Transport Properties of Semiconductors

SVNY85-Sheng S. Li October 2, 25 15:4 7 Transport Properties of Semiconductors 7.1. Introduction In this chapter the carrier transport phenomena in a semiconductor under the influence of applied external fields are presented. Different galvanomagnetic, thermoelectric, and thermomagnetic effects created by the applied electric and magnetic fields as well as the temperature gradient in a semiconductor are discussed in this chapter. The transport coefficients associated with the galvanomagnetic, thermoelectric, and thermomagnetic effects in a semiconductor are derived from the Boltzmann transport equation using the relaxation time approximation. In the event that the relaxation time approximation fails, the solutions for the Boltzmann transport equation could be obtained by using the variational principles. The effect of an applied electric field, magnetic field, or a temperature gradient on the electrons in a semiconductor is to change the distribution function of electrons from its equilibrium condition. As discussed in Chapter 5, in the absence of the external fields, the distribution of electrons in a semiconductor or a metal under equilibrium conditions may be described by the Fermi Dirac distribution function, which is given by 1 f (E) = 1 + e. (7.1) (E E f)/k B T Equation (7.1) shows that in thermal equilibrium the electron distribution function f (E) depends not only on the electron energy but also on the Fermi energy E f, a many-body parameter, and the temperature T. However, under the influence of external fields, f (E) given in Eq. (7.1) may change from its equilibrium distribution function in a semiconductor. This can be best explained by considering the case in which an electric field or a magnetic field is applied to the semiconductor specimen. When an electric field or a magnetic field is applied to the semiconductor, the Lorentz force will tend to change the wave vector of electrons (i.e., F = q( ε + v B) = h d k/dt) along the direction of the applied fields. As a result, the distribution function is modified by the changing wave vector of electrons under the influence of Lorentz force. Furthermore, since f depends on both the energy and temperature as well as the electron concentration, one expects that 171

SVNY85-Sheng S. Li October 2, 25 15:4 172 7. Transport Properties of Semiconductors Figure 7.1. The effect of an applied electric field and a temperature gradient on the electron distribution function in a semiconductor. the nonequilibrium distribution function of electrons will also be a function of the position in space when a temperature gradient or a concentration gradient is presented across the semiconductor specimen. To illustrate the effect of external forces on the electron distribution function, Figures 7.1a and 7.1b show the two-dimensional (2-D) electron distribution functions in the presence of an applied electric field and a temperature gradient, respectively. As shown in Figure 7.1a, when an electric field is applied along the x-direction, the change of electron wave vector in the x-direction is given by k x = qε x τ/ h, where τ is the mean relaxation time of electrons, and ε x is the applied electric field in the x-direction. In this case, the electron distribution function as a whole moves to the right by k x from its equilibrium position. It is noted that the shape of the nonequilibrium distribution function remains unchanged from its equilibrium condition. The fact that the shape of electron distribution function in k-space does not change because of the electric field can be explained by the force acting on each quantum state k. Since the Lorentz force (F x = qε x ) due to the electric field is equal to h k x, the rate of change of k x is the same for all electrons. Consequently, if there is no relaxation mechanism to restore the distribution function to equilibrium, an applied electric field can only cause the distribution function to drift, unaltered in shape, along the k x -direction at a constant velocity (v x = h k x /m ) and the change in crystal momentum as a result of this drift is given by k x = qε x τ/ h.onthe other hand, the change in electron distribution function

SVNY85-Sheng S. Li October 2, 25 15:4 7.2. Galvanomagnetic, Thermoelectric, and Thermomagnetic Effects 173 is quite different when a temperature gradient is applied to a semiconductor. In this case, the nonequilibrium distribution function is shifted to the right by an amount equal to k x for those electrons with energies greater than E f, and shifted to the left by the same amount k x for those electrons with energies less than E f,as is illustrated in Figure 7.1b. The physical mechanisms causing this shift can be explained by using a Taylor series expansion of (E E f ) about the Fermi energy (E f = h 2 k 2 f /2m ). Assuming that E E f E f, one can replace (E E f )by ( h 2 k f /m )(k k f ) and obtain k x = τ hk f m T (k k f ) T x. (7.2) This result shows that the distribution of quantum states at the Fermi surface (i.e., at E = E f )isnot affected by the temperature gradient (i.e., k x = ). From Eq. (7.2) it is noted that for those quantum states with energies greater than E f (i.e., E > E f ) their centers shift in the same direction as the temperature gradient ( k x is positive), whereas for those quantum states with energies smaller than E f (i.e., E < E f ), their centers move in the opposite direction to the temperature gradient ( k x is negative). Section 7.2 describes various galvanomagnetic, thermoelectric, and thermomagnetic effects in a semiconductor. These include the electrical conductivity, Hall effect, Seebeck and Pelter effects, Nernst and Ettinghousen effects, and the magnetoresistance effect. In Section 7.3, the Boltzmann transport equation for the steady-state case is derived. Expressions for the electrical conductivity, electron mobility, Hall coefficient, magnetoresistance, and Nernst and Seebeck coefficients for n-type semiconductors are derived in Section 7.4. Transport coefficients for the mixed conduction case are depicted in Section 7.5. Section 7.6 presents some experimental results on the transport coefficients for germanium, silicon, and III-V compound semiconductors. 7.2. Galvanomagnetic, Thermoelectric, and Thermomagnetic Effects In this section, the galvanomagnetic, thermoelectric, and thermomagnetic effects in a semiconductor are discussed. These effects are created by the transport of electrons (for n-type) or holes (for p-type) in a semiconductor when an external electric field, a magnetic field, or a temperature gradient is applied separately or simultaneously to a semiconductor specimen. The transport coefficients to be described here include the electrical conductivity, thermal conductivity, Hall coefficient, Seebeck coefficient, Nernst coefficient, and the magnetoresistance of an n-type semiconductor. Transport coefficients derived for an n-type semiconductor can also be applied to a p-type semiconductor provided that the positive charge and positive effective mass of holes are used instead. It is noted that for nondegenerate semiconductors, Maxwell Boltzmann (M-B) statistics is used in the derivation of

SVNY85-Sheng S. Li October 2, 25 15:4 174 7. Transport Properties of Semiconductors transport coefficients, while Fermi Dirac (F-D) statistics is used in the derivation of transport coefficients for degenerate semiconductors and metals. 7.2.1. Electrical Conductivity In this section the current conduction due to electrons in an n-type semiconductor is depicted. When a small electric field is applied to the specimen, the electrical current density can be related to the electric field by using Ohm s law, which reads J n = σ n ε = qµ n nε, (7.3) where σ n = qµ n n is the electrical conductivity, µ n is the electron mobility, and n denotes the electron density. The electrical current density can also be expressed in terms of the electron density and electron drift velocity v d along the direction of the applied electric field by J n = qnv d, (7.4) where q is the electronic charge. Comparing Eqs. (7.3) and (7.4) one finds that the electron drift velocity is related to the electric field by v d = µ n ε, (7.5) where µ n is the low-field electron drift mobility, which is defined as the electron drift velocity per unit electric field strength. For metals, µ n can be expressed in terms of the mean collision time τ and the electron effective mass m, µ n = qτ m. (7.6) From Eqs. (7.3) and (7.6), the electrical conductivity σ n for a metal can be expressed in terms of the mean collision time, and the electron effective mass as σ n = q2 nτ. (7.7) m In the collision processes, the transition probability for electron collision is directly related to the density of collision centers, and the collision rate is inversely proportional to the collision time constant. For example, in the case of electron phonon scattering, the number of scattering centers is equal to the phonon population in thermal equilibrium. At high temperatures the average phonon density is proportional to temperature. Consequently, at high temperatures, the collision time τ varies as 1/T, and hence the electrical conductivity σ n varies inversely with temperature T. This prediction is consistent with the observed temperature dependence of electrical conductivity in a metal. Equations (7.6) and (7.7) can also be applied to n-type semiconductors provided that the free-electron mass is replaced by the conductivity effective mass of electrons in the conduction bands m n, and τ is replaced by the average relaxation time τ. Ingeneral, the electron density in a semiconductor is a strong function

SVNY85-Sheng S. Li October 2, 25 15:4 7.2. Galvanomagnetic, Thermoelectric, and Thermomagnetic Effects 175 of temperature, and the relaxation time may depend on both the energy and temperature. A general expression for the current density in an n-type semiconductor can be derived as follows. From Eq. (5.3), the density of quantum states g n (E) for a single-valley semiconductor with a parabolic conduction band can be written as ( ) 4π g n (E) = (2m n )3/2 (E E c ) 1/2. (7.8) h 3 Using Eqs. (7.4) and (7.8), a general expression for the electron current density can be expressed by J n = qnv x = q v x f (E)g n (E)dE, (7.9) where f (E) is the nonequilibrium electron distribution function, which can be obtained by solving the Boltzmann transport equation to be depicted in Section 7.4. The integration of Eq. (7.9) is carried out over the entire conduction band. The minus sign in Eq. (7.9) stands for electron conduction in an n-type semiconductor. For hole conduction in a p-type semiconductor, the plus sign should be used instead. Figure 7.2a shows the applied electric field and the current flow in an n-type Figure 7.2. Longitudinal transport effects in the presence of an electric field or a temperature gradient: (a) electrical conductivity; (b) electronic thermal conductivity; (c) Seebeck effect, S ab = V dc /(T 2 T 1 ); (d) Peltier effect, ab = S ab T ; (e) Thomson effect.

SVNY85-Sheng S. Li October 2, 25 15:4 176 7. Transport Properties of Semiconductors semiconductor specimen. The electrical conductivity for an n-type semiconductor is derived in Section 7.4. 7.2.2. Electronic Thermal Conductivity The electronic thermal conductivity is due to the flow of thermal energy carried by electrons when a temperature gradient is applied across a semiconductor specimen. As shown in Figure 7.2b, when a temperature gradient is created in a semiconductor specimen, a heat flux flow will appear across the specimen. The electronic thermal conductivity K n is defined as the thermal flux density per unit temperature gradient, and can be expressed by Q x K n = ( T/ x), (7.1) Jx = where Q x is the thermal flux density given by Q x = nv x E = v x Ef(E)g n (E)dE. (7.11) Note that the integration on the right-hand side of Eq. (7.11) is carried out over the entire conduction band. Equations (7.9) and (7.11) are the two basic equations that describe the flow of electric current density and heat flux density in an n-type semiconductor, respectively. All the transport coefficients described in this section can be derived from Eqs. (7.9) and (7.11) provided that the nonequilibrium distribution function f (k, r) is known. The steady-state nonequilibrium distribution f (k, r) can be derived by solving the Boltzmann transport equation. It is noted that in thermal equilibrium, both J n and Q x,given by Eq. (7.9) and Eq. (7.11), are equal to zero, and f (k, r) reduces to the equilibrium Fermi distribution function f (E). Figures 7.2 and 7.3 show the plots of various galvanomagnetic, thermoelectric, and thermomagnetic effects in a semiconductor in the presence of the electric field, current density, heat flux, and temperature gradient. Various galvanomagnetic, thermoelectric, and thermomagnetic effects as well as the transport coefficients associated with the applied electric field, magnetic field, and the temperature gradient in a semiconductor are discussed next. 7.2.3. Thermoelectric Coefficients When a temperature gradient, an electric field, or both are applied across a semiconductor or a metal specimen, three different kinds of thermoelectric effects could be observed. They are the Seebeck, Peltier, and Thomson effects. The thermoelectric coefficients associated with each of these effects can be defined according to Figures 7.2c e, in which two pieces of conductors (A and B) are joined at junctions x 1 and x 2.Ifatemperature difference T is established between junctions x 1 and x 2, then an open-circuit voltage V cd is developed between terminals c and d. This is known as the Seebeck effect. In this case, the differential Seebeck

SVNY85-Sheng S. Li October 2, 25 15:4 7.2. Galvanomagnetic, Thermoelectric, and Thermomagnetic Effects 177 Figure 7.3. Galvanomagnetic and thermomagnetic effects in a semiconductor under the influence of an applied electric field, a magnetic field, and a temperature gradient. The polarity shown is for hole conduction: (a) Hall effect, (b) Nernst effect, (c) Ettingshausen effect, (d) Righi-Leduc effect. coefficient, or the thermoelectric power, can be defined by S ab = V T = V dc (T 2 T 1 ). (7.12) If junctions x 1 and x 2 are initially maintained at the same temperature, then by applying a voltage across terminals c and d one observes an electrical current flow through these two conductors. If the result is a rate of heating at junction x 1, then there will be a cooling at the same rate at junction x 2. This is the well-known Peltier effect, the basic principle of thermoelectric cooling. The differential Peltier coefficient is defined by Au: Should the subscript here against V be cd instead of dc? ab = Q x. (7.13) I x The Thomson effect occurs when an electric current and a temperature gradient are applied simultaneously in the same direction on a semiconductor specimen. In this case, the simultaneous presence of the current flow I x and the temperature gradient T/ x in the x-direction will produce a rate of heating or cooling ( Q x / x) per unit length. Thus, the Thomson coefficient can be expressed by τ = ( Q x/ x) I x ( T/ x). (7.14)

SVNY85-Sheng S. Li October 2, 25 15:4 178 7. Transport Properties of Semiconductors Using the thermodynamic principles to the thermoelectric effects shown in Figures 7.2c e, Thomson derived two important equations, later known as the Kelvin relations, that relate the three thermoelectric coefficients. The Kelvin relations can be expressed by ab = S ab T, (7.15) τ a τ b = T ds ab dt, (7.16) Equations (7.15) and (7.16) not only have a sound theoretical basis, but also have been verified experimentally by the measured thermoelectric figure of merit. Equation (7.15) is particularly useful for thermoelectric refrigeration applications. This is due to the fact that the rate of cooling by means of the Peltier effect can be expressed in terms of the Seebeck coefficient, which is a much easier quantity to measure. Equation (7.16) allows one to account for the influence of the Thomson effect on the cooling power of a thermoelectric refrigerator via the variation of Seebeck coefficient with temperature. Since Thomson coefficient is defined for a single conductor, it is appropriate to introduce here the absolute Seebeck and Peltier coefficients for a single conductor. The differential Seebeck and Peltier coefficients between the two conductors are given by (S a S b ) and ( a b ), respectively, where S a and S b are the absolute Seebeck coefficients for conductors A and B, and a and b denote the absolute Peltier coefficients for conductors A and B, respectively. The Kelvin relations for a single conductor given by Eqs. (7.15) and (7.16) can be expressed as = ST, (7.17) τ s = T ds dt, (7.18) where, S, and τ s denote the absolute Peltier, Seebeck, and Thomson coefficients for a single conductor or a semiconductor, respectively. Therefore, using the definitions of thermoelectric coefficients described in this section, the general expressions of thermoelectric coefficients can be derived from the Boltzmann transport equation to be discussed in Section 7.4. 7.2.4. Galvanomagnetic and Thermomagnetic Coefficients When a magnetic field is applied to a semiconductor specimen in addition to an electric field and/or a temperature gradient, the transport phenomena become much more complicated than those without a magnetic field. Fortunately, most of the important galvanomagnetic effects in a semiconductor associated with the applied magnetic fields are focused on the cases in which a magnetic field is applied in the direction perpendicular to the electric field or the temperature gradient. These are usually referred to as the transverse galvanomagnetic effects, which include the Hall, Nernst, and the magnetoresistance effects. In this section, the transverse galvanomagnetic effects in an n-type semiconductor are described.

SVNY85-Sheng S. Li October 2, 25 15:4 7.2. Galvanomagnetic, Thermoelectric, and Thermomagnetic Effects 179 The Hall effect is the most well-known galvanomagnetic effect found in a semiconductor. As shown in Figure 7.3a, when a magnetic field is applied in the z-direction and an electric field is applied in the x-direction, an electric field (known as the Hall field) will be developed in the y-direction of the specimen. The Hall coefficient R H, under isothermal conditions, can be defined by R H = ε y, (7.19) J x B z Jy = where ε y is the Hall field induced in the y-direction, J x is the electric current density flow in the x-direction, and B z is the applied magnetic field in the z-direction. Figure 7.3a shows the schematic diagram of the Hall effect across a semiconductor specimen under isothermal conditions. Note that both the electric current density in the direction of the Hall field (i.e., the y-direction) and the temperature gradient across the specimen are assumed equal to zero. The polarity of the Hall voltage depends on the type of charge carriers (i.e., electrons or holes) in the specimen. This is due to the fact that electrons and holes in a semiconductor will experience an opposite Lorentz force when they are moving in the same direction of the specimen. Therefore, the polarity of the Hall voltage will be different for an n-type and a p-type semiconductor, and the Hall effect measurement is often used to determine the conduction types (i.e., n- or p-type) and the majority carrier concentration in a semiconductor. If a temperature gradient is applied in the x-direction and a magnetic field in the z-direction, then a transverse electric field will be developed in the y-direction of the specimen, as is illustrated in Figure 7.3b. This effect is known as the Nernst effect. The Nernst coefficient, which is thermodynamically related to the Ettingshausen coefficient in the same way as the Seebeck coefficient is related to the Peltier coefficient, is defined by ε y Q n =, (7.2) B z ( T/ x) Jx =J y = where ε y is the Nernst field developed in the y-direction when a temperature gradient is applied in the x-direction and a magnetic field is in the z-direction. Note that the electric current density along the x- and y-direction and the temperature gradient in the y-direction (i.e., ( T/ y = ) are assumed equal to zero. As is shown in Figure 7.3c, if an electric field is applied in the x-direction and a magnetic field in the z-direction, then a temperature gradient will be developed in the y-direction of the specimen. This is known as the Ettingshausen effect. It is this effect that forms the basis of thermomagnetic cooling as a counterpart to the thermoelectric cooling by the Peltier effect discussed earlier. The Ettingshausen coefficient P E is defined by P E = ( T/ y), (7.21) J x B z Jy = T/ x=

SVNY85-Sheng S. Li October 2, 25 15:4 18 7. Transport Properties of Semiconductors where T/ y is the temperature gradient developed in the y-direction of the specimen. Note that in defining the Ettingshausen effect the current density in the y- direction and the temperature gradient in the x-direction are assumed equal to zero. The Ettingshausen coefficient P E and the Nernst coefficient Q n are related by the Bridgeman equation, which is given by P E K n = Q n T, (7.22) where K n is the electronic thermal conductivity defined by Eq. (7.11). The Righi-Leduc effect refers to the creation of a transverse temperature gradient T/ y when a temperature gradient T/ x in the x-direction and a magnetic field B z in the z-direction are applied simultaneously to a semiconductor specimen. The Righi-Leduc coefficient R L can be expressed by R L = ( T/ y) ( T/ x). (7.23) Jx =J y = Note that the current densities J x and J y in the x- and y-direction are assumed equal to zero in Eq. (7.23). In order to derive the general expressions for different transport coefficients depicted above, the nonequilibrium distribution function f (E) ineqs. (7.9) and (7.11) will be solved first from the Boltzmann transport equation using relaxation time approximation, which will be discussed next. 7.3. Boltzmann Transport Equation An analytical expression for the Boltzmann transport equation can be derived for an n-type semiconductor using the relaxation time approximation. The relaxation time approximation assumes that all the collision processes are elastic and can be treated in terms of a unique relaxation time. The elastic scattering requires that change of electron energy during the scattering process must be small compared to the energy of electrons, and the relaxation time is a scalar quantity. Typical examples of elastic scattering processes include the scattering of electrons by the longitudinal acoustical phonons, ionized impurities, and the neutral impurities in a semiconductor. It is noted that the relaxation time τ may be a function of temperature and energy, depending on the types of scattering mechanisms involved, as will be discussed later in Chapter 8. The transport coefficients for an n-type semiconductor to be derived in this section include the electrical conductivity, Hall coefficient, Seebeck coefficient, Nernst coefficient, and the magnetoresistance. According to Liouville s theorem, if f (k, r, t) denotes the nonequilibrium distribution function of electrons at time t,inavolume element of d 3 r d 3 k, and located at (r, k)inther- and k-space, then f (k + k dt, r + ṙ dt, t + dt) represents the distribution function at time (t + dt) within the same volume element. The difference between f (k, r, t) and f (k + k dt, r + ṙ dt, t + dt) must be balanced out by the collision processes that occur inside a semiconductor or a metal. Therefore, the total rate of change of the distribution function with respect to time in the presence

SVNY85-Sheng S. Li October 2, 25 15:4 7.3. Boltzmann Transport Equation 181 of a Lorentz force or a temperature gradient can be written as d f = ( k k f ṙ r f ) + f dt t + f c t, (7.24) where k = dk/dt and ṙ = dr/dt = v denote the change of crystal momentum and the electron velocity, respectively. The first two terms inside the bracket on the right-hand side of Eq. (7.24) represents the external force terms due to the Lorenz force and temperature gradient, the second term is the internal collision term, which tends to offset the external force terms, and the third term is the time-dependent term that exists only for the transient case. Equation (7.24) is the generalized Boltzmann transport equation. In this section, only the steady-state case will be considered. The transport coefficients are derived when the time-independent external forces are applied to a semiconductor specimen. In this case, the third term on the right-hand side of Eq. (7.24) is set equal to zero, and the Boltzmann equation given by Eq. (7.24) becomes k k f + ṙ r f = f t. (7.25) c In general, the nonequilibrium distribution function f (k, r) can be obtained from solving Eq. (7.25), and the transport coefficients of a semiconductor or a metal can be derived once f (k, r)isfound from Eq. (7.25). In order to obtain an analytical expression for f (k, r) from Eq. (7.25), it is necessary to assume that the scattering of charge carriers in a semiconductor is elastic so that the relaxation time approximation can be applied to the Boltzmann equation. According to the classical model, electron velocity is accelerated by the applied electric field over a period of time inside the crystal, while its drift velocity drops to zero through the internal collision process. It is, however, more appropriate to consider the way in which the electron system is relaxed toward its equilibrium distribution once the external perturbation is removed. Therefore, if f (k, r) represents the distribution function of electrons under the influence of an applied electric field and f (E) isthe thermal equilibrium distribution function, then the collision term given by Eq. (7.25) can be expressed in terms of the relaxation time τ as f t = f f. (7.26) c τ Equation (7.26) is the basis of the relaxation time approximation in which the collision term on the right-hand side of Eq. (7.25) is replaced by the difference in the nonequilibrium and equilibrium distribution functions divided by the relaxation time constant τ. The relaxation time constant is usually dependent on the types of scattering mechanisms in a semiconductor. If the external forces are removed, then the nonequilibrium distribution function will decay exponentially to its equilibrium value with a time constant τ governed by the internal scattering processes.

SVNY85-Sheng S. Li October 2, 25 15:4 182 7. Transport Properties of Semiconductors For n-type semiconductors, the relaxation time τ depends on the energy of electrons according to the simple power law: τ = ae s, (7.27) where s is a constant whose value depends on the types of scattering mechanisms involved. Constant a may or may not be a function of temperature, depending on the types of scattering mechanisms. For example, in a semiconductor in which the ionized impurity scattering is dominated, s is equal to 3/2 and a is independent of temperature, while for acoustical phonon scattering, s is equal to 1/2 and a varies inversely with temperature. For neutral impurity scattering, τ is a constant and independent of the electron energy. 7.4. Derivation of Transport Coefficients for n-type Semiconductors In this section, the transport coefficients for n-type semiconductors are derived for the cases when an electric field, a magnetic field, or a temperature gradient is applied to the specimen. Transport coefficients such as electrical conductivity, Hall coefficient, Seebeck and Nernst coefficients, and magnetoresistance can be derived from Eq. (7.24) using the relaxation time approximation. Derivation of the nonequilibrium distribution function from the Boltzmann equation for an n-type semiconductor is depicted first. The Lorentz forces acting on the electrons because of the presence of an electric field and a magnetic field can be expressed by F = q( ε + v B) = h k or k = q h ( ε + v B). (7.28) Now substituting k given by Eq. (7.28) into Eq. (7.25), the first term on the left-hand side of Eq. (7.25) becomes ( ) k q k f = ( ε + v B) k f. (7.29) h The second term on the left-hand side of Eq. (7.25) is due to the presence of a temperature gradient or a concentration gradient in a semiconductor. Using the relaxation time approximation, the collision term is given by Eq. (7.26). Now, substituting Eqs. (7.29) and (7.26) into Eq. (7.25) one obtains ( ) q ( ε + v B) k f v r f = f f (7.3) h τ or ( q ) ( ε + v B) m v f v r f = f f. (7.31) n τ Equation (7.31) is the generalized steady-state Boltzmann equation, which is obtained by using the de Broglie s wave-particle duality relation ( hk = m n v)in Eq. (7.3). To obtain an analytical solution for f (k, r) from Eq. (7.31), certain

SVNY85-Sheng S. Li October 2, 25 15:4 7.4. Derivation of Transport Coefficientsfor n-type Semiconductors 183 approximations must be used. Since the equilibrium distribution function f depends only on energy and temperature, the nonequilibrium distribution function f (k, r) must contain terms that depend only on the electron velocity and energy. Therefore, it is appropriate to write a generalized trial solution for Eq. (7.31) in terms of the equilibrium distribution function and a first-order correction term, which contains both the energy and velocity components. This is given by f = f v P(E) f E, (7.32) where P(E) isanunknown vector quantity, which depends only on the electron energy. For small perturbation case [i.e., ( f f ) f ], each term in Eq. (7.31) can be approximated by [ ] (Ef E) f ν r f ν. r f = ν r T, (7.33) T E ε ν f ε ν f = ε ( ν E) f E = ε (m ν) f E, (7.34) ( ν B) ν f ν [ B P(E)] f E. (7.35) Now, substituting Eqs. (7.33), (7.34), and (7.35) into Eq. (7.31) one obtains ( ) qτ qτ ν ε + [ B P(E)] ν + τ (E f E) ν m r T ν P(E) =. n T (7.36) Equation (7.36) is a generalized steady-state Boltzmann equation in the presence of an applied electric field, a magnetic field, and a temperature gradient. Note that Eq. (7.36) can be further simplified by factoring out the velocity component, and the result yields ( ) qτ P(E) [ B P(E)] = qτ ε + τ (E f E) m r T. (7.37) n T In order to obtain a solution for the unknown vector function P(E) ineq. (7.37), it is assumed that the applied electric fields and temperature gradients are in the x y plane of the semiconductor specimen, and the magnetic field is in the z- direction. Under this assumption, the components for P(E)inEq. (7.37) in the x- and y-direction of the specimen are given respectively by P x (E) + (qτ/m n )B z P y (E) = qτε x + τ (E f E) T T x, (7.38) P y (E) (qτ/m n )B z P x (E) = qτε y + τ (E f E) T T y. (7.39) Solving Eqs. (7.38) and (7.39) for P x (E) and P y (E) one obtains (β δγ) P x (E) = (1 + δ 2 ), (7.4) P y (E) = (γ δβ) (1 + δ 2 ), (7.41)

SVNY85-Sheng S. Li October 2, 25 15:4 184 7. Transport Properties of Semiconductors where δ = qτ B z = ωτ, (7.42) m n [ β = τ qε x + (E ] f E) T, (7.43) T x [ γ = τ qε y + (E ] f E) T. (7.44) T y Thus, the expressions of transport coefficients for an n-type semiconductor depicted in Section 7.2 can be derived using Eqs. (7.31) through (7.44). This is discussed next. 7.4.1. Electrical Conductivity Consider the cases in which the applied electric field and the current flow are in the x- and y-direction of the specimen. By substituting Eq. (7.32) for f (E) into Eq. (7.9), the electric current density components due to electron conduction along the x- and y-direction are given respectively by J x = nqv x = = q J y = q qv x f (E)g(E) de (7.45) v 2 x P x(e)g(e) f E de, v 2 y P y(e)g(e) f de, (7.46) E where P x (E) and P y (E) are obtained from Eqs. (7.4) through (7.44) by setting δ, dt/dx, and dt/dy equal to zero, and the results yield P x (E) = qτε x, (7.47) P y (E) = qτε y. (7.48) From Eqs. (7.45) through (7.48), it is noted that J x and J y vanish if P(E)isequal to zero. To derive the electrical conductivity, it is assumed that the electron velocity is isotropic within the specimen, and hence the square of the velocity components along the x-, y-, and z-direction can be expressed in terms of the kinetic energy of electrons by vx 2 = v2 y = v2 z = 2E, (7.49) 3m n where E is the total kinetic energy of electrons. Equation (7.49) is obtained by using the fact that the electron kinetic energy is equal to (1/2)m n v2, where v 2 = vx 2 + v2 y + v2 z and it is assumed that v x = v y = v z.now, substituting Eqs. (7.47) and (7.49) into Eq. (7.45) one obtains σ n = J x ε x = ( 2q 2 ) 3m n τ Eg(E) f E de. (7.5)

SVNY85-Sheng S. Li October 2, 25 15:4 7.4. Derivation of Transport Coefficientsfor n-type Semiconductors 185 For a nondegenerate semiconductor, the F-D distribution function given by Eq. (7.1) is reduced to the classical M-B distribution function, which reads f exp[(e f E)/k B T ] (7.51) and f E = f k B T. (7.52) Now, substituting Eq. (7.52) into Eq. (7.5), the electrical conductivity can be expressed as ( 2q 2 ) Q n = τ Eg(E) f de 3m n k BT ( 2nq 2 = 3m n k BT ) τ E 3/2 f de E 3/2 f de (7.53) = nq2 τ, m n where τ E 3/2 e ( E/k BT ) de τ = E 3/2 e ( E/k BT ) (7.54) de is the average relaxation time. It is noted that Eq. (7.54) is valid only for the nondegenerate semiconductors in which the M-B statistics is applicable. Equation (7.53) is obtained using the expression of electron density given by ( ) 4π n = f g(e) de = (2m n )3/2 E 1/2 e ( E/k BT ) de. (7.55) h 3 The average kinetic energy of electrons for a nondegenerate n-type semiconductor can be obtained by using the expression Ef g(e)de E = f g(e)de = E 3/2 exp( E/k B T )de E 3/2 exp( E/k B T )de = 3k BT. (7.56) 2 A generalized expression for the average relaxation time to the nth power τ n is given by τ n = τ n Eg(E) f / E de Eg(E) f / E de, (7.57) where n = 1, 2, 3,..., and τ E n = (τ E n )Eg(E) f / E de. (7.58) Eg(E) f / E de It is noted that the electrical conductivity for an n-type semiconductor given by Eq. (7.53) is similar to that of Eq. (7.7) for a metal. The only difference is that the free-electron mass in Eq. (7.7) is replaced by the effective mass of electrons, m n,

SVNY85-Sheng S. Li October 2, 25 15:4 186 7. Transport Properties of Semiconductors and the constant relaxation time τ is replaced by an average relaxation time τ defined by Eq. (7.54). Using the M-B distribution function for f in Eqs. (7.57) and (7.58), the expressions of τ n and τ E n for a nondegenerate semiconductor are given respectively by τ n τ n E 3/2 e E/kBT de =, (7.59) E 3/2 e E/k BT de τ E n = (τ E n )E 3/2 e E/k B T de. (7.6) E 3/2 e E/k BT de Now, solving Eqs. (7.53) and (7.54) one obtains the expression of electrical conductivity for a nondegenerate n-type semiconductor as where ( nq 2 ) τ σ n = (k B T ) s Ɣ (5/2+s), (7.61) Ɣ (5/2) m n Ɣ n (x) = x n 1 e x dx (7.62) is the Gamma function of order n,ɣ n = (n 1)!, and Ɣ 1/2 = π. Since the electrical conductivity is related to the electron mobility by Eq. (7.6), an expression of the electron mobility can be derived from Eqs. (7.6) and (7.61), and the result yields ( ) qτ µ n = (k B T ) s Ɣ (5/2+s). (7.63) Ɣ (5/2) m n It is noted that for acoustical phonon scattering, τ varies as T 1 and s = 1/2, and hence the electron mobility µ n varies with T 3/2.For ionized impurity scattering, s =+3/2 and τ is independent of temperature, the electron mobility varies as T 3/2. Detailed scattering mechanisms in a semiconductor will be discussed in Chapter 8. The electrical conductivity given by Eq. (7.53) was derived on the basis of the single-valley model with spherical constant-energy surface for the conduction band. This applies to most of the III-V compound semiconductors such as GaAs and InP in which the conduction band minimum is assumed to have spherical constant-energy surface (i.e., parabolic band). In this case the conductivity effective mass m n is an isotropic scalar quantity, and n is the total carrier concentration in the single spherical conduction band. For multivalley semiconductors such as silicon and germanium, since their crystal structures possess cubic symmetry, the electrical conductivity remains isotropic. Thus, Eq. (7.53) is still applicable for the multivalley semiconductors provided that the average relaxation time is assumed

SVNY85-Sheng S. Li October 2, 25 15:4 7.4. Derivation of Transport Coefficientsfor n-type Semiconductors 187 isotropic and the conductivity effective mass m n is replaced by [ ( 1 1 m σ = + 2 )] 1 = 3m l 3 m l m l (2K + 1), (7.64) where K = m l /m t is the ratio of the longitudinal and transverse effective masses of an electron along the two main axes of the ellipsoidal energy surface near the conduction band edge. Values of m l and m t can be determined by the cyclotron resonance experiment at 4.2 K. Equation (7.64) is obtained by using the geometrical average of the electron mass along the two main axes of the ellipsoidal energy surface. 7.4.2. Hall Coefficients The general expression of Hall coefficient for a nondegenerate n-type semiconductor with a single-valley spherical energy band can be derived from Eqs. (7.45) and (7.46) using the definition given by Eq. (7.19). Consider the case of small magnetic field (i.e., µb 1) in which the δ 2 term in Eqs. (7.4) and (7.41) is negligible (i.e., δ 2 1). Thus, by substituting P x (E)given by Eq. (7.4) into Eq. (7.45) and letting T/ x equal to zero, one obtains J x = q vx 2 (β γδ)g(e) f E de ( 2q 2 ) [ ( qτ Bz = 3m n k τ E ε x BT m n Similarly, Eq. (7.46) can be expressed as J y = q v 2 y (γ + δβ)g(e) f E de ( 2q 2 ) [ ( qτ Bz = 3m n k τ E ε y + BT m n ) ε y ] g(e) f de. (7.65) ) ε x ] g(e) f de. (7.66) By setting J y = ineq. (7.66), ε x can be expressed in terms of ε y, which can then be substituted into Eq. (7.65) to obtain the expression of Hall coefficient using the definition of R Hn given by Eq. (7.19), and the result yields R Hn = ε ( ) y 3kB T τ 2 Eg(E) f de J x B = z Jy = 2q [ τ Eg(E) f de] 2 = 1 qn τ 2 τ 2, (7.67) where τ is the average relaxation time and τ 2 is the average of relaxation time square, which can be determined by using Eq. (7.57). The minus sign in Eq. (7.67) denotes the electron conduction in an n-type semiconductor. For p-type

SVNY85-Sheng S. Li October 2, 25 15:4 188 7. Transport Properties of Semiconductors semiconductors, the Hall coefficient is given by R Hp = 1 τ 2 qp τ, (7.68) 2 which has a positive Hall coefficient due to the hole conduction. From Eqs. (7.67) and (7.68), the Hall factor γ H can be expressed by γ H = τ 2 τ. (7.69) 2 If the relaxation time is given by τ = ae s, then the Hall coefficient for a nondegenerate n-type semiconductor is given by R Hn = 1 Ɣ (2s + 5/2) Ɣ (5/2) qn [Ɣ (s + 5/2) ] 2 = γ Hn qn, (7.7) where Ɣ n is the Gamma function defined by Eq. (7.62) and γ Hn is the Hall factor for an n-type semiconductor. In general, the Hall factor can be calculated if the scattering mechanisms in the semiconductor are known. The expression of Hall factor for a p-type semiconductor is identical to that of n-type semiconductors discussed above. Another important physical parameter, which is usually referred to as the Hall mobility, can be obtained from the product of electrical conductivity and Hall coefficient. Thus, using Eqs. (7.61) and (7.7) one obtains ( qτ µ Hn = R Hn σ n = m n ) (k B T ) s Ɣ (2s + 5/2) Ɣ (s + 5/2). (7.71) The Hall factor for a nondegenerate semiconductor, which is defined as the ratio of Hall mobility and conductivity mobility, can be obtained from Eq. (7.7) γ Hn = µ Hn = τ 2 µ n τ 2 Au: Please check whether the edited sentence conveys the intented meaning? = Ɣ (2s + 5/2)Ɣ (5/2). (7.72) [Ɣ (s + 5/2) ] 2 Values of γ Hn may vary between 1.18 and 1.93 depending on the types of scattering mechanisms involved. For example, for acoustical phonon scattering with s = 1/2 the Hall factor is equal to 3π/8( 1.18), and for ionized impurity scattering with s = 3/2 the Hall factor was found equal to 315π/512 ( 1.93). For neutral impurity scattering with s =, the Hall factor is equal to 1. Values of the Hall factor given above are obtained for nondegenerate semiconductors with a single-valley spherical energy surface in the conduction band. For multivalley semiconductors such as silicon and germanium where the conduction band valley has an ellipsoidal energy surface, the expression for the Hall factor should be modified to include the mass anisotropic effect. In this case, a

SVNY85-Sheng S. Li October 2, 25 15:4 7.4. Derivation of Transport Coefficientsfor n-type Semiconductors 189 Hall mass factor a is multiplied by the Hall factor given by Eq. (7.69) for the single-valley case to account for the mass anisotropic effect. Thus, the Hall factor for a multivalley semiconductor can be expressed by γ H = µ Hn = τ 2 a µ n τ, (7.73) 2 where a is known as the Hall mass factor, which can be expressed by where ( m ) 2 a = σ = m H 3K (K + 2) (2K + 1) 2, (7.74) m H = m l 3/[K (K + 2)] (7.75) is the Hall effective mass, m σ is the conductivity effective mass defined by Eq. (7.64), and K = m l /m t is the ratio of the longitudinal and transverse effective masses of electrons along the two major axes of the constant ellipsoidal energy surface of the conduction band. For germanium K 2 and a =.785, while for silicon K = 5.2 and a =.864. Thus, the Hall coefficient given by Eq. (7.67) for n-type silicon and germanium should be multiplied by a Hall mass factor given by Eq. (7.74). For p-type silicon, the Hall factor may be smaller than unity because of the warped and nonparabolic valence band structures. In general, it is usually difficult to obtain an exact Hall factor from the Hall effect measurements. In fact, it is a common practice to assume that the Hall factor is equal to one so that the majority carrier density in a semiconductor can be readily determined from the Hall effect measurements. 7.4.3. Seebeck Coefficients The Seebeck coefficient for a single-valley n-type semiconductor with spherical constant-energy surface can be derived from Eqs. (7.4) and (7.45) by letting δ = in Eq. (7.4) and J x = ineq. (7.45), and the result yields ε x S n = ( T/ x) = Jx = ( 1 = qt ( 1 qt )[ τ E τ E f ) [ ] τ E 2 g(e) f / E de τ Eg(E) f / E de E f ]. (7.76) For anondegenerate semiconductor, the Seebeck coefficient given by Eq. (7.76) can be derived using Eqs. (7.59) and (7.6) to find τ E and τ with τ = ae s, and the result yields ( ) 1 S n = [(5/2 + s)k B T E f ]. (7.77) qt

SVNY85-Sheng S. Li October 2, 25 15:4 19 7. Transport Properties of Semiconductors It is noted that values of the Seebeck coefficient given by Eq. (7.77) can be determined if the types of scattering mechanisms and the position of Fermi level are known. For example, if the acoustical phonon scattering is dominant (i.e., s = 1/2), then the Seebeck coefficient is given by ( ) 1 S n = (2k B T E f ). (7.78) qt On the other hand, if the ionized impurity scattering (i.e., s = 3/2) is dominant, then the Seebeck coefficient becomes ( ) 1 S n = (4k B T E f ). (7.79) qt From Eqs. (7.78) and (7.79), it is seen that the Fermi energy for a nondegenerate semiconductor can be determined from the measured Seebeck coefficient provided that the dominant scattering mechanism is known. The minus sign in Eq. (7.77) for the Seebeck coefficient indicates that the conduction is due to electrons in an n-type semiconductor. Thus, measurement of the Seebeck coefficient can be used to determine the conduction type of a semiconductor. For p-type semiconductors, the sign of the Seebeck coefficient given by Eq. (7.76) is positive since the conduction is carried out by holes. Expressions of Seebeck coefficient derived in this section are applicable only for single-valley nondegenerate semiconductors. However, the results can also be applied to multivalley semiconductors such as silicon and germanium provided that the densityof-states effective mass for electrons is modified to account for the multivalley conduction bands. For degenerate semiconductors, the F-D distribution function should be used in deriving the Seebeck coefficients. This is left as an exercise for the readers in the problem session. 7.4.4. Nernst Coefficients The Nernst effect for a nondegenerate n-type semiconductor with a single-valley spherical energy surface in the conduction band is discussed next. The Nernst coefficient Q n, defined by Eq. (7.2), can be derived from Eqs. (7.4) through (7.48). Consider the case of low magnetic fields (i.e., µb = δ 1). Under isothermal conditions, the Nernst coefficient can be derived by letting J x = J y = and T/ y = ineqs. (7.45) and (7.46), which can be expressed as J x = = q vx 2 (β δγ)g(e) f E de ( 2q 2 ) [ = 3m n k τ E ε x (E ( ) ] f E) T ωτε y g(e) f de (7.8) BT T x

SVNY85-Sheng S. Li October 2, 25 15:4 7.4. Derivation of Transport Coefficientsfor n-type Semiconductors 191 and ( 2q 2 ) [ J y = = 3m n k τ E BT ε y (E f E) ωτ T ( ) ] T + ωτε x g(e) f de. x (7.81) Now solving Eqs. (7.8), (7.81), and using Eq. (7.2) for Q n, one obtains ε y Q n = B z ( T/ x) Jx =J y = { ( ) 1 = τ 2 E 2 g(e) f de m T τ Eg(E) f de τ 2 Eg(E) f de τ E } 2 g(e) f de [ τ Eg(E) f de] 2 ( )[ µn τ 2 E = τ 2 ] τ E, (7.82) qt τ 2 τ 3 where µ n = q τ /m n is the electron conductivity mobility. Now substituting τ = ae s into Eq. (7.82), one obtains the Nernst coefficient for a nondegenerate n-type semiconductor as ( ) kb Q n = µ n s Ɣ (2s+5/2)Ɣ (5/2). (7.83) q [Ɣ (s+5/2) ] 2 If the acoustical phonon scattering (s = 1/2) or ionized impurity scattering (s = 3/2) is the dominant scattering mechanism, then Eq. (7.83) is reduced to ( )( ) 3π kb Q n = µ n for s = 1/2 16 q ( )( ) 945π kb = µ n for s = 3/2. (7.84) 124 q It is interesting to note that, in contrast to both the Hall and Seebeck coefficients, the sign of the Nernst coefficient given by Eq. (7.83) depends only on the types of scattering mechanisms (s) rather than on the types of charge carriers. For example, the Nernst coefficient is negative when the acoustical phonon scattering (i.e., s = 1/2) is dominant, and is positive when the ionized impurity scattering (i.e., s =+3/2) is dominant, as shown in Eq. (7.84). Equation (7.83) can be applied to p-type semiconductors provided that the hole mobility is used in the expression. 7.4.5. Transverse Magnetoresistance The transverse magnetoresistance effect describes the change of electrical resistivity when a transverse magnetic field is applied across a semiconductor specimen. For example, if an electric field in the x-direction and a magnetic field in the z- direction are applied simultaneously to a semiconductor specimen, then an increase in resistance with the applied magnetic field may be observed along the direction of current flow. The magnetoresistance in a semiconductor can be derived using Eqs. (7.4) through (7.46). To derive the expression of transverse magnetoresistance in a single-valley semiconductor with spherical energy band, it is assumed

SVNY85-Sheng S. Li October 2, 25 15:4 192 7. Transport Properties of Semiconductors that the specimen is subject to isothermal conditions with T/ x = T/ y = and J y =. Equations (7.4) and (7.41) can be rewritten as where P x (E) = P y (E) = (β δγ) (1 + δ 2 ) (γ + δβ) (1 + δ 2 ) = (β ωτγ ) (1 + ω 2 τ 2 ), (7.85) = (γ + ωτβ) (1 + ω 2 τ 2 ), (7.86) γ = qτε y, β = qτε x, δ = ωτ = qb zτ. (7.87) m n Now, substituting Eqs. (7.85) and (7.86) into Eqs. (7.45) and (7.46) yields ( 2q 2 ) [ (τεx + ωτ 2 ] ε y ) J x = 3m n k Eg(E) f BT (1 + ω 2 τ 2 de, (7.88) ) ( 2q 2 ) [ (τεy ωτ 2 ] ε x ) J y = 3m n k Eg(E) f BT (1 + ω 2 τ 2 de. (7.89) ) Solving Eqs. (7.88) and (7.89) one obtains the transverse magnetoresistance coefficients of a nondegenerate n-type semiconductor for two limiting cases, namely, the low and high magnetic fields cases. (i) The low magnetic field case (i.e., δ = ωτ 1). In this case, the ω 2 τ 2 term in the denominator of Eqs. (7.88) and (7.89) is retained. From Eq. (7.89), ε y may be expressed in terms of ε x by setting J y =, which yields [ ] ωτ 2 Eg(E) f de ε y = ε x. (7.9) τ Eg(E) f de Now, substituting Eq. (7.9) for ε y into Eq. (7.88) and using the binomial expansion for (1 + ω 2 τ 2 ) 1 (1 ω 2 τ 2 )ineq. (7.88) for ωτ 1, one obtains the electrical conductivity σ n for the low magnetic field case σ n = J x ε x ( 2q 2 ) [ = 3m n k τ E(1 ω 2 τ 2 )g(e) f de + ω2 ( ] τ 2 Eg(E) f de) 2 BT τ Eg(E) f de [ ( τ = σ 1 ω 2 3 τ τ 2 2 )], (7.91) τ 2 where σ = nq 2 τ /m n is the electrical conductivity at zero magnetic field. For the low magnetic field case, the electrical conductivity is given by σ n = 1 ( = σ σ = σ 1 σ ), (7.92) ρ n σ

SVNY85-Sheng S. Li October 2, 25 15:4 7.4. Derivation of Transport Coefficientsfor n-type Semiconductors 193 where ρ n is the resistivity of the semiconductor in the presence of a small magnetic field, which can be expressed by ( ρ n = ρ 1 + ρ ) ( σ 1 1 + σ ). (7.93) ρ σ Solving Eqs. (7.91) through (7.93) one obtains ρ = σ [ τ = ω 2 3 ρ σ τ τ 2 2 ] τ 2 [( ) 1 τ = (σ 2 2 ] 2 [ B2 z ) τ 3 ] τ 1 nq τ 2 τ 2 2 [ τ = RH 2 σ 2 3 ] τ B2 z = µ 2 H B2 z 1 τ 2 2 ]. (7.94) [ τ 3 τ τ 2 2 1 The magnetoresistance coefficient can be deduced from Eq. (7.94), which yields ( )( ) ρ 1 ξ = ρ Bz 2 µ 2 = τ 3 τ 1. (7.95) H τ 2 2 For the nondegenerate semiconductor, using τ = τ E s, Eq. (7.95) becomes ξ = Ɣ (3s+5/2)Ɣ (s+5/2) [Ɣ (2s+5/2) ] 2 1. (7.96) From Eq. (7.96), one finds that ξ is equal to.273 for acoustical phonon scattering (i.e., s = 1/2) and.57 for ionized impurity scattering (i.e., s = 3/2). Under low magnetic field condition, the above results show that the transverse magnetoresistance in a nondegenerate semiconductor is directly proportional to the square of the magnetic field. The magnetoresistance data obtained for semiconductors with a single spherical energy band were found in good agreement with the theoretical prediction at low magnetic fields. It should be noted that for semiconductors with spherical energy bands, the longitudinal magnetoresistance (i.e., J n //B) should vanish under the small magnetic field condition. (ii) The high magnetic field case (i.e., δ = ωτ 1). At high magnetic fields, Eqs. (7.88) and (7.89) become ( 2q 2 ) J x = 3m n k E( ε y /ω)g (E) f de, (7.97) BT ( nq 2 ) J y = [ε y τ 1 +ωε x ]. (7.98) m n ω2