Chapter 2 — Semiconductor Physics

Theory, worked examples and 26 previous-year questions with full solutions — IIT-JAM, GATE, JEST and CSIR NET-JRF, 2020 onwards.

Theory

Semiconductors are among the most important classes of materials used in the electronics industry. Diodes, transistors and integrated circuits are all made of semiconductor materials, and before we can understand how these devices operate we must first understand a few of the characteristic properties of the materials themselves. Silicon is by far the most common semiconductor used in discrete devices and integrated circuits, and most of our numerical illustrations will refer to it.

2.1 Intrinsic semiconductors

(a) Classification

Semiconductors are classified as intrinsic or extrinsic. A semiconductor in its purest form is called an intrinsic semiconductor. Figure 2.1(a) shows a two-dimensional representation of a silicon single crystal at \(T=0\) K. Each silicon atom shares its four valence electrons with its four neighbours in covalent bonds, and at absolute zero every valence electron is held in such a bond: the valence band is completely full and the conduction band completely empty. If an electric field is applied, no charge can flow, because every electron remains bound to its own atom.

Figure 2.1 (a) Two-dimensional representation of single-crystal silicon at \(T=0\) K; all valence electrons are bound to the silicon atoms by covalent bonds. (b) The breaking of a covalent bond for \(T>0\) K, creating an electron in the conduction band and a positively charged empty state (a hole).

At room temperature, however, a few electron–hole pairs are generated by thermal energy. As the temperature rises, more and more electrons acquire enough thermal energy to break their covalent bonds and move away from their original positions, as illustrated schematically in Figure 2.1(b). Each electron so released leaves behind a vacant bond, the hole, which behaves as a mobile positive charge.

(b) The energy band picture

The same process is described more economically by the energy band diagram of Figure 2.2(a), in which the vertical scale is electron energy and the horizontal scale is distance through the semiconductor. The top of the valence band has energy \(E_V\) and the bottom of the conduction band has energy \(E_C\). The region between these two energies is called the forbidden band gap; no electron states exist within it in a perfect crystal.

Figure 2.2 (a) Energy band diagram; the vertical scale is electron energy and the horizontal scale is distance through the semiconductor. (b) The generation process, creating an electron in the conduction band and a positively charged empty state in the valence band.

The minimum energy needed to break a covalent bond and raise an electron into the conduction band is the band-gap energy

\[E_g = E_C - E_V .\]

Electrons in the conduction band are called free electrons. Figure 2.2(b) shows an electron that has gained sufficient energy moving from the valence band into the conduction band, leaving a hole behind; this process is called generation of an electron–hole pair.

(c) Conduction in an intrinsic semiconductor

When an electric field is applied across an intrinsic semiconductor, the current through it is carried by both free electrons and holes, as shown in Figure 2.3. Loosely bound electrons near the positive terminal \(A\) are drawn away from their atoms into that terminal, and the holes so created drift towards the negative terminal \(B\) of the battery. The movement of electrons from bond to bond creates new holes near the positive terminal, which in turn drift towards the negative terminal. When the holes reach the negative end, electrons enter the semiconductor from the terminal \(B\) and recombine with them. In the external circuit, therefore, the current is carried by electrons alone, while inside the semiconductor it is carried by both kinds of carrier.

Figure 2.3 Conduction of current in an intrinsic semiconductor.

2.2 Extrinsic semiconductors

Intrinsic semiconductors have a very low conductivity at room temperature and are therefore of little direct use in electronic devices. To increase their ability to conduct, a small, controlled amount of impurity is added to the crystal. This process is called doping, and doped semiconductors are called extrinsic semiconductors. The amount of impurity is extremely small, typically one or two impurity atoms for every \(10^6\) host atoms.

(a) n-type semiconductors

An n-type semiconductor is formed when a small amount of a pentavalent impurity such as arsenic, antimony or phosphorus is added to pure germanium or silicon. A silicon atom has four valence electrons, while antimony has five. As Figure 2.4(a) shows, each antimony atom forms covalent bonds with its four silicon neighbours: four of its valence electrons are used in these bonds, and the fifth is left over, only loosely bound to the antimony atom.

Figure 2.4 n-type semiconductor: (a) formation of covalent bonds, and (b) the charge carriers (mobile free electrons and immobile donor ions).

This loosely bound electron is very easily excited into the conduction band, either thermally or by an applied field. Every antimony atom therefore contributes one conduction electron without creating a hole. Such pentavalent impurities are called donor impurities, because each donates one electron for conduction. Having given up an electron, the donor atom becomes a positively charged ion; but this ion cannot take part in conduction, because it is fixed firmly in the crystal lattice, as indicated in Figure 2.4(b).

The addition of a pentavalent impurity thus increases the number of electrons in the conduction band and hence the conductivity of the material. As a result of doping, the number of free electrons in an n-type semiconductor far exceeds the number of holes. Electrons are therefore called the majority carriers and holes the minority carriers.

(b) p-type semiconductors

A p-type semiconductor is obtained by adding a small amount of a trivalent impurity such as aluminium or boron to the pure semiconductor. A silicon atom has four valence electrons and boron has three, as shown in Figure 2.5(a). The three valence electrons of boron form covalent bonds with three of the four surrounding silicon atoms, leaving one bond incomplete, and this incomplete bond is a hole. A trivalent impurity added to intrinsic silicon thus introduces a large number of holes into the valence band, and these positively charged holes increase the conductivity of the material.

Figure 2.5 p-type semiconductor: (a) formation of covalent bonds, and (b) the charge carriers (mobile holes and immobile acceptor ions).

Trivalent impurities such as boron are called acceptor impurities, because each accepts an electron from the valence band to complete its bonds. In doing so, each boron atom provides a hole for conduction and itself becomes a fixed negative ion. Since the number of holes in p-type material is very much greater than the number of free electrons, holes are the majority carriers and electrons the minority carriers.

2.3 Density of states and the Fermi function

(a) Density of states

To find the actual number of electrons at a given energy, we need two pieces of information: the number of quantum states available at that energy, and the probability that each of those states is occupied. Let \(g(E)\,dE\) be the number of available states per unit volume in the energy range \(E\) to \(E+dE\), and let \(f(E)\) be the probability that a state of energy \(E\) is occupied by an electron. The number of electrons per unit volume in this energy range is then

\[dn(E) = f(E)\,g(E)\,dE .\]

Near the bottom of the conduction band the electrons behave as free particles of effective mass \(m_e^*\), with energy \(E-E_C = \hbar^2k^2/2m_e^*\). In a crystal of unit volume the allowed wave vectors fill \(k\)-space with a density \(1/(2\pi)^3\), and each wave vector accommodates two electrons of opposite spin. The number of states per unit volume with energy below \(E\) is therefore

\[N(E) = 2\cdot\frac{(4/3)\pi k^3}{(2\pi)^3} = \frac{k^3}{3\pi^2} = \frac{1}{3\pi^2}\left[\frac{2m_e^*(E-E_C)}{\hbar^2}\right]^{3/2},\]

and differentiating with respect to \(E\), with \(\hbar = h/2\pi\), gives the density of states in the conduction band,

\[\boxed{g_c(E) = C_n\,(E-E_C)^{1/2},\qquad C_n=\frac{4\pi\,(2m_e^*)^{3/2}}{h^3}},\qquad E\ge E_C .\]

Thus \(g_c(E)\,dE\) is the number of conduction-band states per unit volume between \(E\) and \(E+dE\), and \(g_c(E)f(E)\,dE\) is the number of conduction electrons per unit volume in that range. A similar statement holds for the valence band: \(g_v(E)\,dE\) is the number of valence-band states per unit volume in the range \(E\) to \(E+dE\) for \(E\le E_V\), with

\[\boxed{g_v(E) = C_p\,(E_V-E)^{1/2},\qquad C_p=\frac{4\pi\,(2m_h^*)^{3/2}}{h^3}},\qquad E\le E_V ,\]

where \(m_h^*\) is the hole effective mass. Both functions are sketched in Figure 2.6; each vanishes at its band edge and grows as the square root of the distance into the band.

Figure 2.6 General energy dependence of \(g_c(E)\) and \(g_v(E)\) near the band edges; \(g_c(E)\) and \(g_v(E)\) are the densities of states in the conduction and valence bands respectively.

Integrating the equilibrium distribution of electrons over the conduction band gives the equilibrium electron concentration, and integrating the distribution of empty states over the valence band gives the hole concentration:

\[n = \int_{E_C}^{E_{\text{top}}} g_c(E)\,f(E)\,dE,\qquad p = \int_{E_{\text{bottom}}}^{E_V} g_v(E)\,\bigl[1-f(E)\bigr]\,dE .\]

(b) The Fermi function

The density of states tells us how many states exist at a given energy \(E\); the Fermi function \(f(E)\) tells us what fraction of those states is filled with electrons. Under equilibrium conditions \(f(E)\) is the probability that an available state at energy \(E\) is occupied, and it is given by the Fermi–Dirac distribution

\[\boxed{f(E) = \frac{1}{1+e^{(E-E_F)/k_BT}}},\]

where \(E_F\) is the Fermi energy or Fermi level, \(k_B = 8.62\times10^{-5}\,\text{eV/K}\) is Boltzmann’s constant and \(T\) is the absolute temperature.

Consider first the limit \(T\to0\) K. For every energy \(E<E_F\) the exponent \((E-E_F)/k_BT\to-\infty\), so that \(f(E)\to 1/(1+e^{-\infty})=1\); for every energy \(E>E_F\) the exponent tends to \(+\infty\) and \(f(E)\to 1/(1+e^{+\infty})=0\). At absolute zero, therefore, all states below \(E_F\) are filled and all states above \(E_F\) are empty, and \(f(E)\) is the step function of Figure 2.7(a).

For \(T>0\) K three observations summarise the behaviour of the function. First, at \(E=E_F\) we have \(f(E_F)=\tfrac12\) exactly, whatever the temperature. Secondly, if \(E\ge E_F+3k_BT\), then \(e^{(E-E_F)/k_BT}\gg1\) and

\[f(E)\simeq e^{-(E-E_F)/k_BT},\]

so that above \(E_F+3k_BT\) the occupation probability decays exponentially to zero with increasing energy, and most states lying \(3k_BT\) or more above \(E_F\) are empty. Thirdly, if \(E\le E_F-3k_BT\), then \(e^{(E-E_F)/k_BT}\ll1\) and

\[f(E)\simeq 1-e^{(E-E_F)/k_BT},\]

so that below \(E_F-3k_BT\) the probability \(1-f(E)\) that a state is empty decays exponentially with decreasing energy, and most states lying \(3k_BT\) or more below \(E_F\) are filled. These properties are summarised in the \(T>0\) K plot of Figure 2.7(b): the step is smeared out over a few \(k_BT\) on either side of \(E_F\), symmetrically about the point \(f=\tfrac12\).

Figure 2.7 Energy dependence of the Fermi function: (a) \(T\to0\) K; (b) \(T>0\) K, with the energy coordinate marked in units of \(k_BT\).

The Fermi function is a statistical function associated with electrons in general, and it applies with equal validity to insulators, semiconductors and metals. It applies, however, only under conditions of thermal equilibrium.

2.4 Equilibrium carrier concentrations

(a) Electrons in the conduction band

The number of conduction electrons per unit volume with energies between \(E\) and \(E+dE\) is \(dn = g_c(E)f(E)\,dE\), so that the equilibrium electron concentration is

\[n_0 = \int_{E_C}^{\infty} g_c(E)\,f(E)\,dE ,\]

where the upper limit may be taken as infinity because \(f(E)\) falls off so rapidly that states high in the band contribute nothing. In a non-degenerate semiconductor the Fermi level lies in the gap at least a few \(k_BT\) below \(E_C\), so that \(E-E_F\gg k_BT\) for every \(E\ge E_C\) and the Fermi function reduces to the Boltzmann factor \(f(E)\simeq e^{-(E-E_F)/k_BT}\). Hence

\[n_0 = C_n\int_{E_C}^{\infty}(E-E_C)^{1/2}\,e^{-(E-E_F)/k_BT}\,dE .\]

We write \(E-E_F=(E-E_C)+(E_C-E_F)\) and substitute \(x=(E-E_C)/k_BT\), so that \(dE = k_BT\,dx\):

\[n_0 = C_n\,(k_BT)^{3/2}\,e^{-(E_C-E_F)/k_BT}\int_0^\infty x^{1/2}e^{-x}\,dx = C_n\,(k_BT)^{3/2}\,\frac{\sqrt\pi}{2}\,e^{-(E_C-E_F)/k_BT},\]

since \(\int_0^\infty x^{1/2}e^{-x}dx=\Gamma(3/2)=\sqrt\pi/2\). Inserting \(C_n = 4\pi(2m_e^*)^{3/2}/h^3\), the prefactor becomes

\[\frac{4\pi(2m_e^*)^{3/2}}{h^3}\,(k_BT)^{3/2}\,\frac{\sqrt\pi}{2} = \frac{2\,\pi^{3/2}(2m_e^*k_BT)^{3/2}}{h^3} = 2\left(\frac{2\pi m_e^*k_BT}{h^2}\right)^{3/2}.\]

We therefore obtain

\[\boxed{n_0 = N_C\,e^{-(E_C-E_F)/k_BT}},\qquad \boxed{N_C = 2\left(\frac{2\pi m_e^*k_BT}{h^2}\right)^{3/2}}.\]

The quantity \(N_C\), which varies as \(T^{3/2}\), is called the effective density of states at the conduction-band edge. The result shows that the conduction band behaves as if all its states, \(N_C\) per unit volume, were concentrated at the single energy \(E_C\) and occupied with the Boltzmann probability \(e^{-(E_C-E_F)/k_BT}\).

(b) Holes in the valence band

The probability that a state is empty, that is, occupied by a hole, is

\[1-f(E) = \frac{e^{(E-E_F)/k_BT}}{1+e^{(E-E_F)/k_BT}} \simeq e^{-(E_F-E)/k_BT},\]

the approximation holding when \(E_F-E\gg k_BT\) for all \(E\le E_V\). With \(g_v(E)=C_p(E_V-E)^{1/2}\) the hole concentration is

\[p_0 = C_p\int_{-\infty}^{E_V}(E_V-E)^{1/2}\,e^{-(E_F-E)/k_BT}\,dE .\]

Writing \(E_F-E = (E_F-E_V)+(E_V-E)\) and putting \(x=(E_V-E)/k_BT\), the integral is the same as before, and

\[\boxed{p_0 = N_V\,e^{-(E_F-E_V)/k_BT}},\qquad \boxed{N_V = 2\left(\frac{2\pi m_h^*k_BT}{h^2}\right)^{3/2}},\]

where \(N_V\) is the effective density of states at the valence-band edge.

(c) The Fermi level in an intrinsic semiconductor

An intrinsic crystal must be electrically neutral, and since its carriers are created in pairs, \(n_0=p_0=n_i\). Denoting the intrinsic Fermi level by \(E_{Fi}\), we have

\[N_C\,e^{-(E_C-E_{Fi})/k_BT} = N_V\,e^{-(E_{Fi}-E_V)/k_BT}.\]

Taking logarithms,

\[\ln\frac{N_C}{N_V} = \frac{(E_C-E_{Fi})-(E_{Fi}-E_V)}{k_BT} = \frac{E_C+E_V-2E_{Fi}}{k_BT},\]

so that

\[E_{Fi} = \frac{E_C+E_V}{2} - \frac{k_BT}{2}\ln\frac{N_C}{N_V}.\]

Since \(N_C/N_V = (m_e^*/m_h^*)^{3/2}\), this may also be written

\[\boxed{E_{Fi} = \frac{E_C+E_V}{2} + \frac34\,k_BT\ln\frac{m_h^*}{m_e^*}}.\]

If the effective masses of electrons and holes were equal, \(N_C=N_V\) and \(E_{Fi}=(E_C+E_V)/2\): the intrinsic Fermi level would lie exactly at the centre of the forbidden gap. In practice the correction term is only a small fraction of \(E_g\) at room temperature, so the intrinsic Fermi level lies very close to mid-gap, slightly above it when \(m_h^*>m_e^*\).

(d) The intrinsic carrier concentration

Multiplying the expressions for \(n_0\) and \(p_0\), the Fermi level cancels:

\[n_0p_0 = \left(N_C\,e^{-(E_C-E_F)/k_BT}\right)\left(N_V\,e^{-(E_F-E_V)/k_BT}\right) = N_CN_V\,e^{-(E_C-E_V)/k_BT} = N_CN_V\,e^{-E_g/k_BT}.\]

The product \(n_0p_0\) is therefore a constant for a given material at a given temperature, however the material is doped. In the intrinsic case \(n_0=p_0=n_i\), so that

\[n_i^2 = N_CN_V\,e^{-E_g/k_BT},\qquad \boxed{n_i = \sqrt{N_CN_V}\;e^{-E_g/2k_BT}}.\]

Because \(N_CN_V\propto T^3\), the temperature dependence is

\[\boxed{n_i = B\,T^{3/2}\,e^{-E_g/2k_BT}},\]

where \(B\) is a constant characteristic of the material, \(E_g\) is the band gap in eV and \(k_B = 8.62\times10^{-5}\) eV/K. Equally, from the expression for \(n_0\) applied to intrinsic material,

\[n_i = p_i = N_C\,e^{-(E_C-E_{Fi})/k_BT}.\]

(e) Carrier concentrations referred to \(n_i\); the Fermi level in extrinsic material

Dividing \(n_0 = N_C e^{-(E_C-E_F)/k_BT}\) by \(n_i = N_C e^{-(E_C-E_{Fi})/k_BT}\), and similarly for holes, gives a more convenient pair of expressions:

\[\boxed{n_0 = n_i\,e^{(E_F-E_{Fi})/k_BT},\qquad p_0 = n_i\,e^{(E_{Fi}-E_F)/k_BT}}.\]

These give the position of the Fermi level in an extrinsic semiconductor at once. For n-type material

\[\boxed{E_F = E_{Fi} + k_BT\ln\frac{n}{n_i}},\]

and for p-type material

\[\boxed{E_F = E_{Fi} - k_BT\ln\frac{p}{n_i}}.\]

Donor doping raises the Fermi level above \(E_{Fi}\) towards \(E_C\), and acceptor doping lowers it towards \(E_V\), by an amount that grows only logarithmically with the doping.

Exam tip. Measured from the conduction-band edge, \(E_C-E_F = k_BT\ln(N_C/N_D)\) in n-type material. Increasing \(N_D\) by a factor \(a\) therefore moves \(E_F\) up by \(k_BT\ln a\), independently of the starting doping.

Example 2.1. The intrinsic carrier concentration of silicon (\(E_g = 1.12\) eV) at 300 K is \(1.5\times10^{10}\,\text{cm}^{-3}\). Calculate \(n_i\) for silicon at 400 K and at 500 K. (Take \(k_B = 8.62\times10^{-5}\) eV/K.)

Solution. Since \(n_i\propto T^{3/2}\exp(-E_g/2k_BT)\), the ratio of the concentrations at two temperatures is

\[\frac{n_i(T)}{n_i(300\,\text{K})} = \left(\frac{T}{300}\right)^{3/2}\exp\left[-\frac{E_g}{2k_B}\left(\frac1T-\frac1{300}\right)\right].\]

At 400 K,

\[\frac{n_i(400\,\text{K})}{n_i(300\,\text{K})} = \left(\frac{400}{300}\right)^{3/2}\exp\left[-\frac{1.12}{2\times8.62\times10^{-5}}\left(\frac1{400}-\frac1{300}\right)\right] = 1.540\times224.5 = 345.6,\]

so that

\[n_i(400\,\text{K}) = 1.5\times10^{10}\times345.6 = 5.18\times10^{12}\,\text{cm}^{-3}.\]

Similarly, at 500 K,

\[\begin{aligned}n_i(500\,\text{K}) &= 1.5\times10^{10}\times\left(\frac{500}{300}\right)^{3/2}\exp\left[-\frac{1.12}{2\times8.62\times10^{-5}}\left(\frac1{500}-\frac1{300}\right)\right]\\ &= 1.5\times10^{10}\times2.152\times5779 = 1.87\times10^{14}\,\text{cm}^{-3}.\end{aligned}\]

A rise of 200 K thus increases \(n_i\) by four orders of magnitude, almost entirely through the exponential factor.

2.5 The mass-action law and charge neutrality

(a) The mass-action law

If a pure semiconductor is doped with donors, the number of electrons in the conduction band rises above its intrinsic value and the number of holes in the valence band falls below it. Similarly, doping with acceptors raises the hole concentration and lowers the electron concentration below their intrinsic values. The minority concentration falls because the rate of recombination increases in the presence of a large number of majority carriers. The result of Section 2.4(d) makes this quantitative: under thermal equilibrium the product of the electron and hole concentrations in any non-degenerate semiconductor is a constant, independent of the amount of donor and acceptor doping. This relation, which is confirmed by experiment, is known as the mass-action law,

\[\boxed{n_0p_0 = n_i^2},\]

where \(n_0\) is the number of free electrons per unit volume, \(p_0\) the number of holes per unit volume and \(n_i\) the intrinsic concentration. In calculating the conductivity of a doped semiconductor it is usually sufficient to consider only the majority carriers.

Exam tip. The mass-action law holds only in thermal equilibrium and for non-degenerate material; it fails under carrier injection (a forward-biased junction) or illumination.

(b) Charge neutrality

The mass-action law relates the free-electron and hole concentrations; a second relation follows from the electrical neutrality of the material. When the majority concentration is increased above its intrinsic value by doping, the minority concentration decreases so as to keep the product \(n_i^2\) constant, and the amount of the decrease can be found by combining the mass-action law with the requirement of overall charge neutrality. With all donors and acceptors ionised, the positive charges (holes and ionised donors) must balance the negative charges (electrons and ionised acceptors):

\[\boxed{p_0 + N_D = n_0 + N_A}.\]

Using \(n_0 = n_i^2/p_0\), we have

\[p_0 = N_A - N_D + n_0 = N_A - N_D + \frac{n_i^2}{p_0},\]

\[p_0^2 - (N_A-N_D)\,p_0 - n_i^2 = 0 .\]

This is a quadratic equation of the form \(ax^2+bx+c=0\), and its physically acceptable (positive) root is

\[\boxed{p_0 = \frac{N_A-N_D}{2} + \left[\frac{(N_A-N_D)^2}{4} + n_i^2\right]^{1/2}}.\]

In the same way, eliminating \(p_0\) instead,

\[\boxed{n_0 = \frac{N_D-N_A}{2} + \left[\frac{(N_D-N_A)^2}{4} + n_i^2\right]^{1/2}}.\]

For an intrinsic semiconductor, \(N_A=N_D=0\), and both expressions give \(n_0=p_0=n_i\), as they should.

(c) n-type semiconductors

In an n-type semiconductor \(N_D\gg n_i\) and \(N_A\simeq0\). The hole density is then

\[p_n = -\frac{N_D}{2} + \frac{N_D}{2}\sqrt{1+\frac{4n_i^2}{N_D^2}} .\]

The second term under the radical is much less than unity, so we may expand the square root as \(\sqrt{1+u}\simeq1+u/2\) and neglect higher powers:

\[p_n \simeq -\frac{N_D}{2} + \frac{N_D}{2}\left(1+\frac{2n_i^2}{N_D^2}\right) = \frac{n_i^2}{N_D}.\]

Similarly, the electron density in n-type material is

\[n_n \simeq \frac{N_D}{2} + \frac{N_D}{2}\left(1+\frac{2n_i^2}{N_D^2}\right) = N_D + \frac{n_i^2}{N_D} \simeq N_D .\]

(d) p-type semiconductors

In a p-type semiconductor \(N_A\gg n_i\) and \(N_D\simeq0\), and the same expansion gives

\[n_p \simeq \frac{n_i^2}{N_A},\qquad p_p \simeq N_A + \frac{n_i^2}{N_A}\simeq N_A .\]

The density of majority carriers is thus very nearly equal to the density of impurity atoms at ordinary ambient temperatures, and the minority density is fixed by the mass-action law. We summarise:

\[\boxed{n\text{-type: } n\simeq N_D,\; p\simeq \frac{n_i^2}{N_D};\qquad p\text{-type: } p\simeq N_A,\; n\simeq\frac{n_i^2}{N_A}}.\]

Example 2.2. Determine the resistivity of germanium (a) in the intrinsic condition at 300 K, (b) with a donor impurity of 1 in \(10^7\), (c) with an acceptor impurity of 1 in \(10^8\), and (d) with both these impurities present simultaneously. For germanium at room temperature \(n_i = 2.5\times10^{13}\,\text{cm}^{-3}\), \(\mu_n = 3800\,\text{cm}^2/\text{V\,s}\), \(\mu_p = 1800\,\text{cm}^2/\text{V\,s}\), and the number of germanium atoms per \(\text{cm}^3\) is \(4.4\times10^{22}\).

Solution. We use the conductivity \(\sigma = e(n\mu_n+p\mu_p)\), which is derived in Section 2.6.

(a) In the intrinsic condition \(n=p=n_i = 2.5\times10^{13}\,\text{cm}^{-3}\), so

\[\sigma = en_i(\mu_n+\mu_p) = (1.6\times10^{-19})(2.5\times10^{13})(3800+1800) = 0.0224\,\text{S/cm},\]

and the resistivity is

\[\rho = \frac1\sigma = \frac1{0.0224} = 44.6\,\Omega\,\text{cm}.\]

(b) One donor atom in \(10^7\) germanium atoms gives

\[N_D = \frac{4.4\times10^{22}}{10^7} = 4.4\times10^{15}\,\text{cm}^{-3},\]

and \(n\simeq N_D\). The hole density is

\[p = \frac{n_i^2}{N_D} = \frac{(2.5\times10^{13})^2}{4.4\times10^{15}} = 1.42\times10^{11}\,\text{cm}^{-3}.\]

Since \(n\gg p\), the hole contribution can be neglected, and

\[\sigma \simeq N_De\mu_n = (4.4\times10^{15})(1.6\times10^{-19})(3800) = 2.675\,\text{S/cm},\qquad \rho = \frac1{2.675} = 0.374\,\Omega\,\text{cm}.\]

\[N_A = \frac{4.4\times10^{22}}{10^8} = 4.4\times10^{14}\,\text{cm}^{-3},\]

and \(p\simeq N_A\). Then

\[n = \frac{n_i^2}{N_A} = \frac{(2.5\times10^{13})^2}{4.4\times10^{14}} = 1.42\times10^{12}\,\text{cm}^{-3}.\]

Since \(p\gg n\), \(n\) may be neglected, and

\[\sigma \simeq N_Ae\mu_p = (4.4\times10^{14})(1.6\times10^{-19})(1800) = 0.1267\,\text{S/cm},\qquad \rho = \frac1{0.1267} = 7.89\,\Omega\,\text{cm}.\]

(d) With both impurities present, \(N_D = 4.4\times10^{15}\,\text{cm}^{-3}\) and \(N_A = 4.4\times10^{14}\,\text{cm}^{-3}\). The acceptors compensate part of the donors, and the net donor density is

\[N_D' = N_D - N_A = 4.4\times10^{15} - 4.4\times10^{14} = 3.96\times10^{15}\,\text{cm}^{-3},\]

so that the effective electron density is \(n = N_D' = 3.96\times10^{15}\,\text{cm}^{-3}\). The hole density is

\[p = \frac{n_i^2}{N_D'} = \frac{(2.5\times10^{13})^2}{3.96\times10^{15}} = 1.58\times10^{11}\,\text{cm}^{-3},\]

which is again negligible compared with \(N_D'\). Therefore

\[\sigma \simeq N_D'e\mu_n = (3.96\times10^{15})(1.6\times10^{-19})(3800) = 2.408\,\text{S/cm},\qquad \rho = \frac1{2.408} = 0.415\,\Omega\,\text{cm}.\]

2.6 Drift and diffusion currents

Two basic processes cause electrons and holes to move in a semiconductor: drift, which is motion caused by an electric field, and diffusion, which is the flow caused by variations in concentration, that is, by concentration gradients. Such gradients may be produced by a non-uniform doping distribution or by the injection of electrons or holes into a region.

(a) Drift current density and conductivity

To understand drift, suppose that an electric field is applied to a semiconductor. The field exerts a force on the free electrons and holes, which then acquire a net drift velocity superimposed on their random thermal motion. Consider first an n-type semiconductor with a large number of free electrons, as in Figure 2.8(a). An electric field \(E\) applied in one direction produces a force on the electrons in the opposite direction, because of their negative charge, and they acquire a drift velocity

\[v_{dn} = -\mu_nE ,\]

where \(\mu_n\) is the electron mobility, measured in \(\text{cm}^2/\text{V\,s}\). For lightly doped silicon \(\mu_n\) is typically \(1350\,\text{cm}^2/\text{V\,s}\). The mobility measures how freely an electron can move through the crystal, and the negative sign shows that the electron drift velocity is opposite to the applied field. The electron drift produces a drift current density, in \(\text{A/cm}^2\),

\[J_n = -env_{dn} = -en(-\mu_nE) = +en\mu_nE ,\]

where \(n\) is the electron concentration and \(e\) is the magnitude of the electronic charge. The conventional current is opposite to the flow of negative charge, so the drift current in an n-type semiconductor is in the same direction as the applied field.

Figure 2.8 Directions of the applied electric field and of the resulting carrier drift velocity and drift current density in (a) an n-type semiconductor and (b) a p-type semiconductor.

Next consider a p-type semiconductor with a large number of holes, as in Figure 2.8(b). The field now exerts a force on the holes in its own direction, because of their positive charge, and they acquire a drift velocity

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Chapter 2 Semiconductor Physics