The Devices: P-N Junction, Zener, BJT, MOS Capacitor, MOSFET and Scaling, LED, Photodiode and Solar Cell

Forward bias lowers the barrier by V, so the minority-carrier concentration injected at each depletion edge is multiplied by exp(V/V_T). Solving the minority-carrier diffusion equation from the previous chapter in each quasi-neutral region and adding the two diffusion currents gives the ideal diode equation I = I_S (exp(V/nV_T) − 1), with I_S = qA(D_p p_n0/L_p + D_n n_p0/L_n). Three readings of that equation matter more than the derivation. First, I_S contains the minority-carrier equilibrium concentrations, so it is proportional to n_i² — which means I_S varies as exp(−E_g/kT) and rises by roughly an order of magnitude for every 15 °C in silicon. Second, in reverse bias the exponential dies and I → −I_S, a current independent of voltage, which is what makes a reverse-biased junction a current source rather than a resistor. Third, the ideality factor n is 1 when diffusion dominates and tends toward 2 when recombination inside the depletion region dominates, which is the low-current regime. Practically, at fixed current the forward voltage falls by about 2 mV/°C, and a decade of current change costs 2.303 × 25.9 = 59.6 mV — the '60 mV per decade' every analogue designer quotes.

For small-signal work the diode is a resistance in parallel with two capacitances. The dynamic resistance is r_d = dV/dI = nV_T/I, so at I = 1 mA and n = 1, r_d = 0.0259/10⁻³ = 25.9 Ω — a number worth memorising, because it makes r_d ≈ 26/I(mA) Ω a one-second calculation. The junction (depletion) capacitance C_j = ε_s A/W dominates in reverse bias, falls as (V_bi + V_R)^(−1/2) for an abrupt junction, and is what a varactor exploits. The diffusion (storage) capacitance C_d = τ I/V_T dominates in forward bias and represents the charge stored in the quasi-neutral regions rather than any geometric separation; it is proportional to current, so it is enormous in a forward-biased diode and is why a diode does not switch off instantly. That stored charge has to be removed before the junction can support reverse voltage, which is the reverse recovery time — and, as the previous chapter noted, why fast diodes are deliberately lifetime-killed with gold or platinum. Beyond a critical reverse voltage the junction breaks down by one of two distinct mechanisms, and the syllabus names the diode built to exploit them.

A bipolar junction transistor is an emitter, a base and a collector — n-p-n or p-n-p — and the whole trick is geometric. Forward-bias the emitter-base junction and the emitter injects a large minority-carrier population into the base. Reverse-bias the base-collector junction so its field sweeps up anything that reaches it. Now make the base much thinner than the minority-carrier diffusion length: for a base 0.5 µm wide against the 45.5 µm electron diffusion length computed in the last chapter, the ratio is ninety to one, so almost every injected electron crosses the base by diffusion and is collected rather than recombining. I_C is therefore controlled by the emitter-base voltage and is nearly independent of V_CB — a current-controlled current source, and, crucially, the output current is set by the input junction, which is what makes it an amplifier. The two figures of merit are the common-base gain α = I_C/I_E and the common-emitter gain β = I_C/I_B, related by β = α/(1 − α) and α = β/(1 + β). The four operating modes follow from the two junction biases: both forward is saturation, both reverse is cut-off, emitter forward with collector reverse is the active region, and the reverse of that is reverse-active, which has a wretched gain because the collector was never designed to be an emitter.

Three second-order effects turn the ideal current source into a real device, and all three come from the base. Base-width modulation, the Early effect: raising V_CE widens the base-collector depletion region, which eats into the base from the collector side, shortens the effective base width W_B, steepens the minority-carrier gradient across it and so raises I_C. That gives the active-region characteristics a finite slope, described by an Early voltage V_A — the extrapolated intercept on the V_CE axis — and an output resistance r_o = V_A/I_C. With V_A = 50 V and I_C = 1 mA, r_o = 50 kΩ, so raising V_CE from 2 V to 10 V raises I_C by 8/50 = 16 per cent. Punch-through: push V_CE far enough and the collector depletion region reaches all the way across to the emitter, W_B → 0, and the transistor is simply a short. High-current fall-off: at large I_C the injected minority density in the base rivals the base doping, high-level injection sets in, and β falls — which is why a transistor's β peaks at some mid-range current and droops at both ends. Small-signal, the BJT is characterised by g_m = I_C/V_T, which at 1 mA is 10⁻³/0.0259 = 38.6 mA/V, and r_π = β/g_m = 2.59 kΩ at β = 100. Note the beautiful consequence: the intrinsic gain g_m r_o = (I_C/V_T)(V_A/I_C) = V_A/V_T = 50/0.0259 = 1930, entirely independent of bias current.

Speed in a BJT is set by how long a carrier takes to cross the base. Diffusion across a base of width W_B takes a forward transit time τ_F = W_B²/2D_n, and for W_B = 0.5 µm = 5 × 10⁻⁵ cm with D_n = 20.7 cm²/s that is (5 × 10⁻⁵)²/(2 × 20.7) = 6.03 × 10⁻¹¹ s = 60.3 ps. The corresponding unity-gain frequency is f_T ≈ 1/2πτ_F = 2.64 GHz, and the frequency at which β itself has fallen to one is f_β = f_T/β = 26.4 MHz. The quadratic in W_B is the design lesson: halving the base width quarters the transit time, which is why the entire history of bipolar technology is a history of thinner bases, and why the heterojunction bipolar transistor — which uses a wide-gap emitter to get injection efficiency without needing a lightly doped base — can dope the base heavily, make it very thin, and reach hundreds of gigahertz. The same base thinness, of course, is what brings punch-through closer, so the base doping cannot be reduced to compensate. Every bipolar design is that trade: gain wants a thin lightly doped base, breakdown wants a thick heavily doped one.

A MOS capacitor is a metal or polysilicon gate, a thin oxide, and a semiconductor substrate — an ordinary capacitor except that one of its plates is a semiconductor and can therefore respond in more than one way. Take a p-type substrate at N_A = 10¹⁷ cm⁻³ under a 10 nm oxide. The oxide capacitance per unit area is C_ox = ε_ox/t_ox = 3.9 × 8.854 × 10⁻¹⁴/10⁻⁶ = 3.453 × 10⁻⁷ F/cm², which is 3.45 fF/µm² in the units a layout engineer uses. Now sweep the gate voltage and three regimes appear in order. Make the gate negative and holes — the majority carriers — are attracted to the surface: accumulation, a thin sheet of mobile charge right at the oxide, so the structure is a plain parallel-plate capacitor of value C_ox. Make the gate slightly positive and holes are pushed away, leaving behind immobile negative acceptor ions: depletion, and the capacitance is now C_ox in series with a depletion capacitance ε_s/W_d that shrinks as W_d grows, so the total falls. Push further and the surface potential bends the bands so far that the surface becomes n-type: inversion, a mobile electron sheet at the surface in a substrate that is doped p-type.

Strong inversion is conventionally defined as the point where the surface electron concentration equals the bulk hole concentration, which happens when the surface potential reaches 2φ_F, where φ_F = V_T ln(N_A/n_i) is how far the bulk Fermi level sits from the intrinsic level. For our substrate, φ_F = 0.0259 × ln(10¹⁷/1.5 × 10¹⁰) = 0.407 V — the identical number computed in the last chapter, because it is the identical quantity — so strong inversion needs a surface potential of 2φ_F = 0.814 V. Getting there costs three things, and the threshold voltage is their sum: V_T = V_FB + 2φ_F + Q_dep/C_ox. The flat-band voltage V_FB pays off the work-function difference between gate and substrate plus any fixed oxide charge, and for an n+ polysilicon gate on this substrate it is about −0.9 V. The 2φ_F bends the bands. And Q_dep/C_ox is the gate voltage dropped across the oxide holding up the depletion charge, where Q_dep = √(2qε_s N_A × 2φ_F) = √(2 × 1.6 × 10⁻¹⁹ × 1.036 × 10⁻¹² × 10¹⁷ × 0.814) = 1.643 × 10⁻⁷ C/cm², giving Q_dep/C_ox = 1.643 × 10⁻⁷/3.453 × 10⁻⁷ = 0.476 V. Adding up: V_T = −0.9 + 0.814 + 0.476 = 0.390 V. Cross-check the depletion charge against the geometry: W_dmax = √(2ε_s × 2φ_F/qN_A) = 1.027 × 10⁻⁵ cm = 103 nm, and qN_A W_dmax = 1.643 × 10⁻⁷ C/cm² ✓, the same number by a different route.

Add an n+ source and an n+ drain to the MOS capacitor and the inversion layer becomes a channel with somewhere to go. Above threshold the mobile sheet charge at a point in the channel is Q_n = C_ox(V_GS − V_T − V(x)), and integrating the drift current along the channel gives the triode (linear) expression I_D = µ_n C_ox (W/L)[(V_GS − V_T)V_DS − V_DS²/2]. Raise V_DS and the channel charge at the drain end shrinks until, at V_DS = V_GS − V_T, it reaches zero — the channel pinches off, and beyond that point further V_DS is absorbed in the small depleted region near the drain rather than changing the channel. So in saturation the current stops growing: I_D = ½ µ_n C_ox (W/L)(V_GS − V_T)². Work the reference device: t_ox = 10 nm gives C_ox = 3.453 × 10⁻⁷ F/cm², take the surface mobility µ_n = 400 cm²/V·s (roughly half the bulk value, because carriers in a channel are scattered by the oxide interface), so µ_n C_ox = 138 µA/V². With W/L = 10, V_GS = 1.2 V and the V_T = 0.390 V computed above, the overdrive is V_ov = 0.810 V and I_D = ½ × 138 × 10⁻⁶ × 10 × 0.810² = 454 µA. The transconductance is g_m = µ_n C_ox (W/L)V_ov = 1.12 mA/V, and the cross-check g_m = 2I_D/V_ov = 2 × 454 × 10⁻⁶/0.810 = 1.12 mA/V ✓.

Three departures from the square law are examinable in their own right. Channel-length modulation is the MOSFET's Early effect: beyond pinch-off the depleted region near the drain grows with V_DS, shortening the effective channel, so I_D = ½µ_nC_ox(W/L)V_ov²(1 + λV_DS) and the output resistance is r_o = 1/λI_D. At λ = 0.05 V⁻¹ and I_D = 454 µA that is 44.1 kΩ, giving an intrinsic gain g_m r_o = 49 — forty times worse than the bipolar figure, and the reason analogue MOS design leans so heavily on cascoding. Body effect: the threshold voltage is not a constant of the device but a function of the source-to-body voltage, V_T = V_T0 + γ(√(2φ_F + V_SB) − √(2φ_F)), with γ = √(2qε_sN_A)/C_ox = 0.527 V^½ for our device. Put just V_SB = 1 V on the body and ΔV_T = 0.527 × (√1.814 − √0.814) = 0.527 × (1.347 − 0.902) = 0.234 V, so V_T rises from 0.390 V to 0.624 V, a 60 per cent increase. That is why a stack of series transistors in a CMOS gate does not switch at the voltage the bottom one does, and why the source of an analogue device is tied to its own body wherever the layout permits.

Constant-field (Dennard) scaling is a recipe with one instruction: divide every length and every voltage by the same factor S, and multiply the doping by S. Do that and the electric field inside the device is unchanged, so the physics that could break — breakdown, velocity saturation, oxide reliability — does not get worse. Every other quantity then follows by algebra, and the algebra is worth doing once rather than memorising, because the exam asks for individual entries. Oxide thickness t_ox divides by S, so C_ox per unit area multiplies by S, but the gate area divides by S², so the gate capacitance divides by S. Current I ∝ (W/L)µC_ox V² scales as (1)(S)(1/S²) = 1/S, so current divides by S. Delay CV/I scales as (1/S)(1/S)/(1/S) = 1/S, so circuits get faster by S. Power VI scales as (1/S)(1/S) = 1/S², and since area also divides by S², power density is constant — which is the whole point, and the promise that let the industry double transistor count every couple of years while keeping the same package and the same fan.

An LED is a forward-biased junction in a direct-gap material, arranged so that injected carriers recombine radiatively. The photon energy comes out at roughly the band gap, so the emitted wavelength is fixed by the material through λ(nm) = 1240/E_g(eV) — a relation worth deriving once from E = hc/λ and then using without thought. Gallium arsenide at E_g = 1.42 eV emits at 1240/1.42 = 873 nm, in the near infrared, which is exactly what a remote control and a short-haul fibre link want. To get visible red at 650 nm the gap must be 1240/650 = 1.91 eV, reached by alloying — aluminium gallium arsenide or gallium arsenide phosphide. Blue at 450 nm needs 1240/450 = 2.76 eV, which is why blue and white LEDs are indium gallium nitride: gallium nitride's own 3.4 eV gap would emit at 365 nm, in the ultraviolet, so indium is added to bring the gap down. And silicon, at 1.12 eV, would emit at 1107 nm if it emitted at all — which, being indirect, it effectively does not. The efficiency figures worth knowing are the internal quantum efficiency (photons generated per injected carrier pair), the extraction efficiency (photons that escape rather than being reabsorbed or totally internally reflected at the high-index semiconductor surface), and the product, the external quantum efficiency.

A photodiode runs the same junction backwards: reverse-bias it so its depletion region is wide and its field strong, let photons of energy above the gap create electron-hole pairs inside that region, and the field separates them before they can recombine, delivering a photocurrent proportional to the optical power. The figure of merit is the responsivity R = ηqλ/hc, which in practical units is R = ηλ(µm)/1.24 A/W. For a silicon photodiode with quantum efficiency η = 0.8 at λ = 0.85 µm, R = 0.8 × 0.85/1.24 = 0.548 A/W, so 10 µW of incident light gives 0.548 × 10 = 5.48 µA of photocurrent. Check that by counting photons: at 850 nm each photon carries 1240/850 = 1.459 eV = 2.337 × 10⁻¹⁹ J, so 10 µW is 10⁻⁵/2.337 × 10⁻¹⁹ = 4.28 × 10¹³ photons per second; at 80 per cent efficiency that is 3.42 × 10¹³ electrons per second, or 3.42 × 10¹³ × 1.6 × 10⁻¹⁹ = 5.48 µA ✓. Note what responsivity does with wavelength: it rises linearly with λ, because a longer-wavelength photon carries less energy so the same optical power delivers more photons — and then it collapses to zero past the cutoff wavelength λ_c = 1.24/E_g, which for silicon is 1.107 µm. That single number is why silicon photodiodes are useless at the 1.31 and 1.55 µm wavelengths that optical fibre actually uses, and why those receivers are germanium (λ_c = 1.88 µm) or indium gallium arsenide (λ_c ≈ 1.65 µm).

Two refinements of the photodiode complete the syllabus's list. The PIN photodiode inserts a lightly doped, nearly intrinsic layer between the p+ and n+ regions, which does two things at once: it makes the depletion region wide, so more of the incident light is absorbed inside the field where the carriers will be collected rather than outside it where they must diffuse in slowly; and it lowers the junction capacitance, since C_j = ε_s A/W and W is now the whole intrinsic layer. Both improve speed, and the width is chosen as a trade between absorption (which wants it thick) and transit time (which wants it thin). The avalanche photodiode biases the junction just below breakdown so that each photogenerated carrier triggers impact ionisation and is multiplied by a factor M of ten to a hundred before it reaches the terminal, giving internal gain and an effective responsivity M × R. The catch is that multiplication is a statistical process, so it amplifies the noise as well as the signal and adds an excess noise factor of its own; an avalanche photodiode wins only when the following amplifier's noise, not the photodiode's, is the limit. Both devices also make the point the section began with: nothing new has been introduced. A PIN diode is a junction whose depletion width was chosen deliberately, and an avalanche photodiode is a junction biased into the regime the Zener section already described.