Transistors and Optoelectronic Devices

The p-n 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.

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.

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

α = \frac{I _{C}}{I _{E}} \approx 1 - \frac{1}{2} (\frac{W _{B}}{L _{n}})^{2},

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

β = \frac{I _{C}}{I _{B}} = \frac{α}{1 - α} \approx \frac{2 L _{n}^{2}}{W _{B}^{2}} .

A base width $W_{B} = L_{n} /10$ gives $α = 0.995$ and $β \approx 200$ . 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, $Q = C_{o x} (V_{GS} - V_{T})$ , with $C_{o x}$ the oxide capacitance per unit area. A drain voltage then drives this charge as a current.

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.

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

I_{D} = μ_{n} C_{o x} \frac{W}{L} [(V_{GS} - V_{T}) V_{D S} - \frac{1}{2} V_{D S}^{2}] (linear),

I_{D} = \frac{μ _{n} C _{o x}}{2} \frac{W}{L} (V_{GS} - V_{T})^{2} (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,

λ = \frac{h c}{E _{g}} = \frac{1240 nm \cdot eV}{E _{g}},

so choosing the material chooses the color: gallium arsenide ( $E_{g} = 1.42 eV$ ) 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 $2.7 eV$ ) 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.

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.

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, 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} (e^{e V / k_{B} T} - 1) - 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 (\frac{I _{L}}{I _{0}} + 1) .

The delivered power $P = ∣ I V ∣$ 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.

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.

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 — 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.