By the end of this chapter you should be able to:
- Relate drift velocity to field through mobility, and explain the origin of mobility in terms of mean free time between collisions.
- Account for the temperature and doping dependence of mobility in terms of lattice and ionised-impurity scattering, and identify which dominates where.
- Compute drift current density and conductivity for doped silicon, and state when velocity saturation invalidates the linear relation.
- Write the diffusion current densities for electrons and holes with the correct signs, and explain why the two signs differ.
- State the Einstein relation, verify it numerically for silicon at 300 K, and explain why the same random walk produces both coefficients.
- Assemble the total current density and the continuity equation, and interpret each term.
- Define carrier lifetime and diffusion length, compute the latter from the former, and distinguish band-to-band, Shockley-Read-Hall and Auger recombination.
Chapters 2 and 3 counted the carriers. Counting them says nothing about what they do, and a semiconductor with \(10^{16}\) electrons per cubic centimetre sitting perfectly still carries no current at all. This chapter is about the two — only two — reasons a carrier moves, and about the quantitative machinery that converts a carrier concentration into a current density.
The first mechanism is drift: an electric field exerts a force, and the carrier accelerates until a collision takes its momentum away, so that on average it moves at a speed proportional to the field. That constant of proportionality, the mobility, turns out to be the most doping-sensitive and temperature-sensitive parameter in the whole subject. The second is diffusion: carriers move from where there are many to where there are few, driven not by any force but by the statistics of random motion. Diffusion has no analogue in ordinary circuit theory and it is where most students' intuition fails, yet it is what makes a diode conduct and a transistor amplify. The chapter ends by joining the two — the Einstein relation shows they are the same random walk seen twice — and by asking how long an excess carrier survives before it recombines, which fixes how far it can travel and therefore how thin a transistor base has to be.
1 Drift, Scattering and the Origin of Mobility
Left alone at 300 K, a conduction electron in silicon is not at rest. It carries a thermal kinetic energy of \(\tfrac{3}{2}kT = 0.039\) eV, which for an effective mass of \(0.26\,m_0\) corresponds to a speed of roughly \(2\times10^{7}\) cm/s. It travels at that speed in a straight line until it collides — with a vibrating lattice atom, an ionised impurity or a crystal defect — after which it sets off in a new random direction. The mean free time between collisions, written \(\tau_c\), is of order \(10^{-13}\) s in silicon, so the mean free path is a few tens of nanometres. Because the directions are random, the average displacement over many collisions is zero and there is no current.
Now apply a field \(\mathcal{E}\). Between collisions the electron accelerates at \(a = q\mathcal{E}/m_n^*\), so just before its next collision it has acquired an extra velocity \(a\tau_c\) directed against the field. The collision randomises its direction and the process starts again, so the acquired velocity never accumulates: it is reset every \(\tau_c\). Averaging over many free flights gives a steady drift velocity
\(\mu\) is the mobility, in cm\(^2\)/V·s. Two things make a carrier mobile: long intervals between collisions and a small effective mass. Everything that degrades mobility does so by shortening \(\tau_c\).
The scale of the drift deserves attention, because the picture in most students' heads is wrong. In silicon doped at \(10^{16}\ \text{cm}^{-3}\), \(\mu_n = 1248\ \text{cm}^2/\text{V}\!\cdot\!\text{s}\), so a field of 100 V/cm gives \(v_d = 1.25\times10^{5}\) cm/s. That is less than one per cent of the thermal speed. Conduction is not a stream of carriers marching down the bar; it is a violent random motion on which a very slight directional bias has been imposed, and the entire current of a working device is that slight bias.
The linear relation \(v_d = \mu\mathcal{E}\) is an approximation with a clear limit. It assumes \(\tau_c\) does not depend on the field, which holds only while the drift energy stays small compared with the thermal energy. Above roughly \(10^{4}\) V/cm in silicon the carriers gain enough energy between collisions to excite optical phonons, a very efficient loss channel, and the drift velocity stops rising. It saturates at about \(1\times10^{7}\) cm/s for electrons and \(8\times10^{6}\) cm/s for holes. Saturation is not an obscure limit: a 1 µm channel with 2 V across it sees \(2\times10^{4}\) V/cm, so every modern MOSFET in Chapter 24 operates in it, and the saturation velocity rather than the mobility sets its drive current.
Holes and electrons drift in opposite directions, but their currents add rather than cancel, because the charges are also opposite. This is worth stating once carefully: an electron drifting against \(\mathcal{E}\) carrying charge \(-q\) contributes conventional current along \(\mathcal{E}\), and a hole drifting along \(\mathcal{E}\) carrying \(+q\) does the same. Both drift currents flow in the direction of the field, always.
2 What Mobility Depends On
Mobility is the parameter that makes a data sheet temperature-dependent, and it is worth knowing which of the two competing scattering mechanisms is responsible for a given behaviour.
Lattice, or phonon, scattering is collision with the thermal vibrations of the atoms themselves. Its rate grows with temperature because hotter lattices vibrate more, and the standard result is \(\mu_L \propto T^{-3/2}\) — empirically closer to \(T^{-2.4}\) for electrons in silicon. This mechanism sets the ceiling on mobility in pure material and it cannot be evaded: even a perfect crystal has phonons above 0 K.
Ionised impurity scattering is the Coulomb deflection of a carrier passing a charged dopant ion. Its rate grows with the ion concentration, and it falls with temperature as \(\mu_I \propto T^{+3/2}\), because a faster carrier spends less time near the ion and is deflected less. The opposite temperature signs of the two mechanisms mean that mobility in doped silicon rises with temperature at low \(T\), passes through a maximum, and falls at high \(T\). The two combine as reciprocals, since scattering rates add:
Matthiessen's rule. Whichever mechanism is worse dominates the total, so a single bad scatterer spoils the mobility however good the rest of the crystal is.
| Impurity concentration (cm\(^{-3}\)) | \(\mu_n\) | \(\mu_p\) | \(D_n\) | \(D_p\) |
|---|---|---|---|---|
| \(10^{14}\) | 1358 | 461 | 35.1 | 11.9 |
| \(10^{15}\) | 1345 | 458 | 34.8 | 11.8 |
| \(10^{16}\) | 1248 | 437 | 32.3 | 11.3 |
| \(10^{17}\) | 801 | 331 | 20.7 | 8.55 |
| \(10^{18}\) | 263 | 143 | 6.81 | 3.70 |
| \(10^{19}\) | 116 | 69 | 3.00 | 1.78 |
Mobilities in cm\(^2\)/V·s and diffusion coefficients in cm\(^2\)/s, silicon at 300 K, from the standard Caughey-Thomas fit to measured data. The diffusion coefficients follow from the Einstein relation derived below.
Read the table from the top and the design consequence is unmistakable. Doping from \(10^{16}\) to \(10^{19}\ \text{cm}^{-3}\) multiplies the carrier count by a thousand but divides the mobility by eleven, so the conductivity improves by only ninety-three, not a thousand. Doping is a diminishing return, and it is why a low-resistance region in an integrated circuit is made wide and short rather than simply doped harder.
The knee of the curve, near \(10^{17}\ \text{cm}^{-3}\), is where the two mechanisms are of equal strength at 300 K. Below it the mobility is essentially that of pure silicon and doping is almost free; above it every additional dopant atom costs mobility. Since a typical device uses light doping where it needs speed and heavy doping where it needs low resistance, the knee is in effect the boundary between the two design regimes, and it moves to higher concentration as the sample is cooled.
Three further points. Compensated material is scattered by the total ionised concentration \(N_D + N_A\), not the net, so a compensated region has lower mobility than a lightly doped one with the same carrier count. Electron mobility exceeds hole mobility by about a factor of three at low doping and less than two at high doping, for the reason given in Chapter 2 — valence-band transport is congested. And the mobility that appears here, the conductivity mobility, uses the conductivity effective mass \(0.26\,m_0\) for silicon electrons, not the \(1.08\,m_0\) density-of-states mass of Chapter 3; the two answer different questions and must not be interchanged.
3 Drift Current Density and Conductivity
A current density is charge density times velocity. For electrons, the mobile charge density is \(-qn\) and the velocity is \(-\mu_n\mathcal{E}\), so the two minus signs cancel:
Both are positive for positive \(\mathcal{E}\), confirming that the two drift currents reinforce. Adding them gives Ohm's law in its microscopic form, with the conductivity identified:
Everything a semiconductor does electrically as a bulk material is in this one expression: doping fixes \(n\) and \(p\), scattering fixes \(\mu_n\) and \(\mu_p\), and the product is the conductivity.
In extrinsic material one term overwhelms the other and the expression simplifies. For n-type silicon, \(n \approx N_D\) and \(p = n_i^2/N_D\) is smaller by \(10^{12}\), so \(\sigma \approx qN_D\mu_n\) to twelve significant figures. Minority carriers contribute nothing to bulk conductivity, which is exactly why they can be ignored in a resistor and never in a junction.
Take \(q = 1.602\times10^{-19}\) C, \(n_i = 1.0\times10^{10}\ \text{cm}^{-3}\) at 300 K.
(a) Intrinsic silicon. Here \(n = p = n_i\) and the lightly doped mobilities apply, \(\mu_n = 1350\), \(\mu_p = 480\ \text{cm}^2/\text{V}\!\cdot\!\text{s}\):
so \(\rho_i = 3.41\times10^{5}\ \Omega\cdot\)cm. Pure silicon is a poor insulator, not a conductor.
(b) n-type, \(N_D = 10^{16}\ \text{cm}^{-3}\). From the table \(\mu_n = 1248\ \text{cm}^2/\text{V}\!\cdot\!\text{s}\):
a factor of \(6.8\times10^{5}\) better than intrinsic, achieved with one dopant atom in five million.
(c) A resistor. A bar of that material 100 µm long with a 10 µm × 10 µm cross-section has \(R = \rho L/A = 0.500 \times (100\times10^{-4})/(10^{-6}) = 5.0\ \text{k}\Omega\). Applying 1 V gives a field of \(1/(100\times10^{-4}) = 100\) V/cm and a drift velocity of \(1.25\times10^{5}\) cm/s, so a carrier crosses the bar in 0.80 ns.
(d) p-type, \(N_A = 10^{17}\ \text{cm}^{-3}\). With \(\mu_p = 331\), \(\sigma = 1.602\times10^{-19}\times10^{17}\times331 = 5.30\) S/cm and \(\rho = 0.189\ \Omega\cdot\)cm. Ten times the doping of (b) but only 2.6 times the conductivity, because \(\mu_p\) is both smaller to begin with and further degraded by the heavier doping.
The temperature behaviour of \(\sigma\) now follows from two competing effects and depends on which regime the material is in. In the extrinsic plateau, \(n\) is fixed at \(N_D\) while \(\mu\) falls with temperature, so resistivity rises with temperature, like a metal — the positive temperature coefficient of a doped silicon resistor, exploited in the sensistor. In intrinsic or near-intrinsic material, \(n_i\) climbs exponentially and swamps the mobility fall, so resistivity drops steeply — the negative coefficient of a thermistor. The same piece of silicon does both, in different temperature ranges.
4 Diffusion and the Concentration Gradient
Drift needs a field. Diffusion needs nothing but a non-uniform concentration and a temperature above absolute zero, and it will happen in a sample with no field and no applied voltage at all.
The mechanism is purely statistical. Consider an imaginary plane in the crystal with more carriers on the left than the right. Every carrier is executing its random thermal walk, and the walk knows nothing about which way is which — a carrier on either side is equally likely to cross the plane in either direction. But there are more carriers on the left, so more of them cross to the right than the other way, and there is a net flux from high concentration to low. No force acts, and nothing pushes the carriers; the imbalance is entirely a matter of there being more candidates on one side.
Fick's law states that the flux is proportional to the gradient, and multiplying by the carrier charge gives the current densities:
\(D_n\) and \(D_p\) are the diffusion coefficients, in cm\(^2\)/s. The signs are the single most error-prone detail in this chapter and they are not arbitrary: both fluxes run down their gradients, but electrons carry negative charge, so the electron current runs the opposite way to the electron flux and the sign flips. Get this wrong and a diode will appear to conduct backwards.
Figure 4.3 shows the situation that matters most: excess minority carriers injected at one face of a bar and diffusing inwards while recombining. The profile is exponential, \(\Delta n(x) = \Delta n(0)e^{-x/L_n}\), for a reason developed in the last section, and the exponential is convenient because its derivative is proportional to itself: \(dn/dx = -\Delta n(x)/L_n\), so the diffusion current at any point is proportional to the local excess concentration.
Electrons are injected into p-type silicon so that the excess concentration falls linearly from \(10^{16}\ \text{cm}^{-3}\) at \(x = 0\) to zero at \(x = 2\ \mu\text{m}\). Take \(D_n = 32.3\ \text{cm}^2\)/s. Find the diffusion current density.
The gradient is \(dn/dx = -(10^{16})/(2\times10^{-4}\ \text{cm}) = -5.0\times10^{19}\ \text{cm}^{-4}\), and
Two hundred and fifty amperes per square centimetre, from a gradient set up over two micrometres with no applied field whatever. Compare it with the drift current in the same material under a field of 100 V/cm, which for \(n = 10^{16}\) is \(qn\mu_n\mathcal{E} = 200\) A/cm\(^2\). Diffusion is not a small correction to drift; over the short distances inside a device the two are the same order of magnitude, and near a junction diffusion wins outright. This calculation is, in essence, the forward current of a diode.
One more contrast with drift is worth fixing. Drift current depends on how many carriers there are; diffusion current depends on how unevenly they are distributed. A bar with \(10^{20}\ \text{cm}^{-3}\) carriers uniformly spread carries no diffusion current at all, while one with \(10^{10}\) arranged in a steep gradient carries a measurable one. It is the gradient, never the concentration, that appears in the equation.
5 The Einstein Relation
Drift and diffusion have been treated as separate mechanisms with separate coefficients, but they are limited by the same collisions, so \(\mu\) and \(D\) cannot be independent. The relation between them can be derived without any new physics, by considering a sample in equilibrium with a non-uniform doping.
Suppose an n-type bar is doped more heavily at one end. Electrons diffuse from the heavily doped end towards the lightly doped end, leaving behind uncompensated donor ions and building up negative charge at the far end. That charge separation creates an internal field which drives a drift current back the other way. In equilibrium no net current can flow — a bar sitting on a bench with nothing connected to it cannot pass a current — so the two exactly cancel:
From Chapter 3, \(n = n_i \exp[(E_F - E_i)/kT]\). In equilibrium \(E_F\) is flat, so all the position dependence is in \(E_i\), which moves with the electrostatic potential as \(E_i = -q\phi + \text{constant}\). Differentiating, and using \(\mathcal{E} = -d\phi/dx\):
Substituting into the zero-current condition, \(qn\mu_n\mathcal{E} - qD_n\dfrac{qn}{kT}\mathcal{E} = 0\), and cancelling \(qn\mathcal{E}\):
The Einstein relation. The ratio depends only on temperature, not on the material, the doping or the carrier type. The quantity \(kT/q\) has the units of volts and is called the thermal voltage \(V_T\); at 300 K it equals 0.02585 V, and it will appear in every diode and transistor equation in the rest of the course.
The mobilities and diffusion coefficients in the table of the previous section were measured independently. Test the relation at three doping levels, silicon at 300 K, where \(kT/q = 0.025852\) V.
| Doping (cm\(^{-3}\)) | \(\mu_n\) | \(D_n\) | \(D_n/\mu_n\) (V) | \(\mu_p\) | \(D_p\) | \(D_p/\mu_p\) (V) |
|---|---|---|---|---|---|---|
| \(10^{15}\) | 1345 | 34.77 | 0.02585 | 458 | 11.84 | 0.02585 |
| \(10^{17}\) | 801 | 20.72 | 0.02585 | 331 | 8.55 | 0.02585 |
| \(10^{19}\) | 116 | 3.00 | 0.02585 | 69 | 1.78 | 0.02585 |
The ratio is the same to every digit quoted, across two orders of magnitude of mobility and both carrier types. That constancy is the practical use of the relation: a diffusion coefficient never has to be measured, because it can be computed from a mobility, and mobilities are far easier to measure. At 350 K the ratio would be \(0.02585\times350/300 = 0.03016\) V, and both coefficients would have to be recomputed.
Strictly the relation holds only for non-degenerate material, since it assumed Boltzmann statistics through \(n = n_i e^{(E_F-E_i)/kT}\). Above \(10^{18}\ \text{cm}^{-3}\) the correct form contains a Fermi-Dirac integral ratio that exceeds unity, so \(D/\mu > kT/q\). The correction is about 10 per cent at \(10^{19}\ \text{cm}^{-3}\) and can be ignored at every doping used in this course.
6 Total Current and the Continuity Equation
Adding the drift and diffusion contributions for each carrier gives the four terms that describe all steady transport in a semiconductor:
These, together with Poisson's equation relating \(\mathcal{E}\) to the charge density, are the transport equations. Every device model in this course is a solution of them subject to particular boundary conditions.
An observation that will save a great deal of confusion in Part 2: the total \(J\) must be continuous through a device in the steady state, but \(J_n\) and \(J_p\) individually need not be. In a forward-biased diode the current at the p-side terminal is almost entirely hole current and at the n-side terminal almost entirely electron current, with the changeover happening in the neutral regions through recombination. In a bipolar transistor the ability to convert one into the other in a controlled way is the device.
The transport equations say how carriers move but not how their number changes with time. That requires a bookkeeping statement — carriers in a slab of thickness \(dx\) may be added by generation, removed by recombination, or carried in and out by current — which is the continuity equation. For electrons:
Rate of accumulation equals net flow in, plus generation, minus recombination. The corresponding hole equation carries a minus sign on the current term because of the charge.
For the case that matters most — low-level injection of minority electrons into a field-free p-type region, where transport is by diffusion alone and recombination is proportional to the excess — the equation reduces to
In the steady state the left side vanishes, leaving \(D_n\,d^2\Delta n/dx^2 = \Delta n/\tau_n\), whose decaying solution is
This is where the exponential profile of Figure 4.3 comes from, and it defines the diffusion length \(L_n\): the distance over which an injected population falls to \(1/e\) of its value, and the natural length scale of every minority-carrier device.
It is worth pausing on why the solution is an exponential rather than, say, a straight line. Two processes act on the injected population at once: diffusion spreads it, and recombination removes it in proportion to how much is there. The steady profile is the one for which the spreading exactly replaces what recombination destroys at every point, and the only function whose second derivative is proportional to itself with a positive constant is a decaying exponential. The length \(L_n = \sqrt{D_n\tau_n}\) is then a comparison of the two rates: it is roughly the distance a carrier diffuses in one lifetime, since a random walk covers \(\sqrt{Dt}\) rather than \(vt\).
"Low-level injection" means the injected excess is small compared with the majority concentration, so that the majority carriers can neutralise it without their own concentration changing appreciably and the field stays negligible. It is the assumption behind almost every analytic device result in this course, and it fails in power devices at high current, where high-level injection reduces the effective doping and changes the exponent in the diode equation from \(V/V_T\) to \(V/2V_T\) — a detail Chapter 6 returns to.
7 Lifetime, Diffusion Length and Recombination
An excess carrier does not live for ever. Injected into a region where it is a minority carrier, it wanders until it meets a carrier of the opposite type and recombines. The mean time before that happens is the carrier lifetime \(\tau\), defined by the decay of an excess population after the injection stops:
Lifetime in silicon ranges over six orders of magnitude, from about 1 ns in heavily doped or defect-rich material to several milliseconds in high-purity float-zone wafers, and it is the parameter most sensitive to processing. It is also the one most easily ruined: a single part per billion of gold or iron will cut a millisecond lifetime to nanoseconds.
Three mechanisms compete, and Figure 4.4 shows them. Band-to-band recombination is the direct fall of a conduction electron into a valence-band vacancy, with the energy leaving as a photon. It requires the electron and hole to have the same crystal momentum, which in a direct-gap material like gallium arsenide is easy and in indirect-gap silicon is not, so this is the dominant mechanism in an LED and a negligible one in silicon. Shockley-Read-Hall recombination proceeds in two steps through a trap level near midgap created by a defect or a deep impurity: the trap captures an electron, then captures a hole, releasing the energy to the lattice as heat in two smaller instalments and needing no momentum match. This dominates in silicon at ordinary doping. Auger recombination hands the released energy to a third carrier instead of to a photon or a phonon; because it needs three particles to meet, its rate goes as \(n^2p\) and it becomes dominant only above about \(10^{18}\ \text{cm}^{-3}\), where it sets the ceiling on the injection efficiency of a heavily doped emitter.
The three act in parallel, so their rates add and their lifetimes combine as reciprocals:
The shortest lifetime dominates, which is why a trace of a lifetime-killing impurity controls the result regardless of how good the crystal is otherwise.
Electrons are injected into p-type silicon doped at \(10^{16}\ \text{cm}^{-3}\), where \(D_n = 32.3\ \text{cm}^2\)/s. Find \(L_n\) for lifetimes of 1 µs and 100 ns.
and for \(\tau_n = 100\) ns, \(L_n = \sqrt{32.3\times10^{-7}} = 18.0\ \mu\text{m}\). For holes injected into n-type material with \(D_p = 11.3\ \text{cm}^2\)/s and \(\tau_p = 1\ \mu\text{s}\), \(L_p = 33.6\ \mu\text{m}\).
The design consequence appears in Chapter 16. A bipolar transistor works only if injected minority carriers cross the base before recombining, and the fraction that survives a base of width \(W_B\) is approximately \(1 - W_B^2/2L_n^2\). For \(L_n = 56.8\ \mu\text{m}\), a base 1 µm wide passes \(1 - (1/56.8)^2/2 = 99.985\) per cent of them, which is the transport factor that makes \(\beta\) large. Widen the base to 10 µm and the figure falls to 98.5 per cent, cutting \(\beta\) from several hundred to about sixty. The base of a real transistor is a fraction of a micrometre wide for exactly this reason, and the deliberate gold doping used in older switching diodes to shorten \(\tau\) — and so speed up turn-off — is the same relation used in reverse.
A last connection. The diffusion length is not an independent quantity: it is fixed by \(D\), which comes from \(\mu\) through Einstein, which comes from \(\tau_c\) through scattering. Doping a region harder shortens both the scattering time and the recombination lifetime, so \(L\) falls twice over. Every design decision about doping in Part 4 is a negotiation between conductivity, which wants heavy doping, and diffusion length, which does not.
8 Summary and Key Results
| Quantity | Expression | Value or consequence |
|---|---|---|
| Drift velocity | \(v_d = \mu\mathcal{E}\), \(\mu = q\tau_c/m^*\) | \(1.25\times10^{5}\) cm/s at 100 V/cm; saturates at \(10^{7}\) cm/s above \(10^{4}\) V/cm |
| Lattice scattering | \(\mu_L \propto T^{-3/2}\) | Dominant in lightly doped material; mobility falls as the crystal is heated |
| Impurity scattering | \(\mu_I \propto T^{+3/2}/N_{\text{ion}}\) | Dominant above \(10^{17}\) cm\(^{-3}\); combines by \(1/\mu = 1/\mu_L + 1/\mu_I\) |
| Mobility values | — | \(\mu_n\) 1350 → 116 and \(\mu_p\) 480 → 69 from \(10^{14}\) to \(10^{19}\) cm\(^{-3}\) |
| Conductivity | \(\sigma = q(n\mu_n + p\mu_p)\) | \(2.9\times10^{-6}\) S/cm intrinsic; 2.00 S/cm at \(N_D = 10^{16}\) (\(\rho = 0.50\ \Omega\cdot\)cm) |
| Diffusion current | \(J_{n} = +qD_n\,dn/dx\); \(J_{p} = -qD_p\,dp/dx\) | Sign differs because the charges do; both fluxes run down the gradient |
| Einstein relation | \(D/\mu = kT/q = V_T\) | 0.02585 V at 300 K; verified to four figures at \(10^{15}\), \(10^{17}\) and \(10^{19}\) cm\(^{-3}\) |
| Total current | \(J = q(n\mu_n+p\mu_p)\mathcal{E} + q(D_n\,dn/dx - D_p\,dp/dx)\) | \(J\) is continuous through a device; \(J_n\) and \(J_p\) separately are not |
| Continuity | \(\partial n/\partial t = q^{-1}\partial J_n/\partial x + G - R\) | Steady low-level injection gives \(\Delta n(x) = \Delta n(0)e^{-x/L_n}\) |
| Diffusion length | \(L_n = \sqrt{D_n\tau_n}\) | 56.8 µm for \(\tau_n = 1\ \mu\)s; 18.0 µm for 100 ns; sets the base width in Chapter 16 |
| Recombination | \(1/\tau = 1/\tau_{\text{rad}} + 1/\tau_{\text{SRH}} + 1/\tau_{\text{Auger}}\) | SRH dominates in silicon; radiative in GaAs; Auger above \(10^{18}\) cm\(^{-3}\) |
9 Common Mistakes
\(J_n = +qD_n\,dn/dx\) but \(J_p = -qD_p\,dp/dx\). Both particle fluxes run from high concentration to low, which is what Fick's law says; the currents differ because an electron flux to the right is a conventional current to the left. The drift currents, by contrast, do have the same sign, since there both the charge and the velocity reverse together. Write the flux first, then attach the charge, and the signs come out right every time.
It is proportional to the gradient. A uniformly doped bar with \(10^{20}\) carriers per cubic centimetre carries no diffusion current at all, while a lightly doped region with a steep injected profile carries hundreds of amperes per square centimetre. This is why an equilibrium sample passes no current despite containing enormous carrier populations, and why the current in a forward-biased diode is set by the slope of the minority profile at the depletion edge rather than by how many carriers are there.
Silicon has two different effective masses in common use. The density-of-states mass, \(1.08\,m_0\) for electrons, counts states and belongs in \(N_C\) and in Chapter 3. The conductivity mass, \(0.26\,m_0\), governs acceleration and belongs in \(\mu = q\tau_c/m^*\). Substituting one for the other changes a mobility by a factor of four. The same warning applies to holes, where the corresponding values are \(0.56\,m_0\) and \(0.39\,m_0\).
10 Chapter Review
1. A silicon bar is doped with \(5\times10^{16}\) cm\(^{-3}\) donors, for which \(\mu_n = 950\) cm\(^2\)/V·s. Find its conductivity and resistivity, the field needed to produce a drift velocity of \(10^{5}\) cm/s, and the electron transit time across a 50 µm length at that field.
Extrinsic n-type, so the hole term is negligible: \(\sigma = qN_D\mu_n = 1.602\times10^{-19} \times 5\times10^{16} \times 950 = 7.61\) S/cm, giving \(\rho = 0.131\ \Omega\cdot\)cm. For \(v_d = 10^{5}\) cm/s, \(\mathcal{E} = v_d/\mu_n = 10^{5}/950 = 105\) V/cm. The transit time is \(t = L/v_d = 50\times10^{-4}/10^{5} = 5.0\times10^{-8}\) s = 50 ns. Note that this field is far below the \(10^{4}\) V/cm at which velocity saturation begins, so the linear relation is safe; had the answer come out above that, \(v_d\) would have been pinned near \(10^{7}\) cm/s instead.
2. Explain why the mobility of doped silicon first rises and then falls as the temperature is increased from 50 K to 500 K.
Two scattering mechanisms compete and their temperature dependences have opposite signs. Ionised impurity scattering deflects a carrier that passes close to a charged dopant ion; a faster carrier spends less time in the ion's field and is deflected less, so \(\mu_I \propto T^{+3/2}\) and this mechanism becomes weaker as the sample is heated. Lattice scattering is collision with thermal vibrations, which become more violent as the sample is heated, so \(\mu_L \propto T^{-3/2}\) (nearer \(T^{-2.4}\) in practice for silicon electrons). Since scattering rates add, \(1/\mu = 1/\mu_L + 1/\mu_I\), and the smaller mobility dominates. At low temperature impurity scattering is the bottleneck and \(\mu\) rises with \(T\); at high temperature lattice scattering takes over and \(\mu\) falls. The maximum sits where the two are equal, and it moves to higher temperature as the doping is increased.
3. Verify the Einstein relation for holes in silicon at 300 K given \(\mu_p = 437\) cm\(^2\)/V·s and \(D_p = 11.31\) cm\(^2\)/s, and state the value of \(D_p\) at 350 K if the mobility there is 340 cm\(^2\)/V·s.
The ratio is \(D_p/\mu_p = 11.31/437 = 0.02588\) V, against \(kT/q = (1.381\times10^{-23} \times 300)/(1.602\times10^{-19}) = 0.02585\) V — agreement to three figures, the residual being rounding in the quoted data. At 350 K the thermal voltage is \(0.02585 \times 350/300 = 0.03016\) V, so \(D_p = \mu_p \times kT/q = 340 \times 0.03016 = 10.3\) cm\(^2\)/s. Both \(\mu_p\) and \(D_p\) have fallen, the first because lattice scattering has worsened and the second because it inherits that fall, partly offset by the higher thermal voltage.
4. Holes are injected at one face of an n-type region with an excess concentration of \(10^{14}\) cm\(^{-3}\). Given \(D_p = 11.3\) cm\(^2\)/s and \(\tau_p = 2\ \mu\)s, find the diffusion length, the excess concentration 20 µm in, and the diffusion current density at the injecting face.
\(L_p = \sqrt{D_p\tau_p} = \sqrt{11.3 \times 2\times10^{-6}} = 4.75\times10^{-3}\) cm = 47.5 µm. At \(x = 20\) µm, \(\Delta p = 10^{14}e^{-20/47.5} = 10^{14} \times 0.656 = 6.56\times10^{13}\) cm\(^{-3}\). For the current, the profile is \(\Delta p(x) = \Delta p(0)e^{-x/L_p}\), so \(d\Delta p/dx|_{0} = -\Delta p(0)/L_p = -10^{14}/4.75\times10^{-3} = -2.11\times10^{16}\) cm\(^{-4}\), and \(J_p = -qD_p\,dp/dx = 1.602\times10^{-19} \times 11.3 \times 2.11\times10^{16} = 38.2\) A/cm\(^2\), flowing in the \(+x\) direction. Note that the current is proportional to \(D_p/L_p = \sqrt{D_p/\tau_p}\), so a shorter lifetime gives a larger injected current for the same boundary concentration — a result that reappears as the reverse saturation current of Chapter 6.
5. A silicon sample is deliberately doped with gold to reduce the carrier lifetime from 10 µs to 20 ns. What is done to the diffusion length, and why would a manufacturer want this?
Diffusion length scales as the square root of lifetime, so reducing \(\tau\) by a factor of 500 reduces \(L\) by \(\sqrt{500} = 22.4\). Taking \(D_n = 32.3\) cm\(^2\)/s, \(L_n\) falls from \(\sqrt{32.3\times10^{-5}} = 180\) µm to \(\sqrt{32.3 \times 2\times10^{-8}} = 8.0\) µm. Gold introduces a deep level close to midgap, which is the most effective possible Shockley-Read-Hall trap because it captures electrons and holes with comparable ease. The motive is switching speed. When a forward-biased junction is suddenly reversed, the stored minority carriers in the neutral regions must be removed before the diode can block, and the time this takes — the reverse recovery time of Chapter 6 — is of the order of \(\tau\). Cutting \(\tau\) by 500 cuts the recovery time by roughly the same factor. The price is a higher forward voltage drop and a larger reverse leakage current, since a midgap trap is also an efficient generation centre; the Schottky diode of Chapter 13 achieves fast switching without paying it, by having no stored minority charge at all.