Unit-4: Free electron Fermi Gas

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INTRODUCTION

We can understand a number of important physical properties of metals, particularly the simple metals, in terms of the free electron model. According to this model the most weakly bound electrons of the constituent atoms move about freely through the volume of the metal. The valence electrons of the atoms become the conduction electrons. Forces between conduction electrons and ion cores are neglected in the free electron approximation: all calculations proceed as if the conduction electrons were free to move everywhere within the specimen. The total energy is exclusively the kinetic energy; the potential energy is neglected.

Even in metals for which the free electron model works well, the actual charge distribution of the conduction electrons is known to reflect the strong electrostatic potential of the ion cores. The usefulness of the free electron model is greatest for experiments that depend essentially upon the kinetic properties of the conduction electrons.

The simplest metals are the alkali metals (lithium, sodium, potassium, rubidium, and cesium). Conduction electrons in all metals are observed to act very much like free electrons, except those metals where the d or f shell electrons overlap or fall close in energy to the conduction band. Electrons in d shells tend to be more localized and less mobile than electrons in s and p shells. Other simple metals are Be, Mg, Ca, Sr, Ba, Al, Ga, In, TI, and Pb. Metals that are not simple are the noble metals (copper, silver, gold), the transition metals, and the lanthanide and actinide metals.


Figure 1. Schematic model of a crystal of sodium metal. The atomic cores arc Na + ions; they are immersed in a sea of conduction electrons, the conduction electrons are derived From the 3s valence electrons of the free atoms. The atomic cores contain 10 electrons in the configuration is 1s2 2s2 2p6. In an alkali metal the atomic cores occupy a relatively small part (15 %) of the total volume of the crystal, but in a noble metal (Cu, Ag, Au) the atomic cores are relatively larger and may be in contact with each other. The common crystal structure at room temperature is bcc for the alkali metals and fcc for the noble metals.


Conduction electrons in a simple metal arise from the valence electrons of the constituent atoms. In a sodium atom the valence electron is in a 3s state [Fig. 1.]; in the metal this electron becomes a conduction electron, roving throughout the crystal. A monovalent crystal which contains N atoms will have N conduction electrons and N positive ion cores. The 10 electrons of the Na + ion core fill the 1s, the 2s, and the 2p states in the free ion; the distribution of core electrons is essentially the same in the metal as in the free ion. The ion cores fill only about 15 percent of the volume of a sodium crystal. The interpretation of metallic properties in terms of the motion of free electrons was developed long before the invention of quantum mechanics. The classical theory had several conspicuous successes, notably the derivation of the form of Ohm’s law and the relation between the electrical and thermal conductivity. The classical theory fails to explain the heat capacity and the magnetic susceptibility of the conduction electrons.

There is a further difficulty. From many types of experiments it is clear that a conduction electron in a metal can move freely in a straight path over many atomic distances, undeflected by collisions with other conduction electrons or by collisions with the atom cores. In a very pure specimen at low temperatures the mean free path may be as long as 10 interatomic spacing (more than 1 cm). Why is condensed matter so transparent to conduction electrons? The answer to our question contains two parts:

a. A conduction electron is not deflected by ion cores arranged on a periodic lattice because matter waves propagate freely in a periodic structure. As the propagation of x-rays in periodic lattice.

b. A conduction electron is scattered only infrequently by other conduction electrons. This property we shall show is a consequence of the Pauli Exclusion Principle. By a free electron Fermi gas, we mean a gas of free and non-interacting electrons subject to the Pauli Principle.

THE FREE-ELECTRON GAS

In the free-electron model, the conduction electrons are assumed to be completely free, except for a potential at the surface (Fig. 2.), which has the effect of confining the electrons to the interior of the specimen. According to this model, the conduction electrons move about inside the specimen without any collisions, except for an occasional reflection from the surface, much like the molecules in an ideal gas. Because of this, we speak of a free-electron gas.

Figure 2. The potential in the free-electron model.


Let us look at the model a little more closely. It is surprising that it should be valid at all, because, at first sight, one expects the conduction electrons to interact with the ions in the background, and also with each other. These interactions are strong, and hence the electrons ought to suffer frequent collisions; a picture of a highly non ideal gas should therefore emerge. Why then does the free-electron model work? The answer to this fundamental question was not known to the workers who first postulated the model. We now know the answer, but since it requires the use of quantum mechanics, hence a brief qualitative statement is offered here.

The reason why the interaction between the ions appears to be weak is as follows. Although the electron does interact with an ion through coulomb attraction, quantum effects introduce an additional repulsive potential, which tends to cancel the coulomb attraction. The net potential known as the pseudopotential turns out to be weak, particularly in the case of alkali metals. Another way of approaching this is to note that, when an electron passes an ion, its velocity increases rather rapidly in the ion's neighborhood (Fig. 3.), due to the decrease in the potential. Because of this, the electron spends only a small fraction of its time near the ion, where the potential is strong. Most of the time, the electron is far away in a region in which the potential is weak, and this is why the electron behaves like a free particle, to a certain approximation.


Figure 3. Variation of the local velocity of electrons in space.


We come now to the interaction between the conduction electrons themselves, and the reason for the weakness of this interaction. There are actually two reasons: First, according to the Pauli Exclusion Principle, electrons of parallel spins tend to stay away from each other. Second, even if their spins are opposite, electrons tend to stay away from each other, in order to minimize the energy of the system. If two electrons come very close to each other, the coulomb potential energy becomes exceedingly large, and this violates the tendency of the electron system to have the lowest possible energy. When these two considerations are carried out mathematically, the following situation results: Each electron is surrounded by a (spherical) region which is deficient of other electrons. This region, called a hole, has a radius of about 1A0 (the exact value depends on the concentration of electrons). As an electron moves, its hole-sometimes known as a Fermi hole moves with it. We see now why the interaction between electrons is weak. If we examine the interaction between two particular electrons, we find that other electrons distribute themselves in such a manner that our two electrons are screened from each other. Consequently there is very little interaction between them. Free-electron gas in metals differs from ordinary gas in some important respects. First, free-electron gas is charged (in ordinary gases the molecules are mostly neutral). Free-electron gas is thus actually similar to plasma. Second, the concentration of electrons in metals is large: N{\thicksim} 1029 electrons m − 3. By contrast, the ordinary gas has about 1025 molecules m − 3. We may thus think of free-electron gas in a metal as dense plasma.

Our model of the electron (sometimes called the jellium model) corresponds to taking metallic positive ions and smearing them uniformly throughout a sample. In this way there is a positive background which is necessary to maintain charge neutrality. But, because of the uniform distribution, the ions exert zero fields on the electrons; the ions form a uniform jelly into which the electrons move.

THE DRUDE MODEL

The Drude model is essentially based on the classical kinetic theory of gases. According to Drude the metal must have two types of particles as against only one type in the simplest gases. The discovery of electrons bearing negative charge made it mandatory to accept the presence of positively charged entities (or particles) on the requirement of the charge neutrality condition. It is assumed that when metal atoms come together to form a metal, the valence electrons get liberated and move freely within the metal. The remainder of the atom is a positive ion carrying the major portion of the atomic mass. Drude took these particles as heavy and immobile. Under the action of an external electric field the free electrons referred to as conduction electrons in metals move in the background of immobile positive metal ions. In the absence of a relevant theoretical framework to deal with the free electron gas, Drude took recourse to the methods of kinetic theory of ideal gases for examining metallic conduction without getting deterred by the large electron densities (1022 cm − 3). It is assumed that the reader is acquainted with the postulates of kinetic theory of gases through a prior exposure, and therefore assumptions specific to only free electron gas are being given below:

1. The electron-electron and electron-ion interactions are neglected. The approach is identical to working in the independent electron approximation and the free electron approximation in that order. To be exact, the free electron approximation is not adhered to in strict literal sense because the electrons are considered to remain confined within the metal in the Drude Model and this is possible only if the electron-ion attractive force is active.

2. Under the action of an external electric field, electrons move opposite to the field direction and make collisions with immobile and impenetrable ion cores. An electron bumps from ion to ion and between successive collisions moves in a straight line as determined by the Newton's equations of motion.

3. All electrons move with the RMS speed of a Maxwell-Boltzmann distribution, representing the random or thermal velocity of electrons. The average electron velocity immediately after the collision is zero.

DC ELECTRICAL CONDUCTIVITY OF METALS

Now, we apply Drude theory to derive an expression for dc conductivity. An external static electric field can affect electron velocity during the time interval between two successive collisions. But the gain in velocity is destroyed each time a collision occurs since the average velocity immediately after the collision is zero. A large influence of electric field is reflected in a larger mean free time or relaxation time τ . The probability of collision per unit time is defined as \frac{1}{\tau} . Therefore, the probability that a collision occurs in a small time interval dt is simply \frac{dt}{\tau} .

Taking the acceleration of an electron of mass as \frac{e F}{m} , the mean drift velocity can be written as

    v_d=\left(-\frac{e F}{m}\right)\tau ----------------------------------(1)

Figure 4. (a) An electric field applied to metallic wire. (b) Random versus drift motion of electrons. Cirles represent scattering centers.

If there are a total of n electrons per m3 in the metal, all with constant drift velocity vd , we have the following relation for the net electric current density j

    j=-n e v_d=\left(\frac{ne^2\tau}{m}\right)=\sigma E --------------------------------(2)

where the positive scalar quantity


    \sigma=\frac{ne^2\tau}{m} ------------------------------------(3)

is defined as the electrical conductivity (the reciprocal of the electrical resistivity \boldsymbol{\rho} ). The form of the expression (3) remains the same in all models including those based on quantum physics. The difference lies only in the way n,m , and τ are defined.

We are now beginning to accomplish our aim of highlighting the gains of the Drude model. The first and the foremost of these is the derivation of the Ohm's law contained in (2). The law was initially set up purely on an empirical basis.

It must be, however, stated that the electrical conductivity σ is not universally scalar since in some complicated situations j becomes nonlinear with respect to E making σ to behave as a tensor. The electrical conductivity is often expressed in terms of the drift mobility of electrons

μ as

    σ = neμ ---------------------------------(4)

Where

    \mu=\frac{v_d}{E} ----------------------------------------------(5)

According to the kinetic theory thermal velocity vth, expressed by the following relations:

    \tau=\frac{\land}{v_{th}} --------------------------------------(6)

where \land is the electron mean free path. We also know that

    \frac{3}{2}k_BT=\frac{1}{2}m v^2_{rms}=\frac{1}{2}m v^2_{th} -------------------------------(7)

Using these relations, (3) can be written as

    \sigma=\frac{n e^2\land}{(3 m k_BT)^{1/2}} ----------------------------------(8)

The relation (8) is another achievement of the Drude theory since it gives the right magnitude of electrical conductivity and correctly describes that conductivity increases with decrease in temperature. But the dependence on T^{-\frac{1}{2}} is not in agreement with the observed T − 1 dependence in most of the common metals. The modern theory that resolves this issue attributes this inconsistency to the unrealistic assumption of electron-ion elastic collisions and the use of classical statistics to describe the electrons in the Drude model. The Drude model is further plagued by its inability to account for the low electron heat capacity and the temperature independent behavior of paramagnetic susceptibility of conduction electrons.

By feeding the measured value of conductivity in (3), we obtain the value of relaxation time τ . The estimated value of τ is in the range 10 − 15 to 10 − 14 s. The thermal velocities estimated from (7) are around 105 m s − 1. Using (6), we find that the typical values of mean free path \land are of the order of a few angstroms which compare with the interatomic separations. This in a way supports the electron-ion collisions in Drude theory. But the low temperature measurements on some of the purest and least imperfect crystals show the mean free path as large as a few centimeters. This strongly contradicts the Drude picture in which the electron bumps along from ion to ion. On the other hand, the modern theory tells that electron waves cannot be scattered by a perfectly periodic potential leading to an infinite mean free path in an ideal crystal. The experimental data in no way casts aspersion on the above theoretical view, simply because nobody has ever produced any perfect and ideal crystal. Since every real crystal has deviations from periodicity because of the presence of imperfections and even impurities, the value of the mean free path is limited by the scattering from these centers.

THERMAL CONDUCTIVITY OF METALS

In the Drude model it is assumed that the major contribution to thermal conductivity of metals comes from conduction electrons. This assumption has basis in the general observation that metals are far better conductors of heat than insulators. The participation of metal ions is reflected in the phonon contribution which is neglected in Drude theory on account of its relatively much small measure.


Consider a metal bar whose two ends are maintained at a constant difference of temperature. This situation refers to the steady state when the whole of thermal energy being fed at the hot end is received at the cold end without any net absorption in the bar. Let the temperature gradient along the length of the bar be defined as -\frac{\partial T}{\partial x} meaning thereby that the temperature decreases as length increases by x . For a small temperature gradient, the thermal current j , which is a measure of the thermal energy flowing per unit sectional-area of the bar per unit time, is found to be proportional to the temperature gradient and thus,

    j=K_{el}\left(\frac{\partial T}{\partial x}\right) -------------------------------------(9)

which shows that the thermal energy flows in a direction opposite to that of the gradient.

Let us confine ourselves to the one-dimensional flow of heat energy in which electrons move parallel to the length of the bar. The number of heat carriers in this case (electrons) is conserved while calculating j. Let us concentrate on a group of n electrons per unit volume. At a point x along the length, half of them \left(\frac{n}{2}\right) arrive from the hot end and the rest \left(\frac{n}{2}\right) from the cold end. The electron flowing from the hot end should have met its last collision at (xvxτ) whereas the one from the cold end would have its last collision at (x + vxτ) If the thermal energy per electron in thermal equilibrium at T be denoted by u\left[T(x)\right] , the net thermal current can be written as

    j=\frac{1}{2}n v_x\left[u(T(x-v_x\tau))-u(T(x+ v_x\tau))\right] -------------------------------------(10)

where T is the temperature at x and vx is the x -component of the average electron velocity. In the approximation that the variation in temperature over a mean free path length \land is small, (6.10) can be expanded about x giving,

    j=n e^2_x\tau\frac{\partial u}{\partial T}\left(-\frac{\partial T}{\partial x}\right) ---------------------------------------(11)

The factor n\left(\frac{\partial u}{\partial T}\right) may be replaced by the electron heat capacity per unit volume Cel and v^2_x

by \frac{1}{3}v^2 in (11) to get

    j=\frac{1}{3}C_{el}v^2\tau\left(-\frac{\partial T}{\partial x}\right) ---------------------------------(12)

Comparing (12) with (9), we get .

    K_{el}=\frac{1}{3}C_{el}v^2\tau\frac{1}{3}C_{el}v\land --------------------------------(13)

where v is the RMS speed of an electrons.

The calculation of Cel is made by taking the thermal energy per electron at temperature T as \frac{3}{2}k_B T , in accordance with the law of equipartition of energy. This gives the heat capacity per electron as \frac{3}{2}k_B , and C_{el}=\frac{3}{2}n k_B . Furthermore, the value of v2 is obtained from (7). Making these substitutions in (13), we get

    k_{el}=\frac{3}{2}\left(\frac{n k^2_B\tau}{m}\right)T ---------------------------(14)

Even before the electrons were known, in 1835 Wiedemann and Franz were able to notice that all good conductors of electricity are good conductors of heat too. They established that the ratio of thermal conductivity to electrical conductivity is proportional to the temperature for a large number of metals. Later in 1881 Lorenz (different from Lorentz) observed that the proportionality constant equaling \frac{K}{\sigma T} has almost the same value for most of the common metals. This constant is known as the Lorenz number and the empirical law became famous as the Wiedemann-Franz law. From (3) and (14), Drude obtained

    \frac{K}{\sigma}=\frac{3}{2}\left(\frac{k^2_B}{e}\right)^2 T ---------------------------(15)

The coefficient of T in (15) is a constant. Drude was thus able to derive the Wiedemann-Franz law which is acknowledged as the biggest success of the Drude model. The Lorenz number L as defined in the theory is given by

    L=\frac{K}{\sigma T}=\frac{3}{2}\left(\frac{k^2_B}{e}\right)^2 -------------------------------------------(16)

LORENTZ MODIFICATION OF THE DRUDE MODEL

Lorentz took up the task of modifying the oversimplified Drude model and constructed his theory on the basis of following points:


1. The assumption that all electrons move with the same thermal velocity is abandoned.

2. The classical Maxwell-Boltzmann velocity distribution is perturbed by the presence of an electric field or a thermal gradient. Both of these tend to displace the equilibrium velocity distribution and distort its symmetry.

3. The approach of Boltzmann transport equation is followed to describe the transport of charge and kinetic energy of electrons by a statistical distribution of mobile electrons constituting the electron gas.

To avoid duplication, the derivation of the expression for electrical conductivity will be taken up at a proper stage for the quantum model of the free electron gas, i.e. for the free electron Fermi gas. Nevertheless, we give below the relation as obtained by Lorentz;

    \sigma_L=\left(\frac{8}{3\pi}\right)^{\frac{1}{2}}\frac{n e^2\tau}{(3m k_B T)^{1/2}} --------------------------------------(17)

If we denote the Drude conductivity of (8) as σD then

    \sigma_D=\left(\frac{3\pi}{8}\right)^{\frac{1}{2}}\sigma_L=1.09\sigma_L --------------------------------------(18)


From (17) and (18) we learn that the Lorentz modifications provide neither any noteworthy change in the quantitative measure nor any change in the temperature dependence of electrical conductivity. Lorentz, however, went on to analyze the effect of a uniform magnetic field on current carrying conductors. This effect is known as the Hall Effect, The net force on an electron moving with velocity ν under the action of a static electric field E and a uniform magnetic field B is

    F=-e\left[E+(v\times B)\right] ------------------------------------------(19)

where F is called the Lorentz force.

Lorentz solved the Boltzmann transport equation under the above conditions and obtained the following expression for the Hall coefficient.

    R_H=-\left(\frac{3\pi}{8}\right)\frac{1}{ne} ------------------------------------(20)

The electron density n can be estimated from (20) by feeding the measured value of Hall coefficient in it. The estimate is consistent with the contribution of one conduction electron per atom in sodium. The result is not so good for silver and is worse for other metals. This points to an otherwise established fact that the free electron approximation is best applicable to alkali metals with the lone loosely bound outermost 3s1 electron. The subject is effectively treated by the band theory that also accounts for the observed positive Hall coefficient in certain metals.

ENERGY LEVELS AND DENSITY OF ORBITALS IN ONE DIMENSION

Consider a free electron gas in one dimension, an electron of massm is confined to a length L by infinite barrier (Fig. 5). The wave function Ψn(x) of the electron is a solution of the Schrodinger equation HΨ = εψ ; with the kinetic energy H=\frac{P^2}{2m} , where P is the momentum. So that, the Schrӧdinger equation is given by


    i \hbar \frac{\Psi_n}{dt}=-\frac{\hbar^2}{2m}\frac{d^2\Psi_n}{dx^2}=\epsilon_n\Psi_n ------------------------------(21)

Where εn is the energy of the electron in the orbital. We use the term orbital to denote a solution of the wave equation for a system of only one electron. The term allows us to distinguish between an exact quantum state of the wave equation of a system of N electrons and an approximate quantum state which we construct by assigning the N electrons to N different orbitals, where each orbital is a solution of a wave equation for one electron. The orbital model is exact only if there are no interactions between electrons.

The boundary conditions are Ψn(0) = 0  ;Ψn(L) = 0 , as imposed by the infinite potential energy barrier; they are satisfied if the wave function is sine like with an integral number a of half-wavelengths between 0 and L: ;

Figure 5. First three energy levels and wave functions of a free electron of mass m confined to a line of length L. The energy the levels are labeled according to the quantum number n which gives the number of half-wavelengths in the wave function. The wavelengths are indicated on the wave functions.

    \Psi_n=A sin\left(\frac{2\pi}{\lambda_n}x\right);\frac{1}{2}n\lambda_n=L --------------------------------(22)

where A is a constant. We see that (22) is a solution of (21), because

    \frac{d\Psi_n}{dx}=A\left(\frac{n\pi}{L}\right)cos\left(\frac{n\pi}{L}x\right); and
    \frac{d^2\Psi_n}{dx^2}=-A\left(\frac{n\pi}{L}\right)^2 sin\left(\frac{n\pi}{L}x\right) -------------------------------------(23)

Where the energy εn, is given by

    \epsilon_n=\frac{\hbar^2}{2m}\left(\frac{n\pi}{L}\right)^2 ----------------------------------------(24)

We want to accommodate N electrons on the line. According to the Pauli Exclusion Principle no two electrons can have all their quantum numbers identical. That is, each orbital can be occupied by at most one electron. This applies to electrons in atoms, molecules, or solids. In a linear solid the quantum numbers of a conduction electron orbital are n and ms, where n is any positive integer and m_s=\pm\frac{1}{2} , according to the spin orientation. A pair of orbitals labeled by the quantum number n can accommodate two electrons, one with spin up and one with spin down. If there are six electrons, then in the ground state of the system the filled orbitals are those given in the following table:


More than one orbital may have the same energy. The number of orbitals with the same energy is called the degeneracy.

Let nF denote the topmost filled energy level, where we start filling the levels from the bottom ( n = 1) and continue filling the higher levels with electrons until allN electrons are accommodated. It is convenient to suppose that N is an even number. The condition 2nF = N determinesnf , the value of n for the uppermost filled level. The Fermi energy εF is defined as the energy of the topmost filled level in the ground state. By (24) with n = nF we have in one dimension:

    \epsilon_F=\frac{\hbar^2}{2m}\left(\frac{n_F\pi}{L}\right)^2=\frac{\hbar^2}{2m}\left(\frac{N\pi}{2L}\right)^2 --------------------------------------(25)

EFFECT OF TEMPERATURE ON THE FERMI-DIRAC DISTRIBUTION

The ground state is the state of the system at absolute zero. What happens as the temperature is increased? This is a standard problem in elementary statistical mechanics and the solution is given by the Fermi-Dirac distribution function. The kinetic energy of the electron gas increases as the temperature is increased: some energy levels are occupied which were vacant at absolute zero, and some levels are vacant which were occupied at absolute zero (Fig. 6). The Fermi-Dirac distribution gives the probability that an orbital at energy ε will be occupied in an ideal electron gas in thermal equilibrium;

    f(\epsilon)=\frac{1}{\left(e^{\frac{(\epsilon-\epsilon_F)}{K_B T}}+1\right)} ----------------------------(26)

Figure 6 Fermi-Dirac distribution function at various temperatures.


The quantity εF is a function of the temperature;εF is to be chosen for the particular problem in such a way that the total number of particles in the system comes out correctly — that is, equal to N . At absolute zero εF = μ, because in the limit T\rightarrow 0 the function f(ε) changes discontinuously from the value 1 (filled) to the value 0 (empty) at ε = εF = μ . At all temperatures f(ε) equal to\frac{1}{2} when ε = εF = μ, for then the denominator of (26) has the value 2. The quantity μ is called the chemical potential, and we see that at absolute zero the chemical potential is equal to the Fermi energy. The Fermi energy is the energy of the topmost filled orbital at absolute zero.

The high energy tail of the distribution is that part for which \epsilon-\mu\gg k_B T; here the exponential term is dominant in the denominator of (26), so thatf(\epsilon)\cong e^{(\epsilon-\epsilon_F)lk_B T} . This is the Boltzmann distribution.

FREE ELECTRON GAS IN THREE DIMENSIONS

The free-particle Schrödinger equation in three dimensions is

    i\hbar=\frac{d\Psi_k}{dt}=-\frac{\hbar^2}{2m}\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}\right)\Psi_k(r)=\epsilon_k \Psi_k(r) -------------------------------------------(27)

If the electrons are confined to a cube of edge L , the wave function is the standing wave

    \Psi_n(r)=A sin\left(\frac{\pi n_x}{L}x\right)sin\left(\frac{\pi n_y}{L}y\right)sin\left(\frac{\pi n_z}{L}z\right) ----------------------------------------(28)

where nx,ny,nz are positive integers.

It is convenient to introduce wave functions that satisfy periodic boundary conditions. We now require the wave functions to be periodic in x,y,z with period L, thus

    Ψ(x + L,y,z) = Ψ(x,y,,z) --------------------------------------(29)

and similarly for the y and z coordinates. Wave functions satisfying the free-particle Schrodinger equation and the periodicity condition are of the form of a traveling plane wave:

    Ψk(r) = exp(ik.r) = e(ik.r) ---------------------------------------(30)

provided that the components of the wave vector k satisfy

    k_x=0;\pm\frac{2\pi}{L};\pm\frac{4\pi}{L};\pm\frac{6\pi}{L};... -------------------------------(31)

and similarly forky and kz . That is, any component of k is of the form 2nπ / L , where n is a positive or negative integer. The components of k are the quantum numbers of the problem, along with the quantum numberms for the spin direction. We confirm that these values of k satisfy (29), for

    exp(ik_x(x+L))=exp(i2n\pi(x+L)/L)=exp(i2n\pi x/L)=e^{(ik_x.x)} --------------------------------(32)


On substituting (30) in (27) we have the energy εF of the orbital with wave vector k:

    \epsilon_k=\frac{\hbar^2}{2m}k^2=\frac{\hbar^2}{2m}(k^2_x+k^2_y+k^2_z) ----------------------------------(33)

The magnitude of the wavevector is related to the wavelength by k = 2π / λ .

The linear momentum p may be represented in quantum mechanics by the

operator p=-i \hbar \triangledown, whence for the orbital(30)


    p\Psi_k(r)=-i \hbar \triangledown\Psi-k(r)=\hbar k\Psi_k(r) --------------------------------------(34)

so that the plane wave Ψk , is an eigenfunotion of the linear momentum with the eigenvalue \hbar k. The particle velocity in the orbital k is given by v=\frac{\hbar k}{m} .



Figure 7. In the ground state of a system of N free electrons the occupied orbitals of the system fill a sphere of radiuskF , where \epsilon_F=\frac{\hbar^2 k^2_F}{2m} is the energy of an electron having a wave vector kF.

In the ground state of a system of N free electrons the occupied orbitals may be represented as points inside a sphere in k space. The energy at the surface of the sphere is the Fermi energy; the wave vectors at the Fermi surface have a magnitude kF such that (Fig. 7):

    \epsilon_F=\frac{\hbar^2}{2m}k^2_F --------------------------------(35)

From (31) we see that there is one allowed wave vector that is, one distinct triplet of quantum numbers kx,ky,kz — for the volume element \left(\frac{2\pi}{L}\right)^3 of k space. Thus in the sphere of volume \frac{4}{3}2\pi k^3_F the total number of orbitals is

    2\frac{\frac{4}{3}\pi k^3_F}{\left(\frac{2\pi}{L}\right)^3}=\frac{V}{3\pi^2}k^3_F=N ----------------------------(36)

where the factor 2 on the left comes from the two allowed values of ms , the spin quantum number, for each allowed value of kF . Then

    k_F=\left(\frac{3\pi^2 N}{V}\right)^{\frac{1}{3}} --------------------------------(37)

which depends only on the particle concentration and not on the mass. Using (35),

    \epsilon_F=\frac{\hbar^2}{2m}\left(\frac{3\pi^3 N}{V}\right)^{\frac{2}{3}} --------------------------------(39)

This relates the Fermi energy to the electron concentrationN / V . The electron velocity vF at the Fermi surface is

    v_F=\left(\frac{\hbar k_F}{m}\right)=\left(\frac{\hbar}{m}\right)\left(\frac{3\pi^2 N}{V}\right)^{\frac{1}{3}} ---------------------------------(40)

We now find an expression for the number of orbitals per unit energy range, D(ε) often called the density of states. We use (39) for the total number of orbitals of energy\le\epsilon :

    N=\left(\frac{V}{3\pi^2}\right)\left(\frac{2m\epsilon}{\hbar^2 }\right)^{\frac{3}{2}} ---------------------------------(41)

so that the density of orbitals (Fig. 8) is

    D(\epsilon)=\frac{dn}{d\epsilon}=\left(\frac{V}{2\pi^2}\right)\left(\frac{2m}{\hbar^2}\right)\epsilon^{\frac{1}{2}} ---------------------------------(42)

This result may be obtained and expressed most simply by writing (4) as

    ln N=\frac{3}{2}ln \epsilon+ cons tan t ;   \frac{dN}{N}=\frac{3}{2}\frac{d\epsilon}{\epsilon} ----------------------------------------(43)

Whence

     D(\epsilon)\equiv\frac{dN}{d\epsilon}=\frac{3N}{2\epsilon} -----------------------------(44)

Within a factor of the order of unity, the number of orbitals per unit energy range at the Fermi energy is just the total number of conduction electrons divided by the Fermi energy.

Figure 8. Density of single-particle states as a function of energy, for a free electron gas in three dimensions. The dashed curve represents the density f(ε,T)D(ε) of filled orbitals at a finite temperature, but such that kBT is small in comparison with εF . The shaded area represents the filled orbitals at absolute zero. The average energy is increased when the temperature is increased from 0 to T, for electrons are thermally excited from region 1 to region 2.

HEAT CAPACETY OF THE ELECTRON GAS

Classical statistical mechanics predicts that a free particle should have a heat capacity of \frac{3}{2}k_B , where kB is the Boltzmann constant. If N atoms each give one valence electron to the electron gas, and the electrons are freely mobile, then the electronic contribution to the heat capacity should be \frac{3}{2}N k_B . But the observed electronic contribution at room temperature is usually less than 0.01 of this value. This discrepancy distracted till the discovery of the Pauli Exclusion Principle and the Fermi-Dirac distribution function. Fermi found the correct equation, and he wrote, “One recognizes that the specific heat vanishes at absolute zero and that at low temperatures it is proportional to the absolute temperature.”


When we heat the specimen from absolute zero not every electron gains an energy kBT as expected classically, but only those electrons in orbitals within an energy range kBT of the Fermi level are excited thermally; these electrons gain an energy which is itself of the order of kBT , as in Fig. 8. This gives an immediate qualitative solution to the problem of the heat capacity of the conduction electron gas. If N is the total number of electrons, only a fraction of the order of T / TF can be excited thermally at temperature T , because only these lie within an energy range of the order of kBT of the top of the energy distribution. Each of these NT / TF electrons has a thermal energy of the order of kBT, and so the total electronic thermal energy U is of the order of

    U\approx N \left(\frac{T}{T_F}\right)k_B T -------------------------------(45)

The electronic heat capacity is given by C_{el}=\frac{dU}{dT}\approx N K_F\left(\frac{T}{T_F}\right) and is directly proportional toT , in agreement with the experimental results. At room temperature Cel , is smaller than the classical value \frac{3}{2} N k_B by a factor of the order of 0.01 or less, for T\approx 5\times 10^4 K .

At the low temperatures \frac{k_B T}{\epsilon_F}<0.01 the electronic specific heat can be given as,

     C_{el}=\int\limits_{0}^{\infty}(\epsilon-\epsilon_F)D(\epsilon)\frac{\partial f}{\partial T}d\epsilon --------------------------------------(46)


    \therefore C_{el}\approx D(\epsilon)\int\limits_{0}^{\infty}(\epsilon-\epsilon_F)\frac{\partial f}{\partial T}d\epsilon --------------------------------------------(47)

But f(\epsilon)=\frac{1}{\left(e^{(\epsilon-\epsilon_F)/k_B T}+1\right)} so \frac{\partial f(\epsilon)}{\partial T}=\frac{(\epsilon-\epsilon_F)}{k_B T^2}\frac{e^{(\epsilon-\epsilon_F)/k_B T}}{(e^{(\epsilon-\epsilon_F)/k_B T}+1)^2}


Let \frac{\left(\epsilon-\epsilon_F\right)}{k_B T}=x \rArr d\epsilon=k_B T d x


    \therefore C_{el}\approx k^2_B T D(\epsilon)\int\limits_{-\epsilon_F/k_B T}^{\infty}\frac{x^2 e^x}{(e^x+1)^2}dx ---------------------------------------(48)

The ex factor in the integrand is negligible for x\le-\frac{\epsilon_F}{k_B T} i.e., at low temperatures. Therefore, the lower limit of integration may be safely extended to -\infty transforming the integral into a standard form:


    \int\limits_{-\infty}^{\infty}\frac{x^2 e^x}{(e^x+1)^2}dx=\frac{\pi^2}{3} --------------------------(49)
    C_{el}\approx k^2_B T D(\epsilon)\frac{\pi^2}{3} ----------------------------(50)

But we have D(\epsilon)=\frac{3N}{2\epsilon_F}


    C_{el}\approx k^2_B T \left(\frac{3N}{2\epsilon_F}\right)\frac{\pi^2}{3}=\frac{\pi^2}{2}N k_B\left(\frac{T}{T_F}\right) ---------------------------------(51)

The electronic heat capacity of one and two-dimensional electron gases are also linearly proportional T to T at temperatures T\le T_F . The dimensionality of the gas does not enter the argument except by a numerical factor.

EXPERIMENTAL HEAT CAPACITY OF METALS

At temperatures much below both the Debye temperature and the Fermi temperature, the heat capacity of metals may be written as the sum of electronic and lattice contributions: C = λT + αT3 , where λ and α are constants characteristic of the material. The electronic term is linear in T and is dominant at sufficiently low temperatures. It is convenient to exhibit the experimental values of C as a plot of C / T versus T2 ,


    \frac{C}{T}=\lambda+\alpha T^2 -------------------------------------(52)


Figure 9. Experimental heat capacity values for Cu, plotted as C/T versus T2.

Where the first term represent the electron contribution and the second term is the Debye term contributed by phonons. The phonon or ionic contribution dominates at high temperatures but drops spectacularly at low temperatures where it becomes even smaller than the electron contribution.

The observed values of the coefficient λ are of the expected magnitude, but often do not agree very closely with the value calculated for free electrons of mass m by use of the relation (50). It is common practice to express the ratio of the observed to the free electron values of the electronic heat capacity as a ratio of a thermal effective mass mth to the electron mass m , where mth is defined by the relation


    \frac{m_{th}}{m}\equiv \frac{\lambda(observed)}{\lambda(free)} --------------------------------(53)

This form arises in a natural way because εF, in the expression (50) is inversely proportional to the mass of the electron, whence λαm . The departure from unity involves three separate effects

1. The interaction of the conduction electrons with the periodic potential of the rigid crystal lattice. The effective mass of an electron in this potential is called the band effective mass and is treated later.

2. The interaction of the conduction electrons with phonons. An electron tends to polarize or distort the lattice in its neighborhood, so that the moving electron tries to drag nearby ions along, thereby increasing the effective mass of the electron.

3. The interaction of the conduction electrons with themselves. A moving electron causes an inertial reaction in the surrounding electron gas, thereby increasing the effective mass of the electron.

ELECTRICAL CONDUCTIVITY AND OHM’S LAW

The momentum of a free electron is related to the wave vector by p=m v=\hbar k. In an electric field E and magnetic field B the force F on an electron of charge e is -e\left[E+\frac{1}{c}(v\times B)\right] , so that Newton’s second law of motion becomes (CGS)

    F=m\frac{dv}{dt}=\hbar\frac{dk}{dt}=-e\left[E+\frac{1}{c}(v\times B)\right] ----------------------------------(54)

In the absence of collisions the Fermi sphere (Fig. 10) in k space is displaced at a uniform rate by a constant applied electric field, we integrate with B = 0 to obtain

    k(t)-k(0)=-\frac{e E t}{\hbar} -----------------------------------(55)

If the field is applied at time t = 0 to an electron gas that fills the Fermi sphere centered at the origin of k space, then at a later time t the sphere will be displaced to a new center at

    \delta k=-\frac{e E t}{\hbar} -----------------------------------------(56)

Because of collisions of electrons with impurities, lattice imperfections, and phonons, the displaced sphere may be maintained in a steady state in an electric field. If collision time is τ , the displacement of the Fermi sphere in the steady state is given by (56). The incremental velocity is v=-\frac{e E \tau}{m} . If in a constant electric field E there are n electrons of charge q = − e per unit volume, the electric current density is

    j=n q v=\frac{n e^2\tau E}{m} ------------------------------------------(57)
a
b

Figure 10. (a) The Fermi sphere encloses the occupied electron orhitals in space in the ground state of the electron gas. The net momentum is zero, because for every orbital there is an occupied orbital at - . (b) Under the influence of a constant force acting for a time interval every orbital has its vector increased by . This is equivalent to a displacement of the whole Fermi sphere by . The total momentum is , if there are electrons present. The application of the force increases the energy of the system by .

This is in the form of Ohm’s law. The electrical conductivity σ is defined by j = σE , so that

    \sigma=\frac{n e^2\tau}{m} ----------------------------------(58)

The electrical resistivity \boldsymbol{\rho} is defined as the reciprocal of the conductivity, so that

    \boldsymbol{\rho}=\frac{m}{n e^2 \tau} ---------------------------------(59)

It is easy to understand the result (58) for the conductivity. We expect the charge transported to be proportional to the charge density ne ; the factor e / m enters because the acceleration in a given electric field is proportional to e and inversely proportional to the mass m ; and the time describes the free time during which the field acts on the carrier.

Mean free paths as long as 10 cm have been observed in very pure metals in the liquid helium temperature range.

EXPERIMENTAL ELECTRICAL RESISTIVITY OF METALS

a
b


Figure 11. Electrical resistivity in most metals arises from collisions of electrons with irregularities in the lattice, as in (a) by phonons and in (b) by impurities and vacant lattice sites.

The electrical resistivity of most metals is dominated at room temperature (300 K) by collisions of the conduction electrons with lattice phonons and at liquid helium temperature (4 K) by collisions with impurity atoms and mechanical imperfections in the lattice (Fig. 11). The rates of these collisions are often independent to a good approximation, so that if the electric field were switched off the momentum distribution would relax back to its ground state with the net relaxation time


    \frac{1}{\tau}=\frac{1}{\tau_L}+\frac{1}{\tau_i} ------------------------------(60)

where τL and τ − i are the collision times for scattering by phonons and by imperfections, respectively. The net resistivity is given by


    \boldsymbol{\rho}=\boldsymbol{\rho}_L+\boldsymbol{\rho}_i ----------------------------(61)


where \boldsymbol{\rho}_L is the resistivity caused by the thermal phonons, and \boldsymbol{\rho}_i the resistivity caused by scattering of the electron waves by static defects that disturb the periodicity of the lattice. Often \boldsymbol{\rho}_L is independent of the number of defects when their concentration is small, and often \boldsymbol{\rho}_i is independent of temperature. This empirical observation expresses Matthiessen’s rule.

Although the most characteristic feature of a metal, the electrical resistivity is not at all easy to treat theoretically. The collision time depends on the details of the electron-phonon interaction. One feature that emerges is that τL and τi are different functions of the electron wavevector k . Thus the contributions to the net relaxation time τ(k) are no longer simply additive as in (60). Matthiessen’s rule is not always valid. Calculations of the phonon resistivity \boldsymbol{\rho}_L , have been quite successful for some metals.

MOTION OF CHARGE PARTICLE IN MAGNETIC FIELDS (Hall Effect)

When a magnetic field is applied perpendicular to a conductor carrying current, a voltage is developed across the specimen in the direction perpendicular to both current and the magnetic field. This phenomenon is known as Hall Effect.


Figure 12. The standard geometry for the Hall effect a rod-shaped specimen of rectangular cross-section is placed in a magnetic field , as in (a). An electric field applied across the end electrodes causes an electric current density to flow down the rod. The drift velocity of the electrons immediately after the electric field is applied as shown in (b). The deflection in the direction is caused by the magnetic field. Electrons accumulate on one face of the rod and a positive ion excess is established on the opposite face until, as in (c) the transverse electric field (Hall field) just cancels the force due to the magnetic field.

The magnetic field gives rise to an electric field in a direction mutually orthogonal to the direction of current and magnetic field. The reason for this effect is apparent, when the forces on the current carriers are considered. The electric field Ex which produces the current I , causes a force, of magnitude eEx to act on electron. In the presence of magnetic field, a magnetic force proportional to the magnetic field strength Bz and the electrons average velocity vx also acts on the electron. This force is at right angle to the direction of Bz and vx , and therefore each electron is deflected towards one side of the conductor. When the electron reach the surface of the conductor, an electric charge is built up there, this in turn produces an additional electric field. Under the equilibrium conditions, the sideway force on the moving carriers due to this field is just balance that arising from the magnetic field and the electrons can again move freely down the conductor.

Now the force on the electron in the electromagnetic field in the present condition is given as,

    F_y=-e\left[E_y+\frac{1}{c}(v_x\times B_z)\right] ------------------------------(62)

Hall Voltage and Hall coefficient

In the steady state, the force due to accumulation of electrons becomes equal to the magnetic force and so the flow of electron stops i.e.,Fy = 0


    F_y=0=-e\left[E_y+\frac{1}{c}(v_x\times B_z)\right] -----------------------------(63)


    E_y=\frac{1}{c}v_x B_z ----------------------------------------(64)


Where Ey is called Hall voltage and vx is average drift velocity, therefore, the current density per unit volume can be written as

    j = nevx ---------------------------------------(65)
    v_x=-\frac{j_x}{ne} ------------------------------------(66)

Substituting this value of vx in the Ey we have

    E_y=-\frac{1}{c}\frac{j_x}{ne}B_z --------------------------------(67)

Rearranging the terms in the above equation we have,


    \frac{E_y}{j_x B_z}=R_H=-\frac{1}{nec} ---------------------------------(68)

Where RH is known as Hall coefficient in e.s.u and in e.m.u it will be

    R_H=-\frac{1}{ne} ------------------------------------(69)

The Hall coefficient is determined essentially by the sign of the charge carriers. Means it is to be negative if the conduction is by electron and it is positive if conduction is by holes.

Mobility and Hall angle

When current carrying particles acquire velocity per unit electric field, the velocity is known as mobility and is represented by μ i.e.


    \mu=\frac{v_x}{E_x} ------------------------------------(70)
    v = μEx -----------------------------------------(71)

Substituting this value in equation (64) we have

E_y=\frac{1}{c}\mu E_x B_z=\mu E_x B_z (in emu)

Rearranging the above equation we have

    \mu=\frac{E_y}{E_x}\frac{1}{B_z}=\phi \frac{1}{B_z} ----------------------------(72)

Where \frac{E_y}{E_x}=\phi is the Hall angle, it is also given as,

    φ = μBz --------------------------------------(73)

Importance of Hall Effect:

1. The sign of the charge carriers can be determined

2. The number of charge carriers per unit volume can be calculated from the Hall Coefficient RH .

3. The mobility is measured directly.

4. It can be used to determine the electronic structure of the substance i.e., whether these are metals, insulator or semiconductors.

5. From the Hall voltage we can calculate the applied unknown magnetic field on the substance.

6. It gives the concept of negative mass.

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