Electrical Conductivity in Metals V H Satheeshkumar Department of Physics and Center for Advanced Research and Development Sri Bhagawan Mahaveer Jain College of Engineering Jain Global Campus, Kanakapura Road Bangalore 562 112, India. [email protected] November 22, 2008

Who can use this? The lecture notes are tailor-made for my students at SBMJCE, Bangalore. It is the third of eight chapters in Engineering Physics [06PHY12] course prescribed by VTU for the first-semester (September 2008 - January 2009) BE students of all branches. Any student interested in exploring more about the course may visit the course homepage at www.satheesh.bigbig.com/EnggPhy. For those who are looking for the economy of studying this: this chapter is worth 20 marks in the final exam! Cheers ;-)

Syllabus as prescribed by VTU Free-electron concept; Classical free-electron theory, Assumptions; Drift velocity, Mean collision time, Relaxation time and Mean free path; Expression for drift velocity. Expression for electrical conductivity in metals. Effect of impurity and temperature on electrical resistivity of metals. Failure of classical free-electron theory. Quantum free-electron theory. Fermi - Dirac Statistics. Fermi-energy. Fermi factor. Density of states (with derivation). Expression for electrical resistivity / conductivity. Temperature dependence of resistivity of metals. Merits of Quantum free electron theory.

Reference • Leonid Azaroff, Introduction to Solids, TMH Edition, Tata McGraw-Hill Publishing Company Limited, ISBN- 0-07-099219-3. ———————————— This document is typeset in Free Software LATEX2e distributed under the terms of the GNU General Public License.

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Introduction

The fact that electricity can flow through a substance was discovered by 17th century German physicist Otto von Guericke. Conduction was rediscovered independently by Englishman Stephen Gray during the early 1700s. Gray also noted that some substances are good conductors while others are insulators. The electron theory, which is the basis of modern electrical theory, was first advanced by Dutch physicist Hendrik Antoon Lorentz in 1892. The widespread use of electricity as a source of power is largely due to the work of pioneering American engineers and inventors such as Thomas Alva Edison, Nikola Tesla, and Charles Proteus Steinmetz during the late 19th and early 20th centuries. Thanks to our understanding of how electric conduction happens, now it plays a part in nearly every aspect of modern technology. In this chapter, we discuss the theory of conduction, specifically the theory of classical conduction whose defects were explained by the quantum theory. The modifications that the quantum theory adds to classical conduction not only explains the flaws that arose in the classical theory, but also adds a new dimension to conduction that is currently leading to new developments in the physics world.

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Classical free-electron theory

Around 1900, Paul Drude improved the theory of classical conduction given by Lorentz. He reasoned that since metals conduct electricity so well, they must contain free electrons that move through a lattice of positive ions. This motion of electrons led to the formation of Ohm’s law. The free-moving electrons act just as a gas would; moving in every direction throughout the lattice. These electrons collide with the lattice ions as they move about, which is key in understanding thermal equilibrium. The average velocity due to the thermal energy is zero since the electrons are going in every direction. There is a way of affecting this free motion of electrons, which is by use of an electric field. This process is known as electrical conduction and theory is called Drude-Lorentz theory. The assumptions of the Drude-Lorentz classical theory of free-electrons are the following. • Metals contain free electrons that move through a lattice of positive ions. These free electrons are responsible for electrical conduction when an electric potential is maintained across the conductor. • Electric field produced by lattice ions is considered to be uniform throughout the solid and hence neglected. • The force of repulsion between the electrons and force of attraction between electrons and lattice ions is neglected. • Free electrons in a metal resemble molecules of a gas and therefore the laws of kinetic theory of gases are applicable to free electrons. The motion of an electron is completely random. In the absence of electric field, number of electrons crossing any cross section of a conductor in one direction is equal to number of electrons crossing the same cross section in opposite direction. Therefore net electric current is Zero. This random motion of the electrons is due to thermal energy. Hence, the average kinetic energy of the electron is given by 3 1 2 me vth = kT, 2 2 where, me is the mass of the electron, vth is the thermal velocity, k is the Boltzmann constant and T is the absolute temperature. Therefore, the thermal velocity of free-electrons in a metal at given temperature is given by r 3kT vth = . me • Electric current in the conductor is due to the drift velocity acquired by the electrons in the presence of the applied electric field.

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Drift velocity, Mobility, Mean collision time, Relaxation time and Mean free path

An electric field provides a potential difference along a wire of electrons, which creates a force, F = eE, where e is the charge of an electron and E is the magnitude of the electric field. That force accelerates the electrons, as expected by Newtons second law F = me a. The electrons are given a velocity away from the field, which leads to these collisions with the fixed ions. The collisions rid the electrons of their kinetic energy momentarily, transferring that energy to the ion lattice in the form of heat. As a result of this the conduction electrons move with constant velocity. This is called the drift velocity , usually denoted by vd . We get the expression for vd by equating the above two equations of force, me a = eE. The acceleration can be written as vd /τ ,

me vd = eE, τ eEτ vd = . me In the above expression τ is called the mean collision time. It is the average time taken by an electron between two successive collisions during its random motion. The mobility of electrons is defined as the magnitude of the drift velocity acquired by the electrons in unit electric field. That is vd µ= . E Using the expression for drift velocity in the above expression, we get µ=

1 eEτ · , E me

eτ . me When an electric field is applied, the electrons move with drift velocity vd . If the electric field is switched off, the drift velocity decays exponentially to zero after some time and the electrons will be moving with velocity v0 only because of thermal agitation. The decay process follows the equation, µ=

v0 = vd e−t/τr , where t is the time counted from the instant the field is turned off and τr is called the relaxation time. If t = τr , v0 = vd e−1 , 1 v0 = vd . e Thus the relaxation time is defined as the time during which drift velocity reduces to 1/e times its maximum value after the electric filed is switched off. The relationship between relaxation time and mean collision time is given by τ τr = , 1− < cosθ > where θ is the scattering angle and cosθ is the average value of cosθ taken over very large number of collisions made by electrons. In most of the situations < cosθ >= 0 and hence τr = τ , that means that the mean collision time is same as the relaxation time. The average distance traveled by electrons between two successive collisions during their random motion is called mean free path, denoted by λ.

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Expression for electrical conductivity in metals

Consider a conductor of length L and area of cross section A. The electric current I is proportional to the voltage drop V across the conductor, which is called Ohms Law and is given by V = IR. We rewrite the above equation, V , R I V = , A AR I L V = . A AR L We identify the above quantities as current density J = I/A, electrical conductivity σ = L/RA and electric field E = V /L. Then, J = σE, I=

which is the Ohm’s law in a general form. Since our aim is to find the expression for the electrical conductivity, we use the above equation in the following form J σ= . E Substituting the definition of J back in the above equation, we get σ=

I . AE

Now, we need to find the expression for current I in the conductor. Let n be the number of electrons per unit volume of the conductor and vd be the drift velocity of electrons with charge e. Then, the number of electrons crossing any cross section per unit time is n(vd A). Therefore the current passing through any cross section of the conductor is given by I = nevd A Plugging this expression into that of conductivity, we get σ=

nevd A , AE

nevd . E in the above equation, we have σ=

Using the equation of drift velocity vd =

eEτ me

σ=

ne eEτ · , E me

ne2 τ σ= . me This is the expression of electrical conductivity that we were looking for.

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Effect of impurity and temperature on electrical resistivity of metals

We know that the electrical conductivity of a metal is given by σ=

ne2 τ . me

We write τ in terms of thermal velocity vt h and mean free path λ τ= substituting for vt h =

q

3kT , me

λ , vt h

we get r

λ τ=q

=λ

3kT me

me . 3kT

Plugging τ into the expression for conductivity, we get r ne2 me σ= λ , me 3kT σ=√ Hence, resistivity is given by

ne2 λ . 3me kT

√

3me kT . ne2 λ This equation suggests that the resistivity of a metal must be directly proportional to the square root of temperature. The resistivity of metals is attributed to the scattering of conduction electrons. The scattering of electrons takes place because of two reasons: one due collisions of conduction electrons with the vibrating lattice ions and the other is caused by scattering of electrons by the impurities present in the metal. The resistivity due to scattering of electrons by the lattice vibrations called phonons is denoted by ρp . This increases with temperature. It arises even in a pure conductor and hence called the ideal resistivity. Whereas the resistivity of metals caused by scattering of electrons with the impurities is denoted by ρi . This is independent of temperature and present even at absolute zero of temperature and hence called residual resistivity. Therefore, the total resistivity of a metal can be written as the sum of the two resistivities. This is called Matthiessen’s rule. Mathematically, ρ=

ρ = ρp + ρi Since ρ =

me , we can rewrite the above equation in the following form ne2 τ ρ=

me me + 2 , 2 ne τp ne τi

where τp and τi are the mean collision times of electrons with phonons and impurities respectively. At lower temperatures ρp tends zero as the amplitude of lattice vibrations becomes small which essentially means that all the resistivity will be due to impurities, i.e., ρ = ρi . At higher temperatures ρp increases with temperature however the curve of ρ versus T at room temperature remains more or less linear as shown in the following figure.

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Failure of classical free-electron theory

While the classical understanding of conduction is useful in constructing Ohms law and providing an understanding of the motion of electrons, there is a number of inherent flaws in this theory. 1. From the equation for resistivity, we have √ ρ=

3me kT , ne2 λ

which means that the resistivity of a metal must be directly proportional to the square root of temperature. But it is observed by experiments that resistivity has a linear relationship with temperature. Furthermore, the above stated equation will give a value that is about seven times the measured value of resistivity at a temperature of 300 K. 2. According to the classical theory, the molar heat capacity of free electrons in a metal is However, the experimentally determined molar heat capacity of metals is 10−4 RT .

3 R. 2

3. As seen from the expression for electrical conductivity, σ=

ne2 τ , me

the conductivity is directly proportional to the electron density n. Hence, divalent and trivalent metals should possess much higher electrical conductivity than monovalent metals. This is contrary to the experimental observation that silver and copper are more conducting than zinc and aluminum. 4. There are some flaws from a statistical perspective as well. Applying kinetic theory of gases to 3 the electrons will give an average kinetic energy of kT . But the observed kinetic energy is kT. 2 5. According to quantum mechanics, it is known that electrons share wave-like properties as well, and the classical theory makes no mention of such properties.

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Quantum free-electron theory

By altering the classical theory through the application of the wave properties of electrons, the quantum theory of conduction was formed by Arnold Sommerfeld in late 1920s. It is with this theory that the physics world is constantly moving forward by determining new utilizations of the scattering of electronwave properties throughout a material. The underlying assumptions of the theory are the following. • The electrons can have only discrete values of energy. Those allowed energies for the electron are called the energy levels. • The electrons are distributed among the energy levels according to Pauli’s exclusion principle, which states that there cannot be more than two electrons in any given energy level. • The electrons experience a constant electric potential due to ions located at the lattice points and remain within the solid. • The electron-electron and electron-lattice ion interactions are neglected.

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Density of states

The energy levels of electrons in the case of single atom are sharp as we know in the case of Hydrogen atom. Whereas in the case of a solid, energy levels of electrons will spread out over a range called energy band due to presence of huge number of atoms. Each energy band consists of a number of closely spaced energy levels. To describe the number of states at each energy level that are available to be occupied by the electrons, we introduce the concept of the density of states of a system . The density of states, g(E), is defined as the number of energy levels available per unit volume per unit energy centered at E. The number of states per unit volume between the energy levels E and E + dE is denoted by g(E)dE. From the discussion of an electron in one-dimensional potential well of width L, we know that the allowed energies are given by En =

n 2 π 2 ~2 where n = 1, 2, 3, ..... 2mL2

or

n2 h2 . 8mL2 Since the free-electrons in a solid experience a three dimensional potential well, the above equation takes the form h2 (n2 + n2y + n2z ), E= 8mL2 x where nx ,ny and nz are non-zero positive numbers. Each set of (nx , ny , nz ) indicates the permitted energy vale. Taking h2 E0 = 8mL2 and R2 = n2x + n2y + n2z En =

the above equation will be reduced to E = E0 R2 . The equation R2 = n2x + n2y + n2z represents a sphere of radius R formed by the points (nx , ny , nz ) with nx ,ny and nz as the three mutually perpendicular coordinate axes. Since nx ,ny and nz can take only 1 the positive integers, the above equation represents only of the sphere called the octant as shown in 8 the diagram.

The number of allowed energy values N (E) in a small energy range between E and E + dE is equal 1 th to the product of the of the volume of the sphere between the shells of radius R and R + dr, and 8

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the number of points per unit volume which is 1. Mathematically, 1 N (E)dE = 4πR2 dR × 1, 8 1 N (E)dE = πR2 dR. 2 Since, each such energy value can accommodate two states of electrons according to Pauli’s exclusion principle, the number of allowed energy states between E and E + dE is, 1 N (E)dE = 2 × πR2 dR, 2 N (E)dE = πR2 dR. In the above equation R is abstract quantity which we cannot measure, so we wish to express the entire right hand side in terms of energies using the equation E = E0 R2 . We can express R as r E =R E0 and differentiating it, we have dE = 2E0 R dR. Now, multiplying the above two expressions give us r E dE = 2E0 R2 dR, E0 or 1 R2 dR = 2

s

E dE. E03

Substituting this in the expression for N (E)dE, we get s π E N (E)dE = dE, 2 E03 Using E0 =

h2 8mL2

in the above equation, π N (E)dE = 2

further simplification gives N (E)dE =

s

E h2 3 ( 8mL 2)

dE,

! √ 8 2m3 πL3 √ E dE. h3

In this equation L3 represents the volume of the solid and since the density of states is the number of energy states per unit volume, N (E)dE g(E)dE = , L3 hence the equation ! √ 8 2m3 π √ g(E)dE = E dE. h3 The plot of g(E) versus E is shown below.

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Fermi-energy and Fermi factor

In a single atom there will be many allowed energy levels whereas in a solid each such energy level will spread over a range of few eV 0 s. If there are N number of atoms, there will be N closely spaced energy levels in each energy band of the solid. According to Pauli’s exclusion principle, each such energy level can accommodate two electrons. At absolute zero temperature, two electrons with least energy with opposite spins occupy the lowest available energy level. The next two electrons with opposite spins will occupy next energy level and so on. Thus, the top most energy level occupied by electrons at absolute zero temperature is called Fermi energy level. The energy corresponding to that energy level is called Fermi energy. Fermi energy, EF , is defined as the energy at absolute zero corresponding to the highest filled energy level, below which all energy levels are completely occupied and above which all the energy levels completely empty. Thus Fermi energy represents maximum energy that electrons can have at absolute zero temperature. At absolute zero all energy levels below Fermi energy are completely filled and above it are completely empty. But at any given temperature, the electrons get thermally excited and move up to higher energy levels. As a result there will be many vacant energy levels below as well as above Fermi energy level. Under thermal equilibrium, the distribution of electrons among various energy levels is given by statistical function f (E). The function f (E) is called the Fermi factor and this gives the probability of occupation of a given energy level under thermal equilibrium. The expression for f (E) is, f (E) = 3.2.1

1 e(E−Ef )/kT

+1

.

Variation of Fermi factor with energy and temperature

Variation of Fermi factor with energy and temperature is discussed below. Case 1: For T = 0K and E < EF The Fermi factor is given by f (E) =

1 e(E−Ef )/kT

+1

.

For T = 0K and E < EF , the above expression becomes f (E) =

1 e−∞

+1

=

1 0+1

f (E) = 1. This implies that at absolute zero, all the energy levels below EF are 100% occupied which is true from the definition Fermi energy.

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Case 2: For T = 0K and E > EF The Fermi factor is given by 1

f (E) =

e(E−Ef )/kT

+1

.

For T = 0K and E > EF , the above expression becomes f (E) =

e∞

1 1 = +1 ∞

f (E) = 0. This implies that at absolute zero, all the energy levels above EF are unoccupied which is true from the definition Fermi energy. Case 3: For T > 0K and any E At ordinary temperatures for E = EF , we get f (E) =

1 e(E−Ef )/kT

+1

=

e0

1 1 .= +1 2

For E  EF , the probability starts decreasing from 1 and reaches 0.5 at E = EF , and for E > EF , it further falls off as shown in the figure below.

In conclusion, the Fermi energy is the most probable or the average energy of the electrons in a solid. 3.2.2

Variation of Fermi energy with temperature

We want to find out how the Fermi energy varies with temperature. To achieve this, we consider the following. The number of electrons per unit volume of the solid, n, is nothing but the product of density of states available and the probability of occupancy of electrons among various energy levels up to the Fermi level EF , that is, Z EF

f (E) · g(E) dE.

n= 0

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Case 1: For T = 0K We know that, for T = 0K, we have f (E) = 1 and the Fermi energy for such a case is taken as EF0 . Hence the equation of n takes the form Z EF 0 n= g(E) dE. 0

Substituting for density of states g(E) dE, we have ! √ Z EF 3 0 8 2m π √ E dE, n= h3 0 !Z √ EF0 √ 8 2m3 π n= E dE, h3 0 !  √ E 2 3/2 F0 8 2m3 π × E , n= h3 3 0 !  √  2 8 2m3 π 3/2 n= × EF0 , h3 3 Further simplification gives us EF0

3/2

=

3h3 (8m)3/2 π

h2 8m

  2/3 3 n2/3 π



or  EF0 =

 n

Case 1: For T > 0 K We know that, for T > 0 K, we have f (E) 6= 1 and the Fermi energy for such a case is taken as EF , which is given by "  2 # π 2 kT EF = EF0 1 − . 12 EF0 For smaller temperatures, the second term of the above equation vanishes giving, EF = EF0 , implying that at ordinary temperatures EF and EF0 essentially the same.

3.3

Fermi - Dirac Statistics

A metal has very large number of free electrons and these electrons are distributed among various energy levels in the energy bands. The statistics which governs how the free electrons are distributed among various energy levels is called Fermi-Dirac statistics. This obeys Pauli’s exclusion principle 1 and is applicable to any indistinguishable particles of spin . The distribution of electrons among 2 various available energy levels according Fermi-Dirac statistics under thermal equilibrium is called Fermi-Dirac distribution. Mathematically N (E) dE = f (E) × g(E) dE, where N (E) dE is the number of electrons in unit volume possessing the energies between E and E +dE. The plot of N (E) dE versus E is shown in the following figure for various temperatures.

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3.4

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Expression for electrical resistivity / conductivity

Using the concepts of density of states and Fermi-Dirac statistics Sommerfeld arrived at the following expression for electrical conductivity in metals, σ=

ne2 λ , m ∗ vF

where λ is the mean free path; m∗ is the effective mass and vF is called the Fermi velocity. The Fermi velocity can be found out by equating the Fermi energy to the kinetic energy of the electrons in a metal. That is, 1 2 mv = EF , 2 F r 2EF vF = . m The resistivity of the metal is given by 1 ρ= . σ Therefore, m∗ vF ρ= . ne2 λ

3.5

Merits of Quantum free electron theory

The quantum theory of free electrons solves the flaws of the classical theory which is discussed below. 1. Temperature dependence of resistivity of metals: The resistivity of a metal is given by ρ=

m ∗ vF . ne2 λ

In the above expression only the mean free path λ is the temperature dependent quantity. In the classical theory, the collision was seen as a particle bouncing off another. In the quantum understanding, an electron is viewed as a wave traveling through a medium. If r represents the amplitude of the oscillation of the lattice ions, then the λ is inversely proportional to the area of cross section, i.e., 1 λ ∝ 2. πr But the area of cross section is directly proportional to the absolute temperature, i.e., r2 ∝ T.

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Therefore, λ∝

1 . T

Hence ρ ∝ T, which is exactly the same as the experimental prediction. 2. Specific heat of free electrons: From quantum theory of free electrons, the specific heat of free electrons is given by 2k RT. Cv = EF For a typical value of EF = 5 eV , we get Cv = 10−4 RT, which is in agreement with the experimental results. 3. Dependence of electrical conductivity on electron concentration: The electrical conductivity in metals is given by e2 λ σ = ∗ n. m vF It is clear from this equation that the electrical conductivity depends both the electron concentration and vλF . Now by taking above expression, if we calculate the σ, we get the observed conductivities of the metals. ***