Carrier Concentration and the Fermi Level

1 How Many States Are There to Occupy?

Chapter 1 established that a band contains of order \(10^{22}\) states per cubic centimetre. That total is not what a carrier calculation needs. What it needs is the distribution: how those states are spread over energy, because occupancy depends strongly on energy and the states near the band edge will turn out to be the only ones that matter.

The counting is done by treating a conduction electron as a free particle of effective mass \(m_n^*\) confined to the crystal, so that its allowed wavevectors form a uniform grid in \(k\)-space with one state per volume \((2\pi/L)^3\), doubled for spin. Counting the states inside a shell between \(k\) and \(k + dk\) and converting to energy with \(E - E_C = \hbar^2k^2/2m_n^*\) gives the density of states per unit volume per unit energy:

and the mirror expression \(g_V(E) \propto \sqrt{E_V - E}\) for the valence band, with \(m_p^*\) in place of \(m_n^*\). Both are zero inside the gap, which is the formal statement that the gap is forbidden.

Two features of this result carry all the physics. First, \(g_C(E_C) = 0\): there are no states exactly at the band edge. That is not a numerical accident but a consequence of the three-dimensional counting — the shell of \(k\)-space at \(k = 0\) has zero volume — and it means the density of states grows smoothly out of the edge rather than starting at some finite value. Second, \(g\) increases as the square root of the distance from the edge, so states become more abundant deeper into the band. If availability alone decided matters, the carriers would sit deep inside the band.

They do not, and Figure 3.2 shows why: the occupancy falls exponentially with energy, far faster than \(\sqrt{E - E_C}\) rises, so the product peaks close to the edge and then dies. The peak of \(g_C(E)f(E)\) lies exactly \(kT/2 = 13\) meV above \(E_C\), and 95 per cent of the conduction electrons lie within about \(3kT = 78\) meV of the edge. On the scale of a 1.12 eV gap that is a sliver seven per cent as wide, and it justifies a simplification used everywhere afterwards: the electrical behaviour of a semiconductor is set entirely by what happens within a few \(kT\) of the two band edges. It is why a band diagram drawn as two lines, \(E_C\) and \(E_V\), loses nothing important.

The effective mass enters through \(m^{*3/2}\), so a material with light carriers has few states near its band edge. Gallium arsenide, with \(m_n^* = 0.067\,m_0\), has \(N_C = 4.7\times10^{17}\ \text{cm}^{-3}\) against silicon's \(2.8\times10^{19}\) — sixty times fewer. That, and not any difference in gap, is why GaAs becomes degenerate at a much lower doping than silicon does.

One qualification should be recorded now and returned to in Part 5. The parabolic band assumed here — \(E - E_C \propto k^2\) with a single effective mass — is an approximation to the real band structure, and silicon's conduction band is not a single minimum but six equivalent valleys along the crystal axes. The standard treatment folds the valley count and the anisotropy of each valley into one density-of-states effective mass, \(m_n^* = 1.08\,m_0\) for silicon, which is why that number differs from the \(0.26\,m_0\) conductivity mass used for mobility in Chapter 4. The two masses answer different questions: one counts states, the other governs acceleration. Quoting the wrong one is a common source of factor-of-four errors.

2 The Fermi-Dirac Distribution

Electrons are indistinguishable particles of half-integral spin, and Pauli's principle forbids two of them from occupying the same state. Statistics obeying that constraint give the Fermi-Dirac distribution, the probability that a state of energy \(E\) is occupied at temperature \(T\):

\(E_F\) is the Fermi level, sometimes called the chemical potential. It is a parameter of the distribution, fixed by requiring that the total number of electrons come out right; it is not itself an energy that any particular electron possesses.

Setting \(E = E_F\) gives \(f = 1/(1+e^0) = 1/2\) at every temperature. That is the operational definition, and it holds even when \(E_F\) lies in the gap where there is no state to be half-occupied — the statement is then conditional, about a state that would be there if the crystal allowed it.

At \(T = 0\) the exponent is \(\pm\infty\) and the distribution is a perfect step: every state below \(E_F\) full, every state above it empty. This is the ground state of the electron gas, and it makes \(E_F\) at absolute zero exactly the energy of the highest occupied state. Raising the temperature rounds the step over a width of a few \(kT\), symmetrically about \(E_F\), and does nothing else. States below \(E_F\) lose some occupancy, states above gain the same amount, and the crossing point stays fixed at one half. A useful consequence is that

The probability of finding an electron \(\Delta E\) above the Fermi level equals the probability of finding an empty state — a hole — the same distance below it. The symmetry of the distribution is why electrons and holes end up with symmetric expressions.

Some numerical values, all at 300 K where \(kT = 25.9\) meV:

Once \(E - E_F\) exceeds about \(3kT\) the unity in the denominator is negligible against the exponential and \(f(E) \approx e^{-(E-E_F)/kT}\). This is the Boltzmann approximation, and in a non-degenerate semiconductor it is superb: the conduction-band edge in intrinsic silicon lies 0.57 eV above \(E_F\), which is \(22kT\), so the error is far below any other uncertainty in the problem. The approximation matters because it turns an integral that has no closed form into one that does.

It is worth being clear about what \(E_F\) is and is not, because the term is used loosely elsewhere. In a metal at 0 K it is the highest occupied energy, and calling it "the top of the electron sea" is fair. In a semiconductor it is nothing of the sort: it usually lies in the gap, where no state exists. The general statement, true in both cases, is thermodynamic — \(E_F\) is the chemical potential of the electrons, the energy change when one electron is added to the system at constant temperature and volume. Two systems in contact and in equilibrium must share one value of it, exactly as two bodies in thermal contact must share one temperature. That equalisation is the whole content of Chapter 5, and it is the only property of \(E_F\) that matters for junctions.

3 Carrier Concentrations and the Effective Density of States

The electron concentration is the density of states multiplied by the occupancy, integrated over the conduction band:

With the Boltzmann approximation, \(f(E) \to e^{-(E-E_F)/kT}\), the integral becomes \(\int_0^\infty \sqrt{u}\,e^{-u/kT}du\) after substituting \(u = E - E_C\), which is a standard gamma integral equal to \(\tfrac{1}{2}\sqrt{\pi}\,(kT)^{3/2}\). Collecting the constants:

\(N_C\) is the effective density of states in the conduction band. It has the units of a concentration and the interpretation of one: the whole band behaves as though \(N_C\) states were bunched at \(E_C\), all of them at that single energy. For silicon at 300 K, \(N_C = 2.8\times10^{19}\ \text{cm}^{-3}\).

The same argument on the valence band, where an empty state is a hole and the occupancy is \(1 - f(E)\), gives

Both \(N_C\) and \(N_V\) scale as \(T^{3/2}\), which is the slowly varying prefactor promised in Chapter 2. Multiplying the two expressions makes \(E_F\) cancel, which is the algebraic reason the mass-action law is true:

The product depends on the crystal and the temperature but not on the doping, because doping enters only through \(E_F\) and \(E_F\) has cancelled. Chapter 2 got the same result from rate balance; this derivation shows it is a property of the statistics.

The word "effective" in effective density of states deserves a sentence, because it is doing real work. There are not \(2.8\times10^{19}\) states at \(E_C\); the states are spread over the band as \(\sqrt{E-E_C}\) and there are far more of them in total. What \(N_C\) records is that for the purpose of computing \(n\), a band with a square-root density of states weighted by a Boltzmann tail behaves identically to \(N_C\) states sitting exactly at the edge. The substitution is exact, not an approximation, given Boltzmann statistics — the gamma integral has done the averaging once and for all. It is a considerable simplification: every carrier calculation from here on treats each band as a single energy level with a known multiplicity.

A second point about \(N_C\) and \(N_V\): both carry \(T^{3/2}\), so they rise with temperature, and this is a genuine physical effect and not an artefact. Heating widens the energy range over which the Boltzmann tail is appreciable, so more of the band becomes accessible and the effective count grows. At 400 K, \(N_C = 4.3\times10^{19}\) and \(N_V = 1.6\times10^{19}\ \text{cm}^{-3}\). The growth is slow enough to be swamped by the exponential in every calculation in Part 1, but it must be included when computing \(n_i\) at a temperature far from 300 K.

4 The Fermi Level in Intrinsic Material

In intrinsic material \(n = p\). Writing the two expressions with \(E_F = E_i\), the intrinsic Fermi level, and equating them:

Taking logarithms and rearranging:

The intrinsic level sits at midgap plus a correction that vanishes only if the two effective densities of states are equal.

For silicon at 300 K, \(\ln(N_C/N_V) = \ln(2.8/1.04) = 0.990\), so the correction is \(-\tfrac{1}{2}(0.0259)(0.990) = -12.8\) meV. The intrinsic Fermi level lies 12.8 meV below midgap, giving \(E_C - E_i = 0.573\) eV and \(E_i - E_V = 0.547\) eV. The asymmetry has a simple cause: the conduction band offers more effective states than the valence band (\(m_n^* > m_p^*\)), so to keep \(n = p\) the Fermi level must sit slightly further from \(E_C\) than from \(E_V\), suppressing electrons in favour of holes.

Twelve millielectronvolts is one twentieth of \(kT\) and about one per cent of the gap, so "\(E_i\) is at midgap" is a perfectly good approximation for drawing diagrams. It is worth knowing where it comes from, because the correction grows with temperature (it is proportional to \(T\)) and because in a material with a large mass asymmetry it is not small.

Rewriting the carrier expressions in terms of \(E_i\) rather than the band edges gives the two forms actually used in practice. Since \(n_i = N_C e^{-(E_C-E_i)/kT}\), dividing the general expression for \(n\) by this gives

Two expressions worth memorising. They make the mass-action law obvious — the exponents cancel on multiplication — and they show at a glance that moving \(E_F\) up increases \(n\) and decreases \(p\) by the same factor. A displacement of \(kT\ln 10 = 59.5\) meV changes either concentration by a factor of ten.

These two forms also make one common exam question trivial. If a sample is described as having \(n = 100 p\), then dividing the expressions gives \(n/p = e^{2(E_F-E_i)/kT} = 100\), so \(E_F - E_i = \tfrac{1}{2}kT\ln 100 = 0.0298\) eV, and the individual concentrations follow from \(np = n_i^2\): \(n = 10 n_i = 10^{11}\) and \(p = 10^{9}\ \text{cm}^{-3}\). No knowledge of the doping is needed, because the ratio of the populations and the position of the Fermi level are the same piece of information written two ways.

Note also what these say about the position of \(E_F\) relative to \(E_i\): it is a direct measure of how far the material has been pushed from intrinsic. \(E_F\) above \(E_i\) means n-type, below means p-type, and equal means intrinsic or exactly compensated. In Chapter 5 the bending of the bands across a junction is nothing but this statement applied position by position.

5 How Doping Moves the Fermi Level

Doping does not change \(N_C\), \(N_V\), \(E_g\) or \(n_i\). It changes \(n\) and \(p\), and since \(E_F\) is the parameter that sets both, the Fermi level must move. Take n-type material with complete ionisation, so \(n \approx N_D\), and substitute into \(n = n_i e^{(E_F-E_i)/kT}\):

The central result of the chapter. For p-type material the mirror derivation from \(p \approx N_A\) gives \(E_i - E_F = kT\ln(N_A/n_i)\), so the Fermi level moves below the intrinsic level by the same rule.

The logarithm makes the movement slow. Every decade of doping shifts \(E_F\) by \(kT\ln 10 = 59.5\) meV, so raising \(N_D\) from \(10^{14}\) to \(10^{18}\ \text{cm}^{-3}\) — four decades, a factor of ten thousand in carriers — moves the Fermi level by only 238 meV, about a fifth of the gap. This is why doping can be varied over orders of magnitude while the band diagram still looks much the same.

6 Degenerate Doping and Where the Approximations Break

Everything above rests on the Boltzmann approximation, which needs \(E_C - E_F\) to exceed about \(3kT\). Push the doping high enough and the Fermi level climbs to within \(3kT\) of the conduction-band edge, and beyond that into the band itself. The material is then degenerate.

The threshold is easy to find. Setting \(E_C - E_F = 3kT\) in \(n = N_C e^{-(E_C-E_F)/kT}\) gives

For p-type silicon the corresponding figure is \(N_V e^{-3} = 5.2\times10^{17}\ \text{cm}^{-3}\), lower because \(N_V\) is smaller. Above these concentrations the Boltzmann approximation, the mass-action law and \(E_F - E_i = kT\ln(N_D/n_i)\) all cease to be reliable.

Several things change at once in degenerate material, and it is worth knowing which effects belong together.

• The Fermi level enters the band, so at 0 K the bottom of the conduction band is filled to \(E_F\) exactly as in a metal. Degenerate silicon has a positive temperature coefficient of resistance and looks metallic in its conduction.
• The donor states, being so numerous, overlap and broaden into an impurity band that merges with the conduction band. Ionisation energy effectively vanishes and there is no freeze-out.
• The gap narrows — by roughly 35 meV at \(10^{19}\ \text{cm}^{-3}\) and over 100 meV at \(10^{20}\) — because the dense charged impurities perturb the band edges. This band-gap narrowing raises \(n_i^2\) and is one reason heavily doped emitters have poorer injection efficiency than simple theory predicts.
• Mobility collapses. At \(10^{19}\ \text{cm}^{-3}\) the electron mobility is about 116 cm\(^2\)/V·s against 1350 in lightly doped material, because every ionised impurity is a scattering centre.

The word degenerate is borrowed from statistical mechanics, where a gas is called degenerate when quantum effects — here, the exclusion principle — dominate its behaviour. A non-degenerate carrier gas is dilute enough that the chance of two electrons competing for one state is negligible, which is exactly the condition that makes Fermi-Dirac statistics collapse to Boltzmann. At \(1.4\times10^{18}\ \text{cm}^{-3}\) in silicon the average spacing between conduction electrons has fallen to about 9 nm, comparable with the spread of the electron wavefunctions themselves, and the dilute approximation is finished.

Degenerate regions are not an accident to be avoided; they are used deliberately. The notation \(n^+\) and \(p^+\) on a device cross-section means exactly this: doping at \(10^{19}\) to \(10^{20}\ \text{cm}^{-3}\), chosen so that the region behaves as a low-resistance contact rather than as an active semiconductor. Every ohmic contact in an integrated circuit relies on a degenerate layer under the metal, because the depletion region at a metal-semiconductor junction becomes thin enough for carriers to tunnel through it instead of having to climb the barrier — the mechanism Chapter 13 uses for the Schottky diode, run in reverse. The tunnel diode of the same chapter needs both sides degenerate, and the emitter of a bipolar transistor is doped an order of magnitude above the base precisely to force injection one way.

7 The Fermi Level Against Temperature

The relation \(E_F - E_i = kT\ln(N_D/n_i)\) has temperature in two places, and they pull in opposite directions. The prefactor \(kT\) grows linearly with \(T\), pushing \(E_F\) away from \(E_i\). The logarithm shrinks, because \(n_i\) grows exponentially while \(N_D\) is fixed. The exponential wins comprehensively, so heating always drives \(E_F\) towards midgap.

Figure 3.3 plots the result for three samples, computed from charge neutrality with incomplete ionisation included so that the low-temperature end is honest. Three regions appear, and they are the same three regions Chapter 2 found in the carrier concentration — necessarily so, since the two quantities are two views of one situation.

The heavier doping in the figure illustrates the logarithm at work. Raising \(N_D\) from \(10^{16}\) to \(10^{18}\ \text{cm}^{-3}\) — a hundredfold — lifts the room-temperature Fermi level by only \(2kT\ln 10 = 119\) meV, and the two n-type curves stay roughly that far apart across the whole plot. The more heavily doped sample also stays extrinsic to a higher temperature, simply because \(n_i\) has further to climb before it can compete: at 700 K it is still 0.136 eV above \(E_i\) while the lightly doped sample has effectively arrived there. Doping heavily is one way to buy temperature range, and it is why power devices and high-temperature parts use higher doping than their small-signal equivalents, at the cost of lower breakdown voltage — a trade Chapter 12 quantifies.

At very low temperature the Fermi level lies about halfway between the donor level and the conduction-band edge, which is where it must be for the donors to be half ionised; as \(T\) rises through the freeze-out region it moves down as ionisation completes. Through the extrinsic plateau it falls steadily and almost linearly, because \(n\) is pinned at \(N_D\) while \(kT\ln(N_D/n_i)\) shrinks. Above about 600 K the intrinsic carriers dominate and every curve, whatever its doping, collapses onto \(E_i\).

The engineering content of that last statement is worth spelling out. A junction's built-in potential is the difference between the Fermi levels of the two sides before they are joined, so as the curves converge the barrier shrinks. At 600 K the \(10^{16}\) n-type and \(10^{16}\) p-type curves are separated by only 0.116 eV against 0.714 eV at 300 K, so the junction's ability to block reverse current has largely gone. Chapter 5 computes that potential properly; the point here is that its temperature coefficient, roughly \(-2\) mV/K for silicon, is read straight off this graph, and it is the same \(-2\) mV/K that appears as the temperature coefficient of a diode's forward drop in Chapter 6 and as the input offset drift of a differential pair in Chapter 27.

8 Summary and Key Results

9 Common Mistakes

10 Chapter Review