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Transcript
___________________________________________________________ UNIT 1 TRANSPORT PROPERTIES Structure 1.0 Introduction 1.1 Objectives 1.2 The Boltzman equation 1.3 Electrical conductivity 1.4 1.3.1 Factors affecting electrical conductivity 1.3.2 Calculation of relaxation time 1.3.3 Impurity scattering 1.3.4 Ideal resistance 1.3.5 Carrier mobility General transport coefficients 1,4.1 Thermal conductivity 1.4.2 Thermoelectric effects 1.4.3 Lattice conduction & Phonon drag 1.5 The Hall effect 1.6 The two band model 1.7 Magneto resistance 1.8 Let Us Sum Up 1.9 Check Your Progress: The key _____________________________________________________________________ 1.0 INTRODUCTION We have considered in the free electron theory of metals in which free electrons were treated as an ideal gas of free particles which when in thermal equilibrium, obey Maxwell-Boltzmann statistics. Now the problem is that what happens when a nonequilibrium state is established by allowing electric or thermal currents to flow, i.e., the problem is to investigate how the equilibrium distribution would be modified by small electric or thermal current. It is also necessary to consider the kinetic behavior of the electrons as being that of free particles subjected to instantaneous collisions which serve to return the distribution to equilibrium condition, and to express the final result for electrical and thermal conductivity in terms of mean free path between these randomizing collisions. In a state of steady flow of heat or electricity, the distribution function of velocity components and spatial coordinates of the electrons will be Transport Properties, Semiconductor Crystals and Superconductivity different from that in thermal equilibrium in the absence of flow. The theory of transport phenomena is concerned with determining this distribution function for given external fields. In the calculation of this distribution function, two new, features appears which are of no interest in thermal equilibrium. The first feature is that the external fields accelerate the particles. This acceleration will be reflected in a change of the distribution function away from its equilibrium value. The second feature is that the effect of certain terms in the Hamiltonian of the crystal which have been neglected is now more important. These terms, including the coupling between the electrons and lattice, act to produce transitions, of scattering of electrons between one state and another. The scattering terms have no net effect on the equilibrium state because in equilibrium, the transition rates between any two states exactly balance. However in the presence of external field, the equilibrium state is destroyed and the scattering then does have some effect. In another words, we can say that scattering tends to return the system to equilibrium while the fields to pull system away from equilibrium. _____________________________________________________________________ 1.1 OBJECIVES The main aim of this unit is to study the transport properties. After going through this unit you should be able to: Understand the Boltzman equation and Relaxation time. Know the electrical conductivity, ideal resistance, carrier mobility, transport coefficients and thermal conductivity. Learn the various thermoelectric effects viz., Peltier effect, Tho mson effect etc. and lattice conduction, Phonon drag. Know the Hall effect and the two band model. understand the magneto resistances. _____________________________________________________________________ 1.2 THE BOLTZMAN EQUATION Transport phenomena such as the flow of electric current in solids, involve two characteristic mechanisms with opposite effects: the driving force of the external fields and the dissipative effect of the scattering of the carriers by phonons and defects. The interplay between the two mechanisms is described by the Boltzman 2 Transport Poperties equation. With the help of this equation one may investigate how the distribution of carriers in thermal equilibrium is altered in the presence of external forces and as a result of electron scattering processes. In thermal equilibrium and with no external fields, this distribution function is simply the Fermi distribution f 0 E ( K ) 1 (1) e 1 Let us consider a system of particles that is in dynamical equilibrium under external E ( K ) E F / kT forces. For example, the system may consist of electrons in metal that is acted upon by stationary external electric and magnetic fields. When the steady state current is flowing, the system is in dynamical equilibrium of the type we wish to consider. Suppose x, y, z, are the coordinates of an electron and vx, vy, uz are velocity components, then the distribution function is given by f (x, y, z, ux ,uy, uz,) To derive the equation consider a region of space about the point (x, y, z, ux, uy, uz). The number of particles having position co-ordinates in the range from x to x+dx, y to y+dy, z to z+dz and velocity co-ordinates in the range u, to ux+duy uy+duy uz to uz + duz can be represented by the function . f (x, y, z, , ux ,uy, uz) dx dy dz dux duy duz. (2) There may be variation of the function with time due to the two independent ways: (1) Drift variation: The function may vary because the particles are moving from one region of space to another and are accelerated by external field during motion. Consider the group of particles at an instant t + dt, that are drifted to a cell of phase space corresponding to the co-ordinates (x, y, z, , ux ,uy, uz), The number of particles is the same as were in a cell located at x - u x dt , y u y dt , z u z dt , u x x dt , u x y dt , u z z dt at a time t. Here x , y and z are the components of acceleration. The relationship holds for a small time interval dt for which the collisions have a negligible effect on the distribution. Thus the change due to drift in number of particles having co-ordinates x, y, z and velocity ux, uy and uz in time dt is . (f )d f ( x u x dt , y u y dt , z u z dt , u x x dt , u y dt , u z z dt , t ) f ( x, y, z, u x , u y , u z , t ) (3) Using Taylor,s expression and retaining only first order terms in the limit dt - 0.the above eq. may be written as. 3 Transport Properties, Semiconductor Crystals and Superconductivity f f f f f f ux uy uz x y z dt dx dy z u x u y u z Consequently the rate of change of f caused by drift is f f f f f f df uy uz x y z u x dx dy z u x u y u z dt d (2) Collisions or scattering interactions: This is due to relatively discontinuous changes in velocity that accompany collisions. If u x ,u y , , u z ; u ' x , u ' y , u ' z du ' x du ' y du ' z Represents the probability per unit time that a particle will change its velocity from ux, uy, uz to a value having components in the range extending from u’x to u’x+du’z etc. the total number the velocity of which alters from ux, uy, uz to some other value is a = f(x, y, z, ux, uy, uz) u x, u y , , u z ; u ' x , u ' y , u ' z du ' x du ' y du ' z similarly, the number the velocity of which changes to ux, uy, uz from another values is b f (u"x , u" y , u"z ) (u"x , u" y , u"z u x , u y , u z )du"x , du" y , du"z Thus the rate of change of f caused by collisions is df . b a. dt coll The total rate of change is sum of drift variation and scattering interactions. Hence for df df , 0 dt d dt coll. equilibrium the sum should vanish i.e. Substituting the values from equations , we have ux Or f f f f f f uy uz x y z (b a) 0 x y z u x u y u z ux f f f f f f uy uz x y z (b a) dx dy dz u x u y u z Equation is Bolltzmann’s transport equation. Now we shell consider two cases: (A) When the metal is homogeneous i.e., at the constant temperature in a field free space, then f f f , , 0, x y z or and x , y , z , 0 df dt coll. 0 a = b, 4 Transport Poperties Which shows that the number of particles that leave and enter a given volume of momentum space as a result of collision are equal. (B) For a heterogeneous medium , i.e., if there is temperature gradient. f f f , , 0 x y z Hence df dt coll. 0 _____________________________________________________________________ 1.3 ELECTRICAL CONDUCTIVITY The electrical conductivity of solids was first demonstrated by Stephan Gray in 1729. It is the ability of a material to conduct electricity. The resistance (R) offered by a conductor to the flow of electric charge is found to be directly proportional to the length (l) and inversely proportional to the area of cross section (a) of the conductor. Therefore, l R ( ) a where is the proportionality constant called the electrical resistivity. Then 1/ is the electrical conductivity ( ). The magnitude of the electrical conductivity can be determined by (1) the density of charge carriers, (2) the charge on the carrier, (3) the average drift velocity of the carriers per unit electric field. The electrical conductivity may be defined as the quantity of electricity that flows in unit time per unit area of cross section of the conductor per unit potential gradient. If q is the quantity of electricity that flows through a conductor of cross sectional area q AEt A in time t under a potential gradient E, then Or when t=1, q i AE E ne 2 , 6 K BT ne 2 6 K BT This expression shows that different conductivities of different materials are due to different number of free electrons. 1.3.1 Factors affecting electrical conductivity The main factors affecting the electrical conductivity of solids are (a) temperature (b) defects, e.g., impurities, and (c) electromagnetic radiation. In metals, the charge carriers are electrons; the electronic concentration is large and constant, and is almost unaffected by the presence of impurities. To understand the role of temperature, we have to consider the effect of relaxation time . At low temperature, the value is 5 Transport Properties, Semiconductor Crystals and Superconductivity large since the metal ions vibrate simple harmonically. As the temperature is increased, the metal ions vibrate more vigorously, and an harmonically and thus act as better scattering canters than before; as a result, the mean free path/mobility/relaxation time decreases and decreases with increasing temperature. On the other hand, semiconductors are insulators at temperatures <<Eg/kb. At high temperatures, the thermal agitation promotes electrons into the upper, empty conduction band and the conductivity rises exponentially with increasing temperature. Introduction of some impurities (one part per billion) in an pure semiconductor gives rise to a large increase in the conductivity. This type of conductivity is termed as extrinsic semi conductivity. The electromagnetic radiation can also affect the electrical resistivity of intrinsic semiconductor. This phenomenon is called photoconductivity. Check Your Progress 1 Note: a) Write your answers in the space given below. b) Compare your answers with the ones given at the end of the unit. (1) Derive the Boltzman equation. (2) Explain the electrical conductivity. How does it vary with temperature ? .......................................................................................................................... .......................................................................................................................... .......................................................................................................................... .......................................................................................................................... .......................................................................................................................... 1.3.2 Calculation of relaxation time The relaxation time is closely related to the mean free time between two successive collisions of the electron with lattice. We know from the Newton’s second law of motion, the force on a particle of mass m is F m x , and the force on a charged where m is the effective particle of charge q will be qE x , then m x qEx mass.integration gives the velocity at time t, x qE x t v x0 m The second term is the initial velocity of the electron. We now assume that after a certain time 2 the electrons suffer a collision and that the result of collision is to decrease to zero the excess velocity of the electron acquired from the field. 6 Transport Poperties After the collision, the electron accelerates for another period of time 2 . The time is called relaxation time. The average velocity increment vx of the accelerating atom during its time of flight is simply half its final velocity increment. Jx = Nqvx , vx qE x m N is the no. of electrosns. Jx The current can be written as Then the conductivity is Nq 2 Ex m Nq 2 m Fig. 1. Relaxation time Here the quantity relaxation time is half the time between collisions. Remembering that in the Fermi distribution only the electrons near the Fermi energy can participate in collisions, the relaxation time is then the time between collisions for those electrons.Then the shift of the Fermi sphere is given by or the shift in the velocity of the electrons is given by p0 qEx v qE x m The relaxation time is also related to another parameter , the mean free path of the electrons capable of making collisions. If the velocity gained in the field is negligible as compared with the Fermi velocity, then n f 2 where nf is the speed of the electrons at the top of Fermi distribution. 1.3.3 Impurity scattering By impurities we mean foreign atoms in the solid which are efficient scattering centers, when they have a net charge. Ionized donors and acceptors in a semiconductor are a common example of such impurities. The amount of scattering due to electrostatic forces between the carrier and the ionized impurity depends on the 7 Transport Properties, Semiconductor Crystals and Superconductivity interaction time and the number of impurities. Larger impurity concentrations results in a lower mobility. The dependence on the interaction time helps to explain the temperature dependence. The interaction time is directly linked to the relative velocity of the carrier and the impurity which is related to the thermal velocity of the carriers. This thermal velocity increases with the ambient temperature so that the interaction time increases, the amount of scattering decreases, which results in a mobility increase with temperature. To first order the mobility due to impurity scattering is proportional to, where N1 is the density of charged impurities. Check Your Progress 2 Note: a) Write your answers in the space given below. b) Compare your answers with the ones given at the end of the unit. (2) What do you mean by relaxation time ? (2) Describe the impurity scattering. .......................................................................................................................... .......................................................................................................................... .......................................................................................................................... .......................................................................................................................... .......................................................................................................................... 1.3.4 Ideal resistance If M is the mass of the atom, and (-fx) is the restoring force on the displaced atom, the equation of the atomic oscillator is M d 2x f x 0 dt 2 M d 2x fx 0 , dt 2 1/ 2 where f .x is the amplitude of vibration. M 2 The potential energy of the oscillator will be 1 1 ( M 2 x 2 ) M 4 2 2 x 2 2 2 then the potential energy per degree of freedom will be = 2 2 M 2 x 2 = k BT 2 (1) At high temperature, as the Einstein and Debye frequencies are essentially the same, and most of the modes are the high frequency modes, then 8 Transport Poperties 2 E2 D2 (2) with h D kB D Uing eq. (1) & (2) We have x 2 2T Mk B D2 (3) 1 cons tan t From the kinetic theory man free path is given by x2 n We know that , Obviously tT At low temperature we use the Debye model. At low temperature since the lattice vibrations begin to die out, the scattering cross section would fall and we would therefore expect that their contribution to the resistivity, denote by p will decrease at low temperatures, eventually becoming zero at 0 K. The larger the amplitude of vibration at any temperature, the greater will be p . Since this amplitude depends on the inverse of the Debye temperature D , it is to be expected that p will be less for metals with a high D , and vice-versa. The arrangement of point defects in a crystal resistivity 0 , which they produce, would be expected to be constant. Their contribution to the resistivity is temperature independent, but it does, of course, increase with the impurity concentration. The total resistivity is therefore 0 p T Fig. 2. Variation of the electrical resistivity with temperature. 9 Transport Properties, Semiconductor Crystals and Superconductivity This is shown in Fig. 2. it is very clear from here that at first decreases linearly with T, and at low temperatures , it flattens off to a constant value, equal to 0 , which is called the residual resistivity. It is clear that, for a very pure sample, 0 will be very small, whereas for an impure specimen it will have a high value. The probabilities of electrons being scattered by photons and by impurities are additive, therefore we may write 1 1 p 1 0 in this equation the first term on the right is due to phonons and the second is due to impurities. The former is expected to depend on T and the latter on impurities, but not on T. The 0 p T simple addition of 0 and p T in this equation is often referred to as Matthiessen’s rule. The thermally induced part of the resistivity, p , is sometimes known as the ideal resistivity and the resistance is called ideal resistance, whereas the resistivity due to impurities and defects is summed up in the residual resistivity ( 0 ). 1.3.5 Carrier mobility The conductivity of a solid has been expressed as ne where the mobility is the average velocity by a carrier in a unit electric field. The electrons and holes in a semiconductor are in a rapid random motion because of their thermal energies. It is the additional velocity introduces by an external electric field which constitutes the electric current observed. The net velocity due to the field is called drift velocity. Also eE an electron in a perfect crystal experiences an acceleration in a field E which m implies that its drift velocity continuously increases. This can not be the case because according to Ohm’s law, nevd = constant E j/E This means that the average velocity vd must be a constant. Hence it is necessary to assume that the electron loses energy in collisions with the crystal structure so that it v has a constant average velocity or mobility, d Suppose that the average time E between such collisions is and that at each collision the electron loses all the energy it gained from the field subsequent to the previous collision. Assuming the 10 Transport Poperties independent of electron velocity, then the number of collisions per second is 1/ and v the rate of change of velocity is d . Under steady state conditions this rate of change must be equal to the acceleration due to the field, or eE / m vd / E / . Therefore, e / m . The parameter is called relaxation time. The imperfections disrupt the periodicity of atomic array and cause the electrons to be scattered. The two important causes of scattering in semiconductors are (1) atomic vibrations (phonons) (2) ionized impurity atoms. The effect that these imperfections have on the nobilities of electrons or holes can be determined most easily by calculating the corresponding relaxation time. An actual calculation of the mobility L due to scattering by phonons is the T 3 / 2 temperature dependence. Similarly the mobility due to ionized impurities scattering i is T 3 / 2 temperature dependence. The actual mobility is given by 1 1 L 1 i i.e., 1 a T 3/ 2 bT 3 / 2 _____________________________________________________________________ 1.4 GENERAL TRANSPORT COEFFICIENTS 1.4.1 Thermal conductivity In order to discuss the thermal conductivity of metals we suppose that there exists a temperature gradient across the specimen. The transport of energy in the metals is due to conduction electrons and lattice waves. Here we shall consider the thermal conductivity only due to conduction electrons although lattice conduction may become important under certain circumstances such as low temperature, high magnetic field, large impurity contents etc. We will now consider the application of the Boltzmann equation to effects that involve heat transport. Let there be a thermal gradient dT/dx in a metal and a current density Q. In the measurement of thermal conductivity the specimen is electrically insulated from its surroundings; thus the current vanishes but not the electric fields. This is due to the fact that the temperature gradient produces a drift velocity of the electrons, and a small electric fields. Is set up internally to counteract the drift velocity. Thus the Boltzmann transport equation beside the thermal gradient dT/dx 11 Transport Properties, Semiconductor Crystals and Superconductivity includes a term containing an electric field x . In this case the Boltzmann equation can be written as df f e x f f f 0 0 u x dT x m u x T Or f fo f f T u x e x px T T x When the electric field and T / x are small, we can replace (1) f f f and by o x x p x f o respectively. Substituting in above equation we have p x f f o e xT f o / px u x f o x. Now in view of fo being the Fermi-Dirac distribution function we write and T f o f o T 1 x T x T 1 exp{( E E F ) K BT } X fo f o f o E E F T E F . e x p x E T T X . Now considering that EF is independent of T, we get f E E T f f o x o e x F E T T X (2) We know that electric current density Ic and thermal current density Qx are defined as 2e I x 3 x f f o dp x dp y dp z , h 2e Qx 3 x f f o Edpx dp y dp z h Where E is the energy of an electron. Substituting the value of (f-f0) from above equations, we get and 2e E E T 2 f I x 3 x o e x F dp x dp y dp z E h T T X f 2e E E T 2 Qx 3 x E o e x F dpx dp y dpz E h T T X Assuming that is only a function of the energy and not of the direction of motion, we see that the integrals in equation are functions of energy alone. The triple integrals 2 may be transformed into single integrals by replacing x by 2 / 3 and dpx dpy dpz by 4p 2 dp . thus 12 Transport Poperties Ix 16e(2m)1/ 2 3h 3 16e(2m)1/ 2 Qx 3h 3 E 3 / 2 ( E ) 0 0 f o E E E F T e x T T X dE f o E E E F T e x T T X dE E 5 / 2 ( E ) Introducing a set of integral JN 16e(2m)1 / 2 3h 3 f o dE , N 1,2,3.... 0 E J E J T I x e 2 F 1 e 2 J1 x T T X J E J T Qx 2 F 2 e 2 J 2 x . T T X Now we calculate Qx under the condition Ix = o, because the thermal conductivity of metals is defined as the rate of energy flow divided by thermal gradient when Ix=O. Qx K I x O (T / x) JN From equation when Ix=O., ( E ) E N 1 J J 2 T Qx 2 2 T J1T x Now J J J 22 Comparing the above equations K 1 3 . J1T The value of integral JN is given by JN 2 E N 1/ 2 16 (2m)1/ 2 N 1 2 K BT 2 EF 6 E 2 Substituting equation in equation we get K E E F 2 K B 2TN F 3 m Where N is the density of electrons. In most of the metals, F varies about 1/T, and hence K is nearly temperature independent. If the metal contains impurities, then electron-phonon scattering as well as electron impurity scattering takes place. If the two scatterings are considered to be independent to each other, then the total conductivity K can be represented by 1 1 1 K K1 K ' Where Kl is the contribution arising from electron-lattice scattering and Kl is the contribution from electron impurity scattering. This expression shows that the impurities decreases the thermal conductivity. 13 Transport Properties, Semiconductor Crystals and Superconductivity 1.4.2 Thermoelectric effects The thermoelectric effect is the direct conversion of temperature differences to electric voltage and vice versa. Generally, the term thermoelectric effect encompasses three separately identified effects, the Seebeck effect, the Peltier effect, and the Thomson effect. Let us consider the two metals A and B having different electron density let the electron density in A is greater than electron density in metal B. Now the electronic pressure in A will be greater than in B. Due to the difference in electronic pressure, the electrons diffuse from A to B. This makes A positive and B negative. Thus a potential difference is created at the junction of two metals. When this potential difference reaches a certain value, it prevents the migration of electrons from A to B and a state of equilibrium is set up. This explains that how a potential difference is created at the junction of two metals . We shall now apply this general conclusion to the three effects of thermo electric phenomena. Seebeck effect The seebeck effect is the conversion of temperature differences directly into electricity. In this effect the thermoelectric EMF is created in the presence of a temperature difference between two different metals or semiconductors. The voltage created is of order of several microvolts per kelvin difference. T2 B A vV _ B T1 Fig. 3. Seebeck effect In the circuit, the voltage developed can be derived from: T2 V S B T S A T dT T1 SA and SB are the Seebeck coefficients of the metals A and B as a function of temperature, and T1 and T2 are the temperatures of the two junctions. If the coefficients are constant for the measured temperature range, then the above formula can be written as: V (SB S A ).(T2 T1 ) 14 Transport Poperties Peltier effect In case of Peltier effect an external potential difference is applied to the Junction i.e., current is allowed to flow from A to B. Due to this current flow, there will be a transfer of electrons from B to A. As the electron density in A is greater than in B, hence certain amount of work is done against the electronic pressure difference. This involves the absorption of some energy at the junction which in consequence gets cooled. When the direction of the current is reversed, the electrons flow from A to B which make the energy available at junction in the form of heat i.e., the junction gets heated. The Peltier coefficient is defined as the amount of energy liberated or absorbed when unit charge passes through the junction The expression for can be derived as follows: 1 According to the kinetic theory, the electronic pressure is given by p mn 2 , 3 Where m is the mass of electron, n is the number of electrons per unit volume of the conductor and 2 is the mean square velocity of the electron . We know that p 1 1 mn 2Tor mn 2 T 2 2 Then the above equation Equation can be written as p 2 1 2 n m 2 nT 3 2 3 (1) Now charge per unit volume is ne where e is the charge on the electron. The volume V corresponding to a unit charge is given by v 1 . ne Now the work done when unit charge passes through the junction will be the amount of work done by the electrons in moving from higher to lower electronic pressure. V2 This is equal to Peltier coefficient , hence pdv V1 Where V1 and V2 represent volumes corresponding to unit charge. These are equal to 1/n1e and 1/n2e by relation where n1 and n2 be the electron densities in the two metals respectively. Substituting the values of p from equation (1) in equation we get . V2 V1 2 1 Tdv 3 eV Again V2 1 / n2 e n1 V1 1 / n1e n2 2 T n1 . 3 e n2 15 Transport Properties, Semiconductor Crystals and Superconductivity The relation shows that Peltier coefficient is directly proportional to the absolute temperature. This agrees with the result obtain by thermo dynamical consideration. Thomson effect According to Thomson effect, there is absorption or evolution of heat due to the passage of a current in a single unequally heated conductor. Copper, Silver, zinc, antimony and cadmium have positive Thomson effect because heat is evolved when current flows from hot to cold side while heat is absorbed when current flows from cold to hot side. On the other hand, cobalt, bismuth and platinum have a negative Thomson effect . The electron theory only provides a partial explanation of this effect. Let us consider the case of a conductor whose one end is at higher temperature than the other and a current is passed from hot end to the cold end. Now there will be transfer of electrons from the cold part to the hotter part. We know that the energy of the electron is 1 proportional to its absolute temperature mv2T , hence the energy of the electron 2 moving towards hotter part will be increased. This produces a cooling effect. 1.4.3 Lattice conduction & Phonon drag At any finite temperature, the atoms vibrate about their equilibrium positions. These lattice vibrations may be represented by waves. A lattice vibrational wave in a solid is a repetitive and systematic sequence of atomic displacements viz., longitudinal, transverse, or a combination of the two, which may be characterized by a propagation velocity (v), a wavelength ( ), a wave-vector k , a linear frequency ( ).The energy of lattice vibrational or elastic waves is quantized. This quantum is called a phonon. Transmission of a displacement wave in a solid may be regarded as the movement of one or more phonons, each carrying energy and momentum k . Thermal conduction in non-metallic solids is due to creation or annihilation of a phonon. The of phonons is of the order of 0.1 eV. Phonons are not always in local thermal equilibrium, they move along the thermal gradient. They lose momentum by interacting with electrons and imperfections in the crystal. If the phonon-electron interaction is predominant, the phonons will tend to push the electrons to one end of the material, losing momentum in the process. Phonon drag is an increase in the effective mass of conduction electrons or valence holes due to interactions with the crystal lattice in which the electron moves. As an electron moves past atoms in the 16 Transport Poperties lattice its charge distorts or polarizes the nearby lattice. This effect leads to a decrease in the electronic/ hole mobility, which results in a decreased conductivity. However, as the magnitude of the thermoelectric power increase with phonon drag, it may be beneficial in a thermoelectric material for direct energy conversion applications. The magnitude of this effect is typically appreciable only at low temperatures notable below 200 K. Check Your Progress 3 Note: a) Write your answers in the space given below. b) Compare your answers with the ones given at the end of the unit. (1) Write the various thermoelectric effects and obtain their coefficients ? (2) What are phonons ? Write a short note on phonon drag. .......................................................................................................................... .......................................................................................................................... .......................................................................................................................... .......................................................................................................................... .......................................................................................................................... ____________________________________________________________________ 1.5 THE HALL EFFECT Definition : When a current carrying metallic wire is placed in an external magnetic field in direction perpendicular to the direction of flow of current, an electric field is produced in the wire in direction perpendicular to both the direction of current and direction of magnetic field. This effect is called the Hall effect. The electric field produced in the wire is called the Hall field and the potential difference developed at the ends of wire is called the Hall voltage. Calculation of Hall field and Hall voltage Consider a metallic rod of rectangular cross-section through which current flows by applying an external electric field Ex along its axis (i.e., X-direction). Obviously electrons will drift with velocity Vx in direction opposite to the electric field . Now if a magnetic field Hz is applied in direction perpendicular to the axis of rod. (k.e., in Zdirection), the electrons will experience Lorentz force and will get deflected towards one end of the rod (i.e., in negative Y-direction) . These electrons will not drift in 17 Transport Properties, Semiconductor Crystals and Superconductivity space outside the surface of rod, but they will accumulate on the surface of rod and form the surface charge. Obviously the positive ions get accumulated on the opposite face. Due to charges on the surface, a transverse electric field EH is produced in the rod which is called the Hall field. Due to this electric field, compensating drift of electrons begins in positive Y-direction and an equilibrium of electrons is maintained within the rod (i.e., the force acting on electrons due to the Hall field EH balances the Lorentz force acting on electrons due to the magneticfield). Fig. 4. Creation of Hall field Remember that if in a conductor the charge carriers are holes (or positive charges), then the direction of Hall – electric field EH is reversed. Thus by measuring the Hallvoltage we can determine whether the charge carriers in conductor are electrons or holes.Let the charge carriers in the rod be electrons (charge = -e). Hall electric field EH (or Ey) will be in negative Y-direction. Force on electron due to Hall field EH = eEH ( in positive Y-direction ) And Lorentz force on electron due to magnetic field HZ ^ ^ ^ (e)( iv z ) (k H Z ) J ev x H Z (1) i.e., Lorentz force =evxHz (in-negative Y-direction) EH v x H Z In steady state, (2) e EH ev x H Z , The expression gives the intensity of Hall electric field produced in the rod. If d is the thickness of the rod, Hall potential difference VH = Ed = vxHZd Hall coefficient Let number of electrons (or charge carriers) per unit volume in J the rod be n. Then current density Jx = n(-e)Vx or drift velocity vx x But from ne eqn.(2) The Hall electric field produced in the rod EH = vxHZ Substituting the value of vx from eqn. (1) We get j 1 EH x H Z orE H ( J x H Z ) ne ne 1 RHall EH J x H Z orEH RHall ( J x H Z ) , Where Or ne 18 Transport Poperties The quantity R Hall is called the hall coefficient. Obviously, its unit is volt x metre 3/ amper × weber. From the above expression, it is clear that the sign of Hall coefficient is same as that of the charge carrier in the conductor i.e., tf the charge carrier is electron (negative charge ) , the sign of RHall is negative and if the charge carrier is hole (positive charge), the sign of RHall is positive. Hall Voltage If b is the width of the metallic rod and d is its thickness and a current I flows in the rod, then current density in the rod along X-axis Jx = I / bd Then Hall electric field produced in the rod EH 1 1 I JxHZ Hz ne ne bd Hall voltage VH = EH × d 1 I IH Z H z d RH ne bd b Importance of Hall effect: (i) Measuring the hall voltage in a metal or semi-conductor, we can find the sign of charge carrier in it also we can find the number of charge carriers per unit volume in it. (ii) Measuring the Hall coefficient of a metal, we can find the number of charge carriers per unit volume in it. (iii) The mobility of charge carriers in a metal can be determined. (iv) It can be found that the given substance is a metal or semi-conductor or insulator. It should be remembered that the Hall coefficient is not negative for all the metals, but some metals have the positive Hall coefficient also (i.e., in such metals the charge carriers are holes.) if there are both the holes and electrons as charge carriers, their Hall coefficient can be either positive or negative depending on their relative concentration and mobility. 19 Transport Properties, Semiconductor Crystals and Superconductivity Experimental Determination of Hall Coefficient Fig. 5. Measurement of Hall coefficient Fig.5. Shows the experimental arrangement of the apparatus used for the measurement of Hall coefficient. It consists of a thin metallic rod of width nearly 2-3 cm and length nearly 5-10 cm placed horizontally and with its length along the Xdirection in between the pole pieces of a strong electromagnet. The two ends of the rod C and D are connected with a battery S, a milliammeter (mA), a rheostat Rh and a key K so that current flows in the rod along its length (i.e., in X-directin). A sensitive calibrated potentiometer is connected between the points A and B along the width of the rod (i.e., in Y-direction ) to measure the Hall voltage.. Let breadth of metallic rod be b and thickness be d. Current I flows in the rod due to which current density in the rod along X-axis is Jx = I bd But Hall electric field produced in the rod 1 1 I EH J x H z , EH Hz Hence ne ne bd 1 I IH z H zd , RHall Hall voltage VHall = EH × d, ne bd b V b Hence Hall coefficient RHall Hall IH z Thus knowing the Hall voltage VHall, width of the rod b, current flowing in the rod I, and the magnetic field Hz, we can calculate the Hall coefficient RHall from the above expression. 20 Transport Poperties _____________________________________________________________________ 1.6 TWO BAND MODEL Band model According to band model, the Sommerfeld’s assumption that the free electrons inside the metal move in constant potential well, was considered to be wrong. Actually a free electron inside the metal moves in the electric field of positive ions and of other free electrons. Since the crystal structure itself is periodic, therefore the potential energy of electron also changes periodically with the position coordinate i.e., the motion of electron inside the metal is in a periodic potential well. We know that inside a metal (atomic number Z), the potential energy of free electron at a distance x in the potential field of an atom I is given as U ( x) Ze 2 . Hence a 4ox graph plotted for U(x) versus x is a rectangular hyperbola as shown in Fig. 6. Fig.6. U(x)-x curve in the region of a single atom From the graph, it is clear that for different values of x, the value of U(x) is negative and at x= , the value of U(x) is zero.But inside the metal, atoms are arranged in a definite order, therefore the potential energy of electron in the combined field of atoms I and j can be represented by the complete curve as shown in Fig. 7. In Fig.7. The dotted curves represent the potential energy of electron in the field of atom I and atom j separately, while the complete curve represents the resultant potential energy. But in a metal, there are number of atoms arranged in a definite order in any direction, hence in the length L of the metal, the resultant potential energy of electron in the fields of atoms i. J. k. l. etc. can be represented as shown in Fig. 8. 21 Transport Properties, Semiconductor Crystals and Superconductivity Fig. 7. U(x)-x curve in the region of two atoms From Fig.7. It is clear that at the boundaries of metal (i.e., at its free surfaces), the potential energy suddenly rises and becomes zero, while inside the metal, the potential well is not of uniform depth everywhere, but it is periodic. Fig. 8. U(x)-x curve in the region of many atoms Fig.8. Also shows the different energy levels inside the metal . All the electrons from the lowest energy to Eb are bound with their atoms and they can vibrate only with a very small amplitude, they cannot leave their atoms. The electrons of energy higher than this, with energy in between Eb and EF can move anywhere within the metal, but on reaching at the boundary (or metal surface), they have to face a surface barrier and hence they cannot emerge out of the metal surface. Here EF is the Fermi energy level (i.e., the level of maximum kinetic energy of electrons). Thus, according to band model 22 Transport Poperties (i) Each electron in metal is in the electric field produced due to charge distribution of positive ions and remaining electrons. (ii) Electron is associated with the entire crystal, and not only with an atom. (iii) Electron moves in a periodic potential produced by the ion cores and other electrons inside the crystal. This periodicity vanishes at the free surface of the crystal. The motion of electron inside the crystals like the elastic waves in a continues medium. Kronig – Penny model To explain the behavior of electrons in a solid under a periodic potential, kronig and Penney assumed that the potential energy of an electron can be represented by a periodic array of rectangular potential well as shown in Fig.9. Here the potential peaks obtained from the hyperbolic curves have been assumed to be in form of rectangular peaks. Fig. 9. Periodic array of rectangular potential well Each potential well represents the potential near an atom. If the time period of potential is (a+b), the potential energy is zero in 0<x<a and potential energy is constant (=V0) in – b <x < 0. i.e., In region 0 < x < a, V(x) =0 And in region – b < x < 0, V(x) = V0 (constant ) (1) In both these regions, the Schrodinger wave equation for the wave function n associated with n th energy state En of electron are d 2 n 2m 2 En n 0 (since V= 0 ) (2) dx 2 h d 2 n 2m 2 En V0 n 0 (since V = V0) and in region – b < x < 0, (3) dx 2 h Here, the energy of electron En is very small in comparison to the potential V0. Assuming that as b tends to zero, V0 becomes infinite, Kronig and Penney obtained the following condition for the allowed wave function on solving the above equation : In region 0<x<a 23 Transport Properties, Semiconductor Crystals and Superconductivity (mV0b / h 2 ) sin a + cos a = cos ka (4) Where = 2mEn / h and k is the wave vector . Substituting mV0ba/h2 =P in eqn. The condition for the allowed wave function is P sin a cos a cos a a Fig. 10. Energy bands and forbidden energy gaps Fig.10. Shows a graph between the quantity on the left side of above equation and a for P = 3 /2. Since the maximum and minimum values of the term cos a are respectively +1 and –1, own by thick lines from p to , from q to 2 from r to 3 , from s to 4 …….. in the positive directions and from p’ to - , from q’ to –2 , from r’ to – 3 , from s’ to –4 …… in the negative direction. It is clear that only in these specific ranges of a , the allowed energy levels are continuously obtained i.e., energy bands are obtained in these specific ranges and not the discrete energy states. Obviously there are no energy states possible for electrons in the ranges to q, 2 to r, 3 to s, ……….., i.e., these ranges represent the forbidden energy gap. Fig. 11. Energy spectrum in a crystal Thus from the Kronig–Penney model, we get the following conclusions : 24 Transport Poperties (i) In the energy spectrum of all the electrons present in the metal, there are several energy bands separated by the forbidden energy region. The energy band completely filled with electrons is called the valance band and the energy band which is either completely empty or is partially filled, is called the conduction band . Fig. 12. Energy spectrum with P (ii) From Fig. It is clear that as the value of a increases (or as energy increases because 2mEn / h , the width of the allowed energy band increases (since the width of energy band from p to is less, it is more from q to2 and still more from r to 3 ,…….). allowed energy band decreases and ultimately when the binding energy becomes infinite, the allowed region becomes very narrow At the wave vector k=±n /a (where n= 1,2,3,…)the energy is discontinuous. Fig. 13. E- k curve These values to k give the boundaries of Brillouin zones. Fig shows the E-k curve. For n=1, we get the first Brillouin zone from k=- /a to k=+ /a. The energy in an energy band is a periodic function of k. (iii) The number of total possible wave functions in an energy band is equal to the number of unit cells. (iv) The velocity of free electron is zero at the top and bottom of an energy band and the velocity of free electron is maximum at the point of inflexion of 25 Transport Properties, Semiconductor Crystals and Superconductivity energy band After this point, the velocity of electron decreases with the increase in energy. (v) At T=0 K, the effective number of electron in a completely filled band is zero, _____________________________________________________________________ 1.7 MAGNETO RESISTENCE The magneto resistance is defined as the ratio of change in resistance of a substance due to the application of magnetic field to the resistance in zero field. This effect is due to the fact that, when the magnetic field is applied, the paths of the electrons become curved and the electrons now do not follow the exact direction of the superimposed electric field. When the magnetic field is applied normal to the current flow, the effect is termed as transverse magneto resistance and when the field is applied parallel to current flow, the effect is termed as longitudinal magneto resistance.In case of a magnetic metal the external magnetic field increases the alignment of magnetic moments opposite to the thermal vibrations which decreases the alignment. If the temperature is below Curie point then magneto resistance is decreased and if the temperature is above Curie point, there will be no change in it. At low magnetic field there are changes in resistivity which are apparently associated with magnetostriction ie., elongation or contraction of the metal depending on the direction of magnetic moments. These changes are assumed due to the changes in Fermi surface. Fig. shows the longitudinal and transverse magneto-resistances of nickel at from temperature. The resistivity in the low fields region governed by magnetostrictive effects while at higher field the decrease due to a figment of the magnetic moments is predominant. Fig. 14. Longitudinal and transverse magneto-resistance 26 Transport Poperties The transverse magneto resistance measurements made on a single crystal predict pronounced dependences on magnetic field direction relative to the crystal axes, especially in pure metals. It is observed that the variation of magneto resistance with magnetic field strength may be different for magnetic fields in different crystallographic direction. In some direction it may increase with field, then fall off and saturate at some constant value; on the other hand, in some other directions it may continue to rise even at the highest fields. Mathematical Analysis Let us consider the case of a wire subjected to an electric field x along the x-direction and a magnetic field H normal to the axis of wire i.e., along the z-axis. Now the Lorentz force is given by 1 d 2x 1 e x v y H z vH y e x v H 2 c dt c (as the field is only in z- direction, Hy=0) d 2x 1 m 2 e x .v y H z . dt c 2 d x 1 dy Or (1) m 2 e x . H z . dt c dt d2y 1 Similarly (2) m 2 e x vz H x vx H z . dt c Similarly for z-direction d 2z (3) m 2 0. dt Integrating equation we have dx eH z m e z t y C1 dt c dx e m H z x C2 and dt c Where C1 and C2 are constants of integration . The values of these constants can be obtained from the following conditions: dx ux C1 mux At t=0, x=0, dt F m dx uy dt dx H m e x t e z y mu Thus dt c dy e H z x uy Or dt mc Integrating equation we have t=0, y=0, C2 mu y (4) 27 Transport Properties, Semiconductor Crystals and Superconductivity x As ex t 2 e H z yt u xt C3 . m 2 m c x=0, when t=0, C3=0. ex t 2 e H z yt ut. m 2 m c Substituting the value of x from equation in equation we have x dy eH e t 2 eH z y z x yt u xt u y dt m m 2 mc dy From equation , substituting value of in equation (1) dt d 2x e H z eH Z e x t 2 eH z yt u t u x x y dt 2 m c mc m 2 mc dx e H z eH Z e x t 3 eH z yt 2 ut 2 Integrating it t u t y dt 2 m c mc m 2 mc 2 2 If be the time between two successive collisions, we have (5) (6) dx 1 dx dt dt 0 dt e 2 H z eH z e x 4 eH z y 3 u x 2 2 the u y Here x m 2 c mc 24m 6mc 6 2 term Hzy/6 is very small as compared to 3 / 6 hence it is neglected. Average values of ux and uy are also zero as the carriers have same probability of moving in positive and negative directions. e 2 e3 H 2 z x 4 x . m 2 24m2c 2 dx Ne2 x e 2 H 2 z 3 , . dt 2m 12m 2c 2 The electrical conductivity is given by I Ne2 e 2 H 2 z 3 x . x 2m 12m2c 2 Now current density I x Ne When magnetic field is zero, Hz=0, then electrical conductivity 0 is given by Ne3 0 0. 2m The change of resistively is given by 2 2 p 1 H z 0 p0 0 0 3 c 2 N 2e 2 28 Transport Poperties 1 2 Hz 0 3 cNe 2 2 Hz cNe 2 AH z , 2 p 1 Where A 0 and is very small. 3 cNe 0 2 Check Your Progress 4 Note: a) Write your answers in the space given below. b) Compare your answers with the ones given at the end of the unit. (1) What is Hall effect. What important results are obtained with the help of Hall coefficient ? (2) Write a note on magnetoresistence. .......................................................................................................................... .......................................................................................................................... .......................................................................................................................... .......................................................................................................................... .......................................................................................................................... ____________________________________________________________________ 1.8 Let Us Sum Up After going through this unit, you have achieved the objectives stated earlier in the unit. Let us recall what we have discussed so far. Transport phenomena such as the flow of electric current in solids, involve two characteristic mechanisms with opposite effects: the driving force of the external fields and the dissipative effect of the scattering of the carriers by phonons and defects. The interplay between the two mechanisms is described by the Boltzman equation. With the help of this equation one may investigate how the distribution of carriers in thermal equilibrium is altered in the presence of external forces and as a result of electron scattering processes. In about 1900, long before an exact theory of the solid state was available, Drude describe de metallic conductivity using the assumption of an ideal 29 Transport Properties, Semiconductor Crystals and Superconductivity electron gas in the solid. For an ideal electron gas in an external field E, the dynamics of the electrons is described by the classical equation of motion mv m vD eE The dissipative effect of scattering is accounted for by the friction term mvD / where vD=v-vthermal is the so called drift velocity. The relaxation time is the time constant with which the nonequilibrium distribution relaxes via scattering to the equilibrium state when the external perturbation is switched off. If one end of a long metallic bar or wire is heated, then heat flows spontaneously from one end to the other because of the ensuing temperature gradient. If the gradient is uniform, then the amount of thermal energy crossing a unit area per second (Q) is directly proportional to the temperature gradient (dT/dx)., Q dT dx and then dT Q K dx where the proportionality constant (K) is called the thermal conductivity. knowing the Hall Coefficient it can be found that the given substance is a metal or semi-conductor or insulator. The magneto resistance is defined as the ratio of change in resistance of a substance due to the application of magnetic field to the resistance in zero field. When the magnetic field is applied normal to the current flow, the effect is termed as transverse magneto resistance and when the field is applied parallel to current flow, the effect is termed as longitudinal magneto resistance. It is observed that the variation of magneto resistance with magnetic field strength may be different for magnetic fields in different crystallographic direction. In some direction it may increase with field, then fall off and saturate at some constant value; on the other hand, in some other directions it may continue to rise even at the highest fields. _____________________________________________________________________ 1.9 Check Your Progress: The key 1. (1) See the section 1.2. 30 Transport Poperties (2) See the section 1.3.1. 2. (1) See the section 1.3.2. (2) See the section 1.3.3. 3. (1) See the section 1.4.2. (2) See the section 1.4.4. 4. (1) See the section 1.5 (2) See the section 1.7. 31 Transport Properties, Semiconductor Crystals and Superconductivity REFERENCES AND SUGGESTED READINGS 1.Material Science by Narula, TMH. 2.Solid State Physics by C. Kittel, TMH. 3. Solid State Physics by Pillai. 4. Solid State Physics by Gupta. ********* 32