field. The collector current is nearly the
whole injected emitter current; only the small recombined remainder, plus the hole
injection back into the emitter, must be supplied as **base current**.

$$
% caption: npn transistor in forward-active mode. The forward-biased emitter-base
% junction injects electrons into the thin base; most diffuse across before
% recombining and are collected by the reverse-biased base-collector junction, so
% a small base current controls a large collector current.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % conduction band profile: emitter low, base hump, collector low
  \draw[acc, very thick]
    (0,1.0) -- (1.3,1.0)
    .. controls (1.9,1.0) and (2.0,2.2) .. (2.6,2.2)
    -- (3.4,2.2)
    .. controls (4.0,2.2) and (4.1,0.4) .. (4.7,0.4)
    -- (6.0,0.4);
  % region labels
  \node[anchor=north, font=\scriptsize] at (0.7,0.95) {emitter};
  \node[anchor=south, font=\scriptsize] at (3.0,2.25) {base};
  \node[anchor=north, font=\scriptsize] at (5.3,0.35) {collector};
  % electron flow arrow across
  \fill[acc] (1.1,1.05) circle (2.2pt);
  \draw[acc, ->, thick] (1.25,1.15) .. controls (2.2,1.9) and (3.6,2.35) .. (4.6,0.55);
  \node[acc, anchor=south, font=\scriptsize] at (3.0,1.35) {electrons};
\end{tikzpicture}
$$

The figure of merit is the **current gain**. The fraction of emitter current
reaching the collector is the transport factor; for a base thin compared to the
diffusion length it is

$$
\alpha = \frac{I_C}{I_E} \approx 1 - \frac{1}{2}\left(\frac{W_B}{L_n}\right)^2,
$$

close to but below one. The common-emitter gain relates collector to base current,

$$
\beta = \frac{I_C}{I_B} = \frac{\alpha}{1 - \alpha} \approx \frac{2 L_n^2}{W_B^2}.
$$

> **Definition (Current gain).** The **common-emitter current gain**
> $\beta = I_C/I_B$ of a bipolar transistor is the ratio of controlled collector
> current to controlling base current. It grows as the inverse square of the base
> width, so a base a fraction of a diffusion length thick yields $\beta$ of order
> $100$: a small base signal commands a collector current a hundred times larger.

A base width $W_B = L_n/10$ gives $\alpha = 0.995$ and $\beta \approx 200$. The
thin base is the entire trick — it is why transistor fabrication is a contest to
make the base region as narrow and defect-free as possible, maximizing $L_n$ and
minimizing $W_B$.

## The MOSFET

The metal-oxide-semiconductor field-effect transistor (MOSFET) switches with a
voltage rather than a current, which is why it, not the BJT, fills digital chips. An
n-channel device sits on a p-type substrate with two n$^+$ regions — **source** and
**drain** — separated by a channel region. Over the channel a thin insulating oxide
carries a metal (or polysilicon) **gate**. The gate, oxide, and semiconductor form
a capacitor.

Raising the gate voltage repels holes from the surface and, above a **threshold**
$V_T$, pulls enough electrons to the surface to **invert** it into a thin n-type
channel connecting source to drain. The gate voltage above threshold sets the
channel charge per unit area, $Q = C_{ox}(V_{GS} - V_T)$, with $C_{ox}$ the oxide
capacitance per unit area. A drain voltage then drives this charge as a current.

$$
% caption: n-channel MOSFET cross-section. A positive gate voltage above threshold
% inverts the p-substrate surface under the oxide into an n-channel joining the
% n-plus source and drain; the gate-oxide-channel capacitor controls the channel
% charge and hence the drain current.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % substrate
  \draw[acc, thick, fill=acc!8] (0,0) rectangle (7.0,2.0);
  \node[font=\scriptsize] at (3.5,0.6) {p substrate};
  % source and drain
  \draw[acc, thick, fill=acc!22] (0.4,1.4) rectangle (1.8,2.0);
  \node[font=\scriptsize] at (1.1,1.7) {source};
  \draw[acc, thick, fill=acc!22] (5.2,1.4) rectangle (6.6,2.0);
  \node[font=\scriptsize] at (5.9,1.7) {drain};
  % oxide
  \draw[black, thick, fill=black!6] (1.8,2.0) rectangle (5.2,2.35);
  \node[font=\scriptsize, anchor=west] at (5.25,2.18) {oxide};
  % gate
  \draw[black, thick, fill=black!16] (1.8,2.35) rectangle (5.2,2.75);
  \node[font=\scriptsize] at (3.5,2.55) {gate};
  % inversion channel
  \draw[acc, very thick, dashed] (1.8,1.92) -- (5.2,1.92);
  \node[acc, font=\scriptsize, anchor=north] at (3.5,1.88) {inversion channel};
\end{tikzpicture}
$$

For a small drain voltage the channel is a resistor and the current rises linearly;
as $V_{DS}$ grows, the channel is **pinched off** at the drain end and the current
**saturates**. The two regimes are

$$
I_D = \mu_n C_{ox}\frac{W}{L}\left[(V_{GS} - V_T)V_{DS} - \tfrac12 V_{DS}^2\right] \quad (\text{linear}),
$$

$$
I_D = \frac{\mu_n C_{ox}}{2}\frac{W}{L}(V_{GS} - V_T)^2 \quad (\text{saturation}),
$$

with $W/L$ the channel width-to-length ratio. The saturation current is quadratic
in the gate overdrive $V_{GS} - V_T$, and the gate itself draws no steady
current — it is a capacitor plate, the fact on which the whole technology rests. A logic gate built from complementary n- and
p-channel MOSFETs (**CMOS**) draws current only while switching, which is why a
processor of billions of transistors dissipates watts rather than kilowatts. Making
$L$ smaller raises the current and the switching speed and packs more devices per
area, the engine of five decades of scaling.

## Light-emitting diodes and diode lasers

A forward-biased junction injects electrons and holes into the same region, where
they recombine. In a **direct-gap** semiconductor the conduction-band minimum and
valence-band maximum share the same crystal momentum, so an electron and hole
recombine by emitting a single photon of energy near the gap without needing a
phonon. The emission wavelength is set directly by the gap,

$$
\lambda = \frac{hc}{E_g} = \frac{1240\ \text{nm}\cdot\text{eV}}{E_g},
$$

so choosing the material chooses the color: gallium arsenide ($E_g = 1.42\
\text{eV}$) emits in the near infrared, gallium arsenide phosphide and aluminum
indium gallium phosphide span red and amber, and indium gallium nitride
($E_g$ tuned near $2.7\ \text{eV}$) gives the blue and green that, with a phosphor,
make white light. Silicon, being **indirect**-gap, emits essentially no light —
recombination there needs a phonon and proceeds through non-radiative traps
instead, which is why light emitters are built from compound semiconductors.

$$
% caption: LED operation. Forward bias floods the junction with electrons and
% holes; across a direct gap they recombine radiatively, each pair emitting a
% photon of energy near E_g and wavelength lambda = hc/E_g.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[acc, thick, fill=acc!8] (0,2.4) rectangle (6.0,3.0);
  \node[anchor=west, font=\scriptsize] at (6.05,2.7) {$E_c$};
  \draw[acc, thick, fill=acc!30] (0,0) rectangle (6.0,0.6);
  \node[anchor=west, font=\scriptsize] at (6.05,0.3) {$E_v$};
  % electron in conduction, hole in valence, recombining
  \fill[acc] (2.4,2.4) circle (2.6pt);
  \draw[acc, thick] (2.4,0.6) circle (2.6pt);
  \draw[acc, ->, very thick] (2.4,2.35) -- (2.4,0.7);
  % emitted photon as wavy arrow drawn by segments
  \draw[black, ->, thick]
    (2.6,1.5) -- (3.1,1.65) -- (3.6,1.35) -- (4.1,1.65) -- (4.6,1.35) -- (5.0,1.55);
  \node[black, anchor=west, font=\scriptsize] at (4.0,2.0) {photon};
  \node[acc, anchor=east, font=\scriptsize] at (2.3,1.5) {recombine};
\end{tikzpicture}
$$

Confining the recombination region and placing it between two parallel mirrors —
cleaved crystal facets suffice — turns the LED into a **diode laser**. Above a
**threshold current** the injected carriers sustain a
[population inversion](/condensed-matter/molecular-spectra/lasers-and-masers), and
stimulated emission into the cavity mode produces coherent, monochromatic light.
The double **heterostructure** — a thin low-gap layer between higher-gap cladding —
confines both carriers and photons to the same micron-thin slab, lowering the
threshold enough for continuous room-temperature operation and making the diode
laser the light source of fiber communication and optical storage.

## Photodetectors and solar cells

Run in reverse, a junction is a photon collector. A photon with energy above the
gap absorbed in or near the depletion region creates an electron-hole pair; the
built-in field sweeps the electron to the n side and the hole to the p side,
producing a **photocurrent** $I_L$ proportional to the illumination. A
reverse-biased junction used this way is a **photodiode**; unbiased and optimized
for power, it is a **solar cell**.

Superposing the photocurrent on the Shockley characteristic gives the illuminated
diode equation,

$$
I = I_0\!\left(e^{eV/k_B T} - 1\right) - I_L,
$$

the dark curve shifted down by $I_L$. The operating point of interest lies in the
**fourth quadrant**, $V > 0$ and $I < 0$: the junction delivers power to a load.
Two points bound it. At short circuit ($V = 0$) the current is
$I_{sc} = -I_L$; at open circuit ($I = 0$) the voltage is

$$
V_{oc} = \frac{k_B T}{e}\ln\!\left(\frac{I_L}{I_0} + 1\right).
$$

The delivered power $P = |IV|$ is maximized at an interior knee; the ratio of that
maximum to $I_{sc}V_{oc}$ is the **fill factor**, and the cell's efficiency is the
maximum power divided by the incident solar power.

$$
% caption: The illuminated solar-cell characteristic. Light shifts the dark diode
% curve down by the photocurrent I_L into the fourth quadrant (V positive, I
% negative), where the cell delivers power; the maximum-power point sits at the
% knee between the short-circuit current I_sc and the open-circuit voltage V_oc.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (-0.4,0) -- (4.4,0) node[below] {$V$};
  \draw[->, black] (0,-3.0) -- (0,1.2) node[left] {$I$};
  % dark diode curve (dashed)
  \draw[black, dashed, domain=0:2.05, samples=50, variable=\x]
    plot ({\x*1.6},{0.22*(exp(\x*1.7)-1)});
  \node[black, anchor=south east, font=\scriptsize] at (3.1,0.9) {dark};
  % illuminated curve shifted down by I_L
  \draw[acc, very thick, domain=0:2.13, samples=60, variable=\x]
    plot ({\x*1.6},{0.22*(exp(\x*1.7)-1) - 2.2});
  % short circuit current
  \fill[acc] (0,-2.2) circle (2pt);
  \node[acc, anchor=east, font=\scriptsize] at (-0.05,-2.2) {$I_{sc}$};
  % open circuit voltage
  \fill[acc] (3.32,0) circle (2pt);
  \node[acc, anchor=south west, font=\scriptsize] at (3.25,0.05) {$V_{oc}$};
  % max power point
  \fill[black] (2.7,-1.9) circle (2pt);
  \node[anchor=north east, font=\scriptsize] at (2.75,-1.95) {max power};
  % power rectangle
  \draw[black, dashed] (0,-1.9) -- (2.7,-1.9) -- (2.7,0);
\end{tikzpicture}
$$

> **Worked example.** A silicon cell under one sun has $I_L = 35\ \text{mA/cm}^2$
> and $I_0 = 10^{-11}\ \text{mA/cm}^2$. The open-circuit voltage is
>
> $$
> V_{oc} = (0.0259\ \text{V})\ln\!\left(\frac{35}{10^{-11}}\right) = (0.0259\ \text{V})(28.9) = 0.75\ \text{V},
> $$
>
> below the $1.12\ \text{V}$ gap because the logarithm penalizes the large
> $I_0/I_L$ ratio. With a fill factor near $0.8$ the maximum power is about
> $21\ \text{mW/cm}^2$; against roughly $100\ \text{mW/cm}^2$ of sunlight this is a
> module efficiency near $20\ \%$, close to the single-junction limit set by the
> gap's tradeoff between voltage and absorbed spectrum.

The single silicon gap fixes a ceiling: photons below it pass through unabsorbed,
and photons far above it waste their excess as heat. Stacking junctions of
different gaps — the province of compound-semiconductor
[heterostructures and nanostructures](/condensed-matter/nanostructures/quantum-wells-wires-and-dots) —
harvests more of the spectrum and pushes past it, the direction the next module
takes as it moves from bulk crystals to engineered low-dimensional systems.