Semiconductor Electronics

When a very large number of atoms come together to form a solid, the sharply defined energy levels of an isolated atom split and merge into continuous energy bands. The highest band that is normally filled with electrons is the valence band; the next higher band, where electrons can move freely and carry current, is the conduction band. Between them lies the forbidden energy gap, E_g, a range of electron energies that cannot exist in the crystal.

Solids are classified by the size of this gap. In metals the valence and conduction bands overlap (or the conduction band is partly filled), so electrons conduct freely even at low temperature. In insulators, E_g is large (typically above 3 eV; diamond has E_g ≈ 5.4 eV), so essentially no electron is thermally excited across it at room temperature. Semiconductors have a small gap (silicon ≈ 1.1 eV, germanium ≈ 0.7 eV) — small enough that some electrons gain enough thermal energy at room temperature to jump into the conduction band, leaving behind vacant bonds called holes in the valence band. A hole behaves as a mobile, effectively positive charge carrier.

In a chemically pure crystal, called an intrinsic semiconductor, every thermally excited electron leaves exactly one hole behind, so the electron concentration n_e equals the hole concentration n_h, both equal to the intrinsic concentration n_i. Conductivity of an intrinsic semiconductor is low and rises rapidly with temperature, unlike a metal whose resistance rises with temperature.

Useful semiconductor behaviour comes from doping — deliberately adding a controlled trace of impurity atoms. Adding a pentavalent impurity (phosphorus, arsenic, antimony) to silicon or germanium gives an n-type semiconductor: four of the five valence electrons of each impurity atom form covalent bonds with neighbouring Si/Ge atoms, and the fifth electron is very loosely bound, occupying a donor energy level just below the conduction band. This electron is easily donated to the conduction band, so electrons become the majority carriers and holes the minority carriers. Adding a trivalent impurity (boron, aluminium, indium) instead creates a p-type semiconductor: the impurity atom can complete only three covalent bonds, leaving one bond incomplete, which readily accepts an electron from a neighbouring atom. This creates an acceptor energy level just above the valence band and generates a mobile hole, so holes become the majority carriers and electrons the minority carriers. In both cases the crystal remains electrically neutral overall — the extra carriers are mobile, but the impurity ions left behind are fixed in the lattice.

Even in a doped (extrinsic) semiconductor, the product of electron and hole concentrations remains fixed at a given temperature: n_en_h = n_i², known as the law of mass action.

Worked Example: Silicon is doped with a donor concentration N_d = 1×10²⁰ m^-3, with n_i = 1.5×10¹⁶ m^-3 at 300 K. Since N_d ≫ n_i, almost every donor atom is ionised, so n_e ≈ N_d = 1×10²⁰ m^-3. Using the law of mass action, the hole concentration is n_h = n_i²/n_e = (1.5×10¹⁶)² / (1×10²⁰) = 2.25×10¹² m^-3. Electrons outnumber holes by nearly eight orders of magnitude, confirming this is a strongly n-type sample.

A p-n junction is formed when a p-type and an n-type region are created within a single continuous semiconductor crystal (commonly by diffusing acceptor and donor impurities from opposite faces). Near the junction, the sharp concentration difference causes majority carriers to diffuse across it: electrons diffuse from the n-side into the p-side and holes diffuse from the p-side into the n-side. As they cross, electrons and holes recombine near the junction, leaving behind fixed, uncompensated ion cores — positive donor ions on the n-side and negative acceptor ions on the p-side. This region, stripped of mobile charge carriers, is the depletion region.

The exposed ion charges set up an internal electric field pointing from the n-side to the p-side, which opposes further diffusion of majority carriers (though it assists a small drift of minority carriers across the junction in the opposite sense). At equilibrium, diffusion and drift currents balance, and the field corresponds to an internal potential barrier V₀ (about 0.7 V for silicon, 0.3 V for germanium at room temperature).

When an external voltage is applied, the diode's behaviour depends on the direction (bias):

• Forward bias: the p-side is connected to the positive terminal and the n-side to the negative terminal. The applied field opposes the internal field, so the potential barrier is lowered and the depletion region narrows. Once the applied voltage approaches V₀ (the knee/threshold voltage), majority carriers flood across the junction and the current rises very rapidly (approximately exponentially) with small further increases in voltage.
• Reverse bias: the p-side is connected to the negative terminal and the n-side to the positive terminal. The applied field adds to the internal field, so the barrier height increases and the depletion region widens, blocking majority-carrier flow almost completely. Only a very small, nearly voltage-independent reverse saturation current I₀ flows, carried by the few thermally generated minority carriers. If the reverse voltage is increased beyond a critical breakdown voltage, the current rises sharply (used deliberately in Zener diodes).

Because the I-V relationship is strongly non-linear, a diode does not have a single fixed resistance; instead we define a dynamic (AC) resistance at an operating point as the ratio of a small change in voltage to the resulting small change in current.

Worked Example: In forward bias, the current through a diode rises from 2 mA to 7 mA when the voltage across it changes by 50 mV. The dynamic resistance is r_d = ΔV/ΔI = 0.050 V / (5×10^-3 A) = 10 Ω, a typical low value for a forward-biased silicon diode operating well above the knee voltage. In reverse bias, by contrast, a similar voltage change produces almost no change in the (already tiny) current, so the dynamic resistance there is enormously larger (of order MΩ), reflecting the diode's near-ideal blocking action.

Because a p-n junction diode conducts appreciably only when forward biased and blocks current when reverse biased, it acts like a one-way valve for current. This property is used to convert alternating current (AC), which reverses direction every half cycle, into a unidirectional (though pulsating) output — a process called rectification.

In a half-wave rectifier, a single diode is connected in series with the AC source and the load. During the half-cycle that forward-biases the diode, it conducts and current flows through the load, reproducing (approximately) that half of the input waveform. During the other half-cycle, the diode is reverse biased and blocks conduction, so the output is zero. The result is a series of one-directional pulses occurring at the same frequency as the input, with long "dead" gaps in between — a poor approximation to steady DC.

A full-wave rectifier overcomes this using either a centre-tapped transformer with two diodes, or four diodes in a bridge arrangement, so that current is driven through the load in the same direction during both half-cycles of the input. Whichever diode is forward biased in a given half-cycle conducts, and the two conduction paths are arranged so the load current never reverses. Consequently the output consists of a pulse for every half-cycle of the input, i.e. the ripple (pulsation) frequency of a full-wave rectifier is twice the input frequency, compared to the half-wave rectifier's ripple frequency, which equals the input frequency. This makes the full-wave output smoother, gives it a higher average (DC) value for the same peak input, and makes the residual ripple easier to remove with a capacitor filter connected across the load.

Worked Example: A step-down transformer feeds a peak voltage V_m = 15 V to a full-wave bridge rectifier operating from 50 Hz mains. (a) The ripple frequency of the output is twice the input frequency: 2 × 50 Hz = 100 Hz. (b) Assuming ideal diodes (no forward voltage drop), the average (DC) output voltage is V_dc = 2V_m/π = (2 × 15)/3.1416 ≈ 9.55 V, compared to only V_m/π ≈ 4.77 V that a half-wave rectifier would give for the same peak input.

Ordinary junction diodes are designed to avoid reverse breakdown, but some diodes are built specifically to exploit it or to interact with light.

A Zener diode is a heavily doped p-n junction diode designed to operate safely and repeatedly in reverse breakdown. Because of the heavy doping, the depletion region is very thin, and at a well-defined reverse voltage V_Z the junction breaks down (via Zener tunnelling for lower voltages, or avalanche multiplication for higher voltages), after which the voltage across the diode stays almost constant over a wide range of current. This makes it ideal as a voltage regulator: the Zener is connected in reverse bias, in series with a resistor R_s across the unregulated input, with the load connected in parallel with the Zener. Any increase in input voltage (or decrease in load current) is absorbed as extra current through the Zener rather than extra voltage across the load, so the load voltage remains close to V_Z.

A photodiode is a p-n junction operated in reverse bias, enclosed so that light can fall on the junction. Photons with energy at least equal to the band gap generate electron-hole pairs; the reverse-bias field sweeps these apart before they can recombine, producing a photocurrent that increases almost linearly with the intensity of incident light. Reverse bias is used (rather than zero or forward bias) because it gives a faster response and a current that tracks light intensity more linearly.

A light-emitting diode (LED) is a heavily doped p-n junction, usually of a direct-band-gap compound semiconductor (such as GaAs or GaP-based alloys), operated in forward bias. Injected electrons and holes recombine near the junction, releasing the recombination energy as photons instead of heat; the emitted wavelength is set by the material's band gap. LEDs are compact, low-voltage, low-power, fast-switching and need no warm-up time, unlike incandescent bulbs.

A solar cell is also a p-n junction, but requires no external bias at all: when light falls on it, electron-hole pairs generated near the junction are separated by the built-in field itself, so the device generates an emf directly (the photovoltaic effect) and can deliver power to an external circuit. Its performance is described by the open-circuit voltage V_oc, short-circuit current I_sc, and the maximum-power point (V_m, I_m) on its I-V characteristic. The fill factor compares the actual maximum power obtainable to the (unattainable) ideal product V_ocI_sc, and the efficiency compares the maximum electrical power output to the incident optical power.

Worked Example: An LED is fabricated from a compound semiconductor with band gap E_g = 1.9 eV. Using hc ≈ 1240 eV·nm, the emitted wavelength is λ = hc/E_g = 1240/1.9 ≈ 652.6 nm, which falls in the red region of the visible spectrum — consistent with typical red LEDs.

A bipolar junction transistor consists of three alternately doped regions forming two back-to-back p-n junctions: the emitter (heavily doped, to supply a large number of majority carriers), the base (made very thin and lightly doped), and the collector (moderately doped and physically larger, to collect carriers and dissipate heat). An npn or a pnp arrangement is possible; the npn transistor is more commonly used because electrons (which move faster than holes) form the main current.

For normal (active-region) operation, the emitter-base junction is forward biased and the collector-base junction is reverse biased. Forward bias at the emitter-base junction injects a large number of majority carriers from the emitter into the base. Because the base is thin and lightly doped, only a small fraction of these recombine there (contributing to the small base current); the vast majority diffuse across the narrow base and are swept into the collector by the reverse-biased collector junction's field. This is transistor action: a small base current controls a much larger collector current.

By Kirchhoff's current law applied to the transistor, the emitter current splits into the base and collector currents: I_E = I_B + I_C. Two current-gain factors describe this: the common-base current gain α_dc = I_C/I_E (always slightly less than 1, since I_C is slightly less than I_E), and the common-emitter current gain β_dc = I_C/I_B (much greater than 1, since I_B is a very small fraction of I_E). The two are related by β = α/(1−α).

A transistor's behaviour is summarised by two characteristic curves (for a given configuration, commonly common-emitter): the input characteristic (I_B vs V_BE at constant V_CE), which resembles the forward characteristic of a diode, and the output characteristic (I_C vs V_CE for different fixed values of I_B), which shows three regions: cutoff (both junctions reverse biased, I_C ≈ 0), the active region (I_C nearly constant for a given I_B, rising only slightly with V_CE, and closely proportional to I_B), and saturation (both junctions forward biased, V_CE small, I_C large and limited mainly by the external circuit).

Worked Example: A transistor operates with I_E = 5 mA and I_B = 0.1 mA. Then I_C = I_E − I_B = 5 − 0.1 = 4.9 mA. The common-base gain is α = I_C/I_E = 4.9/5 = 0.98, and the common-emitter gain is β = I_C/I_B = 4.9/0.1 = 49. Check: β = α/(1−α) = 0.98/0.02 = 49, which matches.

The two extreme regions of the output characteristic give a transistor its role as an electronic switch. When the base-emitter voltage is below the threshold needed to forward bias the emitter junction, the transistor sits in cutoff: both junctions are effectively reverse biased, negligible collector current flows, and the output (collector) voltage stays close to the supply voltage ("high"/OFF state for current flow, but output logic-high). Driving enough base current instead pushes the transistor into saturation, where both junctions are forward biased, the collector-emitter voltage drops to a small value, and collector current is limited mainly by the external load resistor ("ON" state, output logic-low). Switching rapidly between these two states, rather than operating anywhere in between, is the basis of transistor use in digital circuits.

Operating instead in the active region, with a fixed DC bias point (Q-point) chosen roughly in the middle of the usable range, the transistor can amplify small AC signals. In the common-emitter (CE) configuration, the input AC signal is applied between base and emitter, and the amplified output is taken across a load resistor R_C connected between collector and the supply. A small AC change in base current ΔI_B produces a much larger change in collector current ΔI_C = β_acΔI_B, where β_ac = ΔI_C/ΔI_B (measured at constant V_CE) is the AC current gain. This changing collector current develops a changing voltage across R_C, giving voltage amplification; combined with the current gain, the CE configuration also gives substantial power gain, which is why it is the most widely used amplifier configuration. A characteristic feature of the CE amplifier is that the output voltage is 180° out of phase with (i.e. inverted relative to) the input voltage.

Worked Example: A CE amplifier has β_ac = 80, collector load R_C = 4 kΩ, and AC input (base) resistance R_i = 800 Ω. If the input AC voltage changes by ΔV_i = 5 mV: the base current change is ΔI_B = ΔV_i/R_i = 5 mV / 800 Ω = 6.25 μA; the collector current change is ΔI_C = β_acΔI_B = 80 × 6.25 μA = 500 μA = 0.5 mA; the voltage gain is A_v = β_ac(R_C/R_i) = 80 × (4000/800) = 80 × 5 = 400; and the resulting output voltage change is ΔV_o = A_vΔV_i = 400 × 5 mV = 2 V (inverted in phase relative to the input).