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We can understand many physical properties of metals, and not only of the simple metals, in terms of the free electron model. According to this model, the valence electrons of the constituent atoms become conduction electrons and move about freely through the volume of the metal.

Even in metals for which the free electron model works best, the charge distribution of the conduction electrons reflects the strong electrostatic potential of the ion cores. The utility of the free electron model is greatest for properties that depend essentially on the kinetic properties of the conduction electrons. The interaction of the conduction electrons with the ions of the lattice is treated in Chapter 7.

The simplest metals are the alkali metals-lithium, sodium, pot ssium, cesium, and rubidium. In a free atom of sodium the valence electron is in a 3s state; in the metal this electron becomes a conduction electron. We speak of the 3s conduction band.

A monovalent crystal which contains N atoms will have N conduction electrons and N positive ion cores. The Na⁺ ion is 0.98 Å. Whereas one-half of the nearest-neighbor distance of the metal is 1.83 Å.

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. (These are not failures of the free electron model, but failures of the Maxwell distribution function.)

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 10⁸ interatomic spacings (more than 1 cm).

Why is condensed matter so transparent to conduction electrons? The answer to the q. ion 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. (b) A conduction electron is scattered only infrequently by other conduction electrons. This property is a consequence of the Pauli exclusion principle. By a free electron Fermi gas, we mean a gas of free electrons subject to the Pauli principle.

ENERGY LEVELS IN ONE DIMENSION

Consider a free electron gas in one dimension, taking account of quantum theory and of the Pauli principle. An electron of mass m is confined to a length L by infinite barriers (Fig. 2). The wavefunction ψ_a(x) of the electron is a solution of the Schrodinger equation Hψ = Îψ; with the neglect of potential energy we have H = p²|2m. where p os the momentum. In quantum theory p may be represented by –ih d/dx, so that

,

Where Î_n 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 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 ψ_n (0) = 0; ψ_n (L) = 0, as imposed by the infinite potential energy barriers. They are satisfied if the wavefunction is sinelike with an integral number n of half-wavelengths between 0 and L:

,

 ; ,

Where A is a constant. We see that (2) is a solution of (1), because

Whence the energy Î_n is given by

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.

Figure 2 First three energy levels and wavefunctions of a free electron of mass m confined to a line of length L. the energy levels are labeled according to the quantum number n which gives the number of half-wavelengths in the wavefunctios. The wavelengths are indicated on the wavefunctions. The energy Î_n of the level of quantum number n is equal to (h²/2m) (n/2L)².

In a linear solid the quantum numbers of a conduction electron orbital are n and m_s, where n is any positive integer and the magnetic quantum number m_s = ± , according to 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 table:

n	ms	Electron occupancy	n	ms	Electron occupancy
1		1	3		1
1		1	3		1
2		1	4		0
2		1	4		0

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

Let n_F denote the topmost filled energy level, where we start filling the levels from the bottom (n = 1) and continue filling higher levels with electrons until all N electrons are accommodated. It is convenient to suppose that N is an even number. The condition 2n_F = N determines n_F, the value of n for the uppermost filled level.

The Fermi energy Î_F is defined as the energy of the topmost filled level in the ground state of the N electron system. By (3) with n = n_F we have in one dimension:

Provided that the components of the wavevector k satisfy.

and similarly for k_y and k_z.

any component of k is of the form 2np/L, where n is a positive or negative integer. The components of k are the quantum numbers of the problem, along with the quantum number m_s for the spin direction. We confirm that these values of k, satisfy (8) for exp[ik_x(x + L)] = exp[i2np(x + L)/L]

= exp(i2np/L) exp(i2np) = exp(i2np/L) = exp(ik_x)

On substituting (9) in (6) we have the energy Î_k of the orbital with wave-vector k:

 

The magnitude of the wave-vector is related to the wavelength l by k = 2p/l.

The linear momentum p may be represented in quantum mechanics by the operator p = -ihÑ, whence for the orbital (9)

So that the plane wave ψ_k is an eigenfunction of the linear momentum with the eigenvalue h_k. The particle velocity in the orbital k is given by v = hk/m.

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 k_F such that (Fig. 4)

From (10) we see that there is one allowed wave-vector that is, one distinct triplet of quantum numbers k_x, k_y, k_z –for the volume element (2p/L)³ of k space. Thus in the sphere of volume 4pk³_F/3 the total number of orbitals is

Where the factor 2 on the left comes from the two allowed values of m_s, the spin quantum number, for each allowed value of k. Then

Which depends only on the particle concentration.

Using (14)

This relates the Fermi energy to the electron concentration N/V. The electron velocity v_F at the Fermi surface is

Calculated values of k_F, v_F, and Î_F are given in Table 1 for selected metals; also given are values of the quantity T_F which is defined as Î_F / k_B. (The quantity T_F has nothing to do with the temperature of the electron gas!)

Figure 4 in the ground state of a system of N free electrons the occupied orbitals of the system fill a sphere of radius k_F, where Î_F = ħ²k²_F/2m is the energy of an electron having a wavevector k_F.

Figure 5 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(Î,T) D(Î) of filled orbitals at a finite temperature, but such that k_BT is small in comparison with Î_F. 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.

We now find an expression for the number of orbitals per unit  energy range, D(Î), called the density of states.[1] We use (17) for the total number of orbitals of energy £ Î:

So that the density of states (Fig 5) is

This result may be obtained and expressed more simply by writing (19) as

Whence

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

HEAT CAPACITY OF THE ELECTRON GAS

           

The question that caused the greatest difficulty in the early development of the electron theory of metals concerns the heat capacity of the conduction electrons. Classical statistical mechanics predicts that a free particle should have a heat capacity of k_B, where k_B 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 Nk_B, just as for the atoms of a monatomic gas. But the observed electronic contribution at room temperature is usually less than 0.01 of this value.

This discrepancy distracted the early workers, such as Lorentz: how can the electrons participate in electrical conduction processes as if they were mobile, while not contributing to the heat capacity? The question was answered only upon the discovery of the Pauli exclusion principle and the Fermi 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 ~k_BT as expected classically, but only those electrons in orbitals within an energy range  k_BT of the Fermi level are excited thermally; these electrons gain an energy which is itself of the order of  k_BT, as in Fig. 5. 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/T_F can be excited thermally at temperature T, because only these lie within an energy range of the order of k_BT of the top of the energy distribution.

Each of these NT/T_F electrons as a thermal energy of the order of k_BT. The total electronic thermal kinetic energy U is of the order of U = (NT/T_F) k_BT.

The electronic heat capacity is given by

and is directly proportional to T, in agreement with the experimental results discussed in the following section. At room temperature C_el is smaller  than the classical value Nk_B by a factor of the order of 0.01 or less, for T_F ~ 5 x 10⁴ K.

we now derive a quantitative expression for the electronic heat capacity valid at low temperatures k_BT << Î_F. The increase DU @ U(T) – U(0) in the total energy (Fig. 5) of a system of N electrons when heated from 0 to T is

Here f(Î) is the Fermi-Dirac function and D(Î) is the number of orbitals per unit energy range. We multiply the identity.

by Î_F to obtain

We use (26) to rewrite (24) as

The first integral on the right-hand side of (27) gives the energy needed to take electrons from Î_F to the orbitals of energy Î > Î_F, and the second integral gives the energy needed to bring the electrons to Î_F from orbitals below Î_F. Both contributions to the energy are positive.

The product f(Î)D(Î)dÎ in the first integral is the number of electrons elevated to orbitals in the energy range dÎ at an energy Î. The factor [1- f(Î)] in the second integral is the probability that an electron has been removed from an orbital Î. The function DU is plotted in Fig. 6. In Fig. 3 we plotted the Fermi-Dirac distribution function versus Î for six values of the temperature. The electron concentration of the Fermi gas was taken such that Î_F/k_B = 50,000 K, characteristic  of the conduction electrons in a metal.

The heat capacity of the electron gas is found on differentiating DU with respect to T. The only temperature-dependent term in (27) is f(Î), whence we can group terms to obtain

At the temperatures of interest in metals t/Î_F < 0.01, and we see from Fig. 3 that (Î-Î_F) df/dT has large positive peaks at energies near Î_F. It is good

Figure  6 Temperature dependence of the energy of a noninteracting fermion gas in three dimensions. The energy is plotted in normalized form as DU/NÎ_F, where N is the number of electrons. The temperature is plotted as k_BT/Î_F.

Figure 7 Plot of the chemical potential m versus temperature k_BT for a gas of noninteracting fermions in three dimensions. For convenience in plotting, the units of m and k_BT are 0.763 Î_F.

Approximation to evaluate the density of states D(Î) at Î_F and take it outside of the integral:

Examination of the graphs in Figs. 7 and 8 of the variation of m with T suggests that when k_BT << Î_F we ignore the temperature dependence of the chemical potential m in the Fermi-Dirac distribution function and replace m by

Figure 8 Variation with temperature of the chemical potential m, for free electron Fermi gases in one and three dimensions. In common metals t/Î_F = 0.01 at room temperature, so that m is closely equal to Î_F . These curves were calculated from series expansions of the integral for the number of particles in the system.

We set

And it follows from (29) and (30) that

We may safely replace the lower limit by -¥ because the factor e^x in the integrand is already negligible at x = -Î_F/t if we are concerned with low temperatures such that Î_F/t ~ 100 or more. The integral² becomes

²The integral is not elementary, but may be evaluated from the more familiar result

On differentiation of both sides respect to the parameter a.

Figure 9 Experimental heat capacity values for potassium, plotted as C/T versus T². (After W.H. Lien and N. E. Philips)

Whence the heat capacity of an electron gas is

From (21) we have

For a free electron gas with k_BT_F = Î_F. Thus (34) becomes

Recall that although T_F is called the Fermi temperature, it is not an actual temperature, but only a convenient reference notation.

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 electron and phonon contributions: C = gT + AT³, where g and A 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:

C/T = gT + AT²

For then the points should lie on a straight line with slope A and intercept g. Such a plot for potassium is shown in Fig. 9. Observed values of g, called the Sommerfeld parameter, are given in table 2.

The observed values of the coefficient g 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 (34). 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 m_th to the electron mass m, where m_th is defined the relation

This form arises in a natural way because ÎF is inversely proportional to the mass of the electron, whence g ¥ m. Values of the ratio are given in table 2. The departure from unity involves three separate effects:

• 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.
• 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.
• 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.

Heavy Fermions. Several metallic compounds have been discovered that have enormous values, two or three orders of magnitude higher than usual, of the electronic heat capacity. The heavy fermion compounds include. UBe₁₃, CeA_l3, and CeCu₂Si₂. It has been suggested that f electrons in these compounds may have inertial masses as high as 1000 m, because of the weak overlap of wavefunctions of f electrons on neighboring ions (see Chapter 9, “tight binding”). References are given by Z. Fisk, J. L. Smith, and H. R. Ott, physics Today, 38, S-20 (January, 1985). The heavy fermion compound form a class of superconductors known as “exotic superconductors”.

ELECTRICAL CONDUCTIVITY AND OHM’S LAW

The momentum of a free electron is related to the wavevector by mv = ħk. In an electric field E and magnetic field B the force F on an electron of charge –e is –e[E + (1/c)v x B], so that Newton’s second law of motion becomes

(CGS)

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 pbtain

k(t) – k(0) = -eEt/ħ

Figure 10 (a) The Fermi sphere encloses the occupied electron orbitals in k space in the ground state of the electron gas. The net momentum is zero, because for every orbital k there is an occupied orbital at –k. (b) Under the influence of a constant force F acting for a time interval t every orbital has its k vector increased by dk = Ft/ħ. This is equivalent to a displacement of the whole Fermi sphere by dk. The total momentum is Nħdk, if there are N electrons present. The application of the force increases the energy of the system by N(ħdk)²/2m.

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

dk = -eEt/ħ

Notice that the Fermi sphere is displaced as a whole.

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 t, the displacement of the Fermi sphere in the steady state is given by (41) with t = t. The incremental velocity is v = -eEt/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 = nqv = ne²tE/m

this is Ohm’s law.

The electrical conductivity s is defined by j = sE, so that

The electrical resistivity r is defined as the reciprocal of the conductivity, so that

r = m/ne²t

Values of the electrical conductivity and resistivity of the elements are given in table 3. It is useful to remember that s in Gaussian units has the dimensions of frequency.

It is easy to understand the result (43) for the conductivity. We expect the charge transported to be proportional to the charge density ne; the factor e/m enters because the because the acceleration in a given electric field is proportional to e and inversely proportional to the mass m. The time t describes the free time during which the field acts on the carrier. Closely the same result for the electrical  conductivity is obtained for a classical (Maxwellian) gas of electrons, as realized at low carrier concentration in many semiconductor problems. The mathematics of this similarity is developed in the section on transport theory in TP, Chapter 14.

It is possible to obtain crystals of copper so pure that their conductivity at liquid helium temperatures (4 K) is nearly 10⁵ times that at room temperature; for these conditions t = 2 x 10^-9 s at 4 K. The mean free path l of a conduction electron is defined as.

 

Where v_F is the velocity at the Fermi surface, because all collisions involve only electrons near the Fermi surface. From table 1 we have v_F = 1.57 x 10⁸ cm s^-1 for Cu; thus the mean free path is

l(4 K) = 0.3 cm ; l(300 K) = 3 x 10^-6 cm

Mean free paths as long as 10 cm have been observed in very pure metals in the liquid helium temperature range. For electron-electron collisions, see Eq. (10.63).

Experimental Electrical Resistivity of Metals

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

Where t_L and t_i are the collision times for scattering by phonons and by imperfections, respectively

The net resistivity is given by

r = r_L + r_{i ,}

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.

Figure 12 Resistance of potassium below 20 K, as measured on two specimens by D. MacDonald and K. Mendelssohn The different intercepts at 0 K are attributed to different concentrations of impurities and static imperfections in the two specimens.

Where r_L is the resistivity caused by the thermal phonons, and r_i is the resistivity caused by stattering of the electron waves by static defects that disturb the periodicity of the lattice. Often r_L is independent of the number of defects when their concentration is small, and often r_i is independent of temperature. This empirical observation expresses Matthiessen’s rule. Which is convenient in analyzing experimental data (Fig. 12).

The residual resistivity, r_i (0), is the extrapolated resistivity at 0 K because r_L vanishes at T à 0. The lattice resistivity, r_L (T) = r -  r_i(0), is the same for different specimens of a metal, even though r_i (0) may itself vary widely. The resistivity ratio of a specimen is usually defined as the ratio of its resistivity at room temperature to its residual resistivity. It is a convenient approximate indivator of sample purity: for many materials an impurity in solid solution creates a residual resistivity of about 1 mohm-cm, corresponding to an impurity ratio may be as high as 10⁶, whereas in some alloys (e.g., manganin) it is as low as 1.1.

The temperature-dependent part of the electrical resistivity is proportional to the rate at which an electron collides with thermal phonons and thermal electrons (Chapter 10). The collides with phonons is proportional to the concentration of thermal phonons. One simple limit is at temperatures over the Debye temperature q. Here the phonon concentration is proportional to the temperature T, so that r µ T for T > q. A sketch of the theory is given in Appendix J.

Umklapp Scattering

Umklapp scattering of electrons by phonons (Chapter 5) accounts for most of the electrical resistivity of metals at low temperatures. These are electron phonon scattering processes in which a reciprocal lattice vector G is involved, so that electron momentum change in the process may be much larger than in a normal electron-phonon scattering process at low temperatures. (In an umklapp process the wavevector of one particl may be “flipped over”.)

Consider a section perpendicular to [100] through two adjacent Brillouin zones in bcc potassium, with the equivalent Fermi spheres inscribed within each (Fig. 13). The lower half of the figure shows the normal electron-phonon collision k’ = k + q, while the upper half shows  possible scattering process k’ = k + q + G involving the same phonon and terminating outside the first Brillouin zone, at the point A. This point is exactly equivalent to the point A’ inside the original zone, where AA’ is a reciprocal lattice vector G.

This scattering is an umklapp process, in analogy to phonons. Such collisions are strong scatterers because the scattering angle can be close to p, and a single collision can practically restore the electron to its ground orbital.

When the Fermi surface does not intersect the zone boundary, there is some minimum phonon wavevector q_o for umklapp scattering. At low enough temperatures the number of phonons available for umklapp scarring falls as exp (-q_U/T), where q_U is a characteristic temperature calculable from the geometry of the Fermi surface inside the Brillouin zone. For a spherical Fermi surface with one electron orbital per atom inside the bcc Brillouin zone, one can show by geometry that q_o = 0.267 k_F.

Figure 13 Two Fermi spheres in adjacent zones: a construction to show the role of phonon umklapp processes in electrical resistivity.

The experimental data for potassium have the expected exponential form with q_U = 23 K compared with Debye q = 91 K. At the very lowest temperatures (below about 2 K in potassium) the number of umklapp processes is negligible and the lattice resistivity is then caused obly by small angle scattering, which is the normal scattering.

Bloch obtained an analytic resulf for the normal scattering, with rL µ T⁵/q⁶ at very low temperatures. This is a classic limiting result. These normal processes contribute to the resistivity in all metals, but they have not yet been clearly isolated for any metal because of the large competing effects of imperfection scattering, electron-electron scattering, and umklapp scattering.

MOTION IN MAGNETIC FIELDS

By the argument of (39) and (41) we are led to the equation of motion for the displacement dk of a Fermi sphere of particles acted on by a force F and by friction as represented by collisions :

The free particle acceleration term is (ħd/dt) dk and the effect of collisions (the friction) is represented by ħdk/t, where t is the collision time.

Consider now the motion of the system in uniform magnetic field B. The Lorentz force on an electron is

If mv = ħdk, then the equation of motion is

An important situation is the following: let a static magnetic field B lie along the z axis. Then the equations of motion are

The results in SI are obtained by replacing c by 1.

In the steady state in a static electric field the time derivatives are zero, so that the drift velocity is

Where w_c = eB/mc is the cyclotron frequency, as discussed in Chapter 8 for cyclotron resonance in semiconductors.

Hall effect

The hall field is the electric field developed across two faces of a conductor, in the direction j x B, when a current j flows across a magnetic field B. Consider a rod-shaped specimen in a longitudinal electric field E_x, and a transverse magnetic field, as in Fig. 14. If current cannot flow out of the rod in the y direction we must have dv_y = 0. From (52) this is possible only if there is a transverse electric field

is called the Hall coefficient. To evaluate it on our simple model we use j_x = ne2tE_x/m and obtain

Figure 14 The standard geometry for the Hall effect; a rod-shaped specimen of rectangular cross section is placed in a magnetic field Bz as in (a) An electric field E_x, applied across the end electrodes causes an electric current density ¦_x to flow down the rod. The drift velocity of the negatively-charged electrons immediately after the electric field is applied as shown in (b). The deflection in the y 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 Lorentz force due to the magnetic field.

This is negative for free electrons, for e is potitive by definition.

The lower the carrier concentration, the greater the magnitude of the Hall coefficient. Measuring R_H is an important way of measuring the carrier concentration.

The symbol R_H denotes the Hall coefficient (54), but it is sometimes used with a different meaning, that of Hall resistance in two-dimensional problems. When we treat such problems in Chapter 19, we shall instead let

r_H = BR_H  = E_y/j_x

denote the Hall resistence, where j_x is the surface current density.

The simple result (55) follows from the assumption that all relaxation times are equal, independent of the velocity of the electron. A numerical factor of order unity enters if the relaxation time is a function of the velocity. The expression becomes somewhat more complicated if both electrons and holes contribute to the conductivity. The theory of the Hall effect again becomes simple in high magnetic fields such that w_ct >> 1, where w_c is the cyclotron frequency and t the relaxation time. (See QTS, pp. 241-244).

In table 4 observed values of the Hall coefficient are compared with values calculated from the carrier concentration. The most accurate measurements are made by the method of helicon resonance which is treated as a problem in Chapter 10. In the table “conv.” Stands for “conventional.”

The accurate values for sodium and potassium are in excellent agreement with values calculated for one conduction electron per atom, using (55). Notice, however, the experimental values for the trivalent elements aluminum and indium; these agree with values calculated for one positive charge carrier per atom and thus disagree in magnitude and sign with values calculated for the expected three negative chare carriers.

The problem of an apparent positive sign for the charge carriers arises also for be and As, as seen in the table. The anomaly of the sign was explained by Peiers (1928). The motion of carriers of apparent positive sign, which Heisen berg later called “holes”, cannot be explained by a free electron gas, but finds a natural explanation in terms of the energy band theory developed in Chapter 7-9. Band theory also accounts for the occurrence of very large values of the Hall coefficient, as for As, Sb, and Bi.

THERMAL CONDUCTIVITY METALS

In Chapter 5 we found an expression L = Cvl for the thermal conductivity of particles of velocity v, heat capacity C per unit volume, and mean free path l. The thermal conductivity of a Fermi gas follows from (36) for the heat capacity, and with

Here l = v_Ft; the electron concentration is n, and t is the collision time.

Do the electrons or the phonons carry the greater part of the heat current in a mental? In pure metals the electronic contribution is dominant at all temperatures. In impure metals or in disordered alloys, the electron mean free path is reduced by collision with impurities, and the phonon contribution may be comparable with the electronic contribution.

Ratio of Thermal to Electrical Conductivity

The Wiedemann-Franz law states that for metals at not too low temperatures the ratio of the thermal conductivity to the electrical conductivity is directly proportional to the temperature, with the value of the constant of proportionality independent of the particular metal. This result was important in the history of the theory of metals, for it supported the pictures of an electron gas. It can be explained by using (43) for s and (56) for K

The Lorenz number L is defined as

And according to (57) should have the value

This remarkable result involves neither n nor m. It does not involve t if he relaxation times are identical for electrical and thermal processes. Experimental values of L at 0^oC and at 100^oC as given in table 5 are in good agreement with (59). At low temperature (T<<q) the Lorenz number tends to decrease; see the book by Ziman.

NANOSTRUCTURES

The term nanostructure denotes a condensed matter structure having a minimum dimension approximately between 1 nm (10 Å) and nm (100 Å). These structures may be fine particles, fine wires, or thin films. Fine particles typically contain between 10 and 1,000 atoms. Semiconductor technology (see section “superlattices” in Chapter 8) has made it possible to fabricate small pools of electrons called in various ways: single-electron transistors, quantum dots, artificial atoms, Coulomb islands, or quantum corrals (Chapter 19). The unusual physical properties of nanostructures compared with bulk solids are attributed to several factors involving properties treated above and in later chapters:

• The ratio of the number of atoms on the surface to the number of atoms in the interior may be of the order of unity.
• The ratio of surface energy to total energy may be of the order of unity.
• The conduction or valence electrons are confined to a small length or volume, so that the quantum wavelength of the lowest electronic state is constricted and consequently the minimum wavelength is shorter than in the bulk solid.
• A wavelength or boundary condition shift will change the optical absorption spectrum (Chapter 11).
• Assemblies of nanoclusters of metals may have great hardness and yield strength, because it is difficult to create and to move dislocations (Chapter 20) in spatially confined regions.
• Magnetic monolayers, as of alternating films of ferromagnetic iron and of paramagnetic chromium, may have the magnetization (Chapter 15) of the iron films coupled by tunneling of the magnetization through the chromium barriers.

Problems

1. Kinetic energy of electron gas. Show that the kinetic energy of a three-dimensional gas of N free electrons at 0 K is

2. Pressure and bulk modulus of an electron gas. (a) Derive a relation connecting the pressure and volume of an electron gas at 0 K. Hint : Use the result of problem 1 and the relation between Î_F and electron concentration. The result may be written as p = (U_o/V). (b) Show that the bulk modulus B = -V(¶p/¶V) of an electron gas at 0 K is B = 5p/3 = 100 U_o/9V. (c) Estimate for potassium, using table 1, the value of the electron gas contribution to B.
3. Chemical potential in two dimensions. Show that the chemical potential of a Fermi gas in two dimensions is given by :

For n electrons per unit area. Note: The density of orbitals of a free electron gas in two dimensions is independent of energy: D(Î) = m/pħ², per unit area of specimen.

4. Fermi gases in astrophysics. (a) Given M_¤ = 2 x 10³³ g for the mass of the Sun, estimate the number of electrons in the Sun. in a white dwarf star this number of electrons may be ionized and contained in a sphere of radius 2 x 10⁹ cm; find the Fermi energy of the electrons in electron volts. (b) The energy of an electron in the relativistic limit Î >> mc² is related to the wavevector as Î = pc = ħkc. Show that the Fermi energy in this limit is Î_F = ħc(N/V)^1/3, roughly. (c) If the above number of electrons were contained within a pulsar of radius 10 km, show that the Fermi energy would be = 10⁸ eV. This value explains why pulsars are believed to be composed largely of neutrons rather than of protons and electrons, for the energy release in the reaction n à p + e^- is only 0.8 x 10⁶ eV, which is not large enough to enable many electrons to for a Fermi sea. The neutron decay proceeds only until the electron concentration builds up enough to create a Fermi level 0.8 x 10⁶ eV, at which point the neutron, proton, and electron concentrations are in equilibrium.
5. Liquid He³. The atom He³ has spin  and is a fermion. The density of liquid is He³ 0.081 g cm^-3 near absolute zero. Calculate the Fermi energy Î_F and the Fermi temperature T_F.

The quantity defined by

1. Frequency dependence of the electrical conductivity. Use the equation m(dv/dt + v/t) = -eE for the electron drift velocity v to show that the conductivity at frequency w is

Where s (0) = net/m.

2. Dynamic magnetoconductivity tensor for free electrons. A metal with a concentration n of free electrons of charge –e is in a static magnetic field Bz. The electric current density in the xy plane is related to the electric field by

Where = 4 pne²/m. (b) Note from a Maxwell equation that the dielectric function tensor of the medium is related to the conductivity tensor as Î = 1 + i (4p/w)s. Consider an electromagnetic wave with wavevector k = kz. Show that the dispersion relation for this wave in the medium is

At a given frequency there are two modes of propagation with different wavevectors and different velocities. The two modes correspond to circularly polarized waves. Because a linearly polarized wave van be decomposed into two circularly polarized waves, it follows that the plane of polarization of a linearly polarized wave will be rotated by the magnetic field.

3. Cohesive energy of free electron Fermi gas. We define the dimensionless length r_s as r_o/a_H. Where r_o is the radius of a sphere that contains one electron, and a_H is the Bohr radius ħ²/e²m, (a) Show that the average kinetic energy per electron in a free electron Fermi gas at 0 K is 2.21/r²_s, where the energy is expressed in rydbergs, with 1 Ry = me²/2ħ². (b) Show that the coulomb energy of a point positive charge e interacting with the uniform electron distribution of one electron in the volume of radius r_o is -3e²/2r_o, or -3/r_s in rydbergs. (c) Show that the coulomb self-energy of the electron distribution in the sphere is 3e²/5r_o, or 6/5r_s in rydbergs. (d) The sum of (b) and (c) gives -1.80/r₂ for the total coulomb energy per electron. Show that the equilibrium value of r_s is 2.45. Will such a metal be stable with respect to separated H atoms?
4. Static magnetoconductivity tensor. For the drift velocity theory of (51), show that the static current density can be written in matrix form as

In the high magnetic field limit of w_ct >> 1, show that

In his limit s_yx = 0, to order 1/w_ct. The quantity s_yx is called the Hall conductivity.

5. Maximum surface resistance. Consider a square sheet of side L, thickness d, and electrical resistivity r. The resistance measured between opposite edges of the sheet is called the surface resistance: R_sq = rL/Ld = r/d, which is independent of the area L² of the sheet. (R_sq is called the resistance per square and is expressed in ohms per square, because r/d has the dimensions o ohms.) If we express r by (44), then R_sq = m/nde²t. Suppose now that the minimum value of the collision time is determined by scattering from the surfaces of the sheet, so that t » d/v_F,  where v_F is the Fermi velocity. Thus the maximum surface resistivity is R_sq » mv_F/nd²e². Show for a monatomic metal sheet one atom in thickness that R_sq » ħ/ e² = 4.1 kW, where 1 kW is 10³ ohms.
6. Small metal spheres. Consider free electrons in a spherical square well potential of radius a, with infinitely high boundary. (a) Show that the wave function of an orbital of angular momentum l and projection m has the form

Where the radial wave function has the form

And Y is a spherical harmonic. Here J is a Bessel function of half-integral order and satisfies the boundary condition J_l+1/z(ka) = 0. The roots given the energy eigenvalues Î of the levels above the bottom of the well, where Î = ħ²k²/2m. (b) Show that the order of the levels above the ground orbital is.

Where s, p, d, f, g, h denote l = 0, 1, 2, 3, 4, 5.

7. Density of states – nanometric wire. (a) Consider a nanometric wire in the form of a rectangular parallelepiped, with tow sides L_x » L_y » 1 nm and the long axis L_z = 1 cm. The single particle eigenstates of the system may be written as

The energy of the eigenstate is, with n = n_x, n_y

Where v is the electron velocity along the z-axis. Here A = (2pħ)²/2m and N = . Then dE = 2ANdN, show that the density of states D_n at fixed n, with account of the two spin orientations and the two ± values of N, is

(b) Sum over the values of n for which E ³ Î_F to show that

Where q (x) is the Heaviside unit function, zero for x < 0 and unity for x > 0.

8. Quantization of conductance. The current in the nanometric wire of problem 12 is I = (N₊ - N_-)ev_F, where N₊ - N_- = D(E_F)eV, where V is the bias voltage. Show that the current may be written as

Where n_occ is the number of occupied states n_x, n_y, hence the quantized conductance is (2e2/pħ2) n_occ

REFERENCES

D.N. Langenberg, Resource letter OEPM1 on the ordinary electronic properties of metals,” Amer. J. Phys. 36, 777 (1968), An excellent early bibliography on transport effects, anomalous skin effect, Azbel-Kaner cyclotron resonance, magnetoplasma waves, size effects, conduction electron spin resonance, optical spectra and photoemission, quantum oscillations, magnetic breakdown, ultrasonic effects, and the Kohn effect.

G.T. Meaden, Electrical resistance of metals, Plenum, New York, 1965.

C. M. Hurd, The Hall effect in metals and alloys, Plenum, 1972.

C. L. Chien and C. R. Westgate, eds., Hall effect and its applications, Plenum, 1980.

C. M. Hurd, Electrons in metals, Wiley, 1975.

S. Kagoshima, et al., One-dimensional conductors, Springer, 1988.

J. Ziman, Electrons and phonons, Oxford, 1960.

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