+Ze. n = N/V = 6.02 x x (Z Z c ) m /A, (1.1) Avogadro s number

Transcription

1 In 1897, J. J. Thomson disovered eletrons. In 1905, Einstein interpreted the photoeletri effet In Rutherford proved that atoms are omposed of a point-like positively harged, massive nuleus surrounded by a sea of eletrons. - Drude onstruted his theory of eletrial and thermal ondution in metals by (1) onsidering the eletrons to be a gas of negatively harged partiles traversing in a medium of uniformly distributed positive ions, and () applying the kineti theory of gas to the eletron sea. Below is a shemati diagram of Drude s model of metals: (In 19, Bohr was awarded the Nobel Prize for his ontribution to the understanding of the struture of atoms. In late 195, the Shrödinger equation was formulated.) Core eletrons Z e Z e +Ze Z e +Ze +Ze Z e +Ze Positively harged ions Eletron sea due to the deloalized valene eletrons, (Z Z ) from eah atom. There is one important parameter of the model: (1) Eletron number density, n n = N/V = 6.0 x 10 3 x (Z Z ) m /A, (1.1) Avogadro s number where N is the total number of eletrons in the metal, V is the volume, m is the mass density, and A is the mass number. Note that only the valene eletrons ((Z Z ) per atom) ontribute to n. () Average eletron separation, r s. V/N = 1/n = (4/3) r s 3 r s = [3/(4n)] 1/3 (1.) 1

2 The value of n varies from /m 3 for Cs to /m 3 for Be among different metals. The values of r s also vary between those of these two metals, from 5.6a 0 to 1.87a 0 (where a 0 is the Bohr radius = 4 0 / me = m), with the majority lying between a 0 and 3a 0. You may notie the very high eletron density found in metals, whih is about 1000 times that of lassial gases at normal temperature and pressure. In spite of this and in spite of the foreseeable strong eletron-eletron (e-e) and eletron-ion (e-ion) eletrostati interations, Drude boldly treated the dense eletron gas by the kineti theory of gas. Below are the basi assumptions he used: 1. Between ollisions, the interation of a given eletron, both with other eletrons and with the ions, is negligible. The former approximation is known as the independent eletron approximation while the latter is the free eletron approximation. Correspondingly, in the absene of externally applied eletri or magneti fields, eah eletron is taken to move in a straight line between ollisions. In the presene of an applied field, however, eah eletron is taken to move aording to the Newton s laws of motion.. Collisions are onsidered to be instantaneous events that alter the veloity of an eletron. Drude attributed them to ollisions with the presumably stationary ions (rather than ollisions with other eletrons as in ordinary gases). It turns out e-e sattering is indeed one of the least important of the several sattering mehanisms in a metal under normal onditions. However, the piture that e-ion sattering is the major sattering mehanism is also not aurate. 3. An eletron is assumed to experiene a ollision with a probability per unit time of 1/. In other words, the probability of an eletron undergoing a ollision in an infinitesimal time interval, dt, is just dt/. The time is variously known as the relaxation time, ollision time, or mean free time. In Drude s model, is assumed to be independent of the eletron position and is independent of time. 4. Eletrons are assumed to ahieve thermal equilibrium with their surroundings only through ollisions. In partiular, immediately after eah ollision, an eletron is taken to emerge with a veloity totally unorrelated with its veloity before ollision. Moreover, it is randomly direted and assumes a magnitude aountable by loal thermal equilibrium at the loation where the ollision took plae (i.e., mv final / = (3/)k B T loal ). Thus the hotter the region in whih a ollision ours, the faster an eletron will emerge on average from the ollision. In the following, we shall examine how Drude used these assumptions to build models prediting the eletroni transport properties of metals, and how well the preditions desribe the experiments or reality.

3 1.1 DC Eletrial Condutivity of a Metal Aording to Ohm s law: E = j, where j is the urrent density (= urrent per unit area). Note that it is a vetor quantity. E is the eletri field (= V/L where V is the applied voltage differene giving rise to the urrent, and L is the distane over whih the voltage differene is applied.) To relate j to the average veloity, v, of the eletrons, onsider a metal wire, with a ross-setional area, A, and eletron number density n, and suppose that the eletrons in the wire are moving at a uniform veloity, v, as shown below: v vdt A After a given time lapse, dt, the eletrons would have traversed a distane of vdt through the wire. This gives the number of eletrons rossing an area A in time dt to be nva dt. Sine eah eletron arries a harge of e, this rossing of eletrons gives rise to a urrent density (i.e. harge per unit area per unit time): j = env. (1.3) Next, we examine how j is related to E due to Drude s assumptions. Consider an arbitrary eletron, and let t be the time elapsed sine its last ollision. Suppose its veloity right after the last ollision is v o. After time t, this eletron would have aumulated a veloity of v o eet/m based on Newton s seond law. Sine Drude assumed that the eletrons emerge in random diretions from a ollision, there will be no ontribution to the average veloity from v o. It follows that the urrent density must ome entirely from the average of eet/m. Denote the average elapsed time between ollisions or the relaxation time by, we find from eqn. (1.3) that j = env avg = (ne /m)e. (1.4) Using Ohm s law, and that eletrial ondutivity, = 1/, we have: j= E; ne /m. (1.5) Rearranging this equation, we have m/ne. Typial values of for metals at room temperature are of the order of Ohm-m. Therefore, it is more onvenient to express in terms of in Ohm-m instead of (in Ohm-m) as follows: 3

4 = m m 10 m ne -6-8 m kg 1 4 r S 10 3 (0.510 m) 16 8 ( ) C 10 m 3 a r. 10 s S a0 This result suggests that the relaxation time is of the order of to10-14 se. To see whether this is a reasonable estimate, we use it to estimate the mean free path, l ( v 0. Using the equipartition law, (½)mv 0 = (3/)k B T, where k B is the Boltzmann onstant and T is absolute temperature, we have 3-1 3k T JK 300 K 31 m kg ms -1 = ms -1 B v0 This gives l 1 10 Å at T = 300K. Note that the additional ontribution to v from E is ee/m (~ 1.7 m/s), thus is negligible. At lower temperatures, the Drude model predits a smaller still value of l. In later hapters, you will see that the above estimate of v 0 is an order of magnitude less than the atual value at room temperatures. Furthermore, at very low temperatures, v 0 gets 10 times bigger than the room-temperature value and is independent of T. This raises the mean free path to >1000 Å, i.e., ~1000 times the interioni spaing. This is strong evidene that the eletrons do not simply bump off the ions. In the absene of a theory of the eletron ollisions and hene the relaxation time,, it beomes important to find preditions of the Drude model that are independent of the value of. There are several -independent quantities that are derivable by the Drude model and still of fundamental interest today. Below, we shall disuss three suh quantities, namely the d/a eletrial ondutivities in the presene of a (spatially) uniform d magneti and a uniform d/a eletri field and the ratio of the thermal to eletrial ondutivities (Wiedemann-Franz law). 4

5 1. Hall Effet and Magnetoresistane Consider a metal blok subjet to an externally applied uniform eletri field pointing in the +x diretion (= E x x), and a uniform magneti B field pointing in the +z diretion as shown below (Hall s experiment, 1879): B -ev B Suppose that the applied eletri field brings about an eletroni urrent density of j x in the +x diretion. In the presene of the applied magneti field B, eah eletron onstituting the urrent experienes a Lorentz fore of ev B (= evb(x z) = evb y) ating along the y-diretion. Sine the two sides of the metal blok along y are not onneted, eletrons defleted by the Lorentz fore towards the y side annot esape, resulting in an aumulation of the eletrons there. Beause of eletrial neutrality, there must be the same amount of positive harges aumulated on the +y side of the blok, leading to an eletri field, E y, pointing in the y-diretion. In the steady state, the field E y stops further defletion of the eletrons by the Lorentz fore. This happens when the fore from E y equals to the Lorentz fore or E y = vb. There are two quantities of interest. One is the magnetoresistane, (B): The other quantity is the Hall oeffiient, R H : (B) = E x /j x. (1.6) R H = E y /(j x B). (1.7) Notie that for eletrons, E y points in the y-diretion and R H is negative. But if the moving harges were positive, whih was unthinkable in Drude s time, the Lorentz fore (now = +evb (x z)) would still be pointing in the y-diretion. But beause the moving harges are positive, the Lorentz fore leads to aumulation of positive harges in the y side, and negative harges in the +y side of the metal blok. This auses the transverse field E y to be direted in the +y-diretion and the Hall oeffiient R H to be positive. It 5

6 turns out some metals (e.g. Be, Mg, In, Al) exhibit positive values of R H at a large magneti field of 1 Tesla. To proeed, one observes that the average veloity, v(t) of the eletrons at time t is related to its average momentum, p(t) by v = p(t)/m. We are aspired to alulate the average momentum of the eletrons at a later time t + dt. If an eletron does not enounter a ollision before this time, it would be aelerated by the total fore, f(t), arising from the applied E and B fields. Aording to Drude s assumptions, eletrons as suh amount to a fration (1 - dt/) of the total. By using this and Newton s seond law, we have p(t + dt) = (1 - dt/)[p(t) + f(t) dt + O(dt) ]. = p(t) (dt/)p(t) + f(t) dt + O(dt). (1.8) Note that we have negleted the ontribution from the rest of the eletrons that have undergone a ollision within dt. It is beause the diretions of motion of these eletrons would be randomized after the ollision and so the average momentum (a vetor) of these eletrons would be zero. The orretion term O(dt) omes from the fat that the fore is atually varying from f(t) at time t to f(t + dt) at time (t + dt) with the amount of variation being of order dt. Equation 1.8 an be rearranged to give: Taking the limit dt 0, we have: p(t + dt) p(t) = (dt/)p(t) + f(t) dt + O(dt). (1.9) dp(t)/dt = p(t)/ + f(t). (1.10) This simply states that the effet of ollisions is to introdue a fritional damping term to the equation of motion of the eletrons. We may now use this equation to alulate the magnetoresistane and Hall oeffiient. Given Hall s onditions, eqn. (1.10) an be rewritten as: dp p p e E B. (1.11) dt m dp In steady state, 0. Hene dt eb px 0 eex py. (1.1) m eb py 0 eey px. (1.13) m Write eb / m, and multiply the above equations by ne / m, then use j nep / m, one obtains: 0Ex jy jx, (1.14) 6

7 0Ey jx jy, (1.15) where 0 is the Drude model DC ondutivity in the absene of a magneti field. The Hall field is determined by the ondition that there be no transverse urrent j y. Setting j y = 0 in eq. (1.15), we have: R H Ey 1. (1.16) j B ne x Equation (1.16) predits that R H depends on no parameters of the metal, exept the density of the harge arriers. Moreover, R H should only be negative, as expeted from Drude s presumption that the harge arriers are eletrons. It turns out eqn. (1.16) provides good agreement with the alkali metal only. As pointed out above, some metals show positive R H. Appendix: Derivation of : Consider an eletron moving in a irle with radius r in the x-y plane, under a magneti field B field pointing in +z. F B ev B = -evbr The Lorentz fore provides the entripetal fore that sustains the irular motion: evb = mv /r eb = mv/r = m = eb/m We an similarly determine the magnetoresistane by taking j y = 0 in eq. (1.14), whih give: Ex 1 ( B ). (1.17) j This result shows that the magnetoresistane is the same as the zero-field resistivity, whih agrees to some degree with experiment. x AC Eletrial ondutivity of a metal Consider an applied AC eletri field of the form: E ( t) E ( )exp( it). (1.18) The equation of motion for an eletron under the influene of this eletri field is: 7

8 dp dt p ee. (1.19) We assume a solution to Eq. (1.19) of the form: p ( t) p ( )exp( it). (1.0) and substitute Eqs. (1.18) and (1.0) into Eq. (1.19). It leads to: p( ) ip( ) ee( ) ee( ) p( ). 1/ i Further use j() = nep()/m, we have: ( ne / m) E( ) j( ). 1/ i Writing this result in the form j() = ()E(), we may find: a ondutivity ne ( ), 0. (1.1) 1 i m R / 0 I / 0 Note that Eq. (1.1) orretly redues to the Drude DC ondutivity when frequeny is equal to zero. An important appliation of this result is the propagation of EM waves in a metal. At first sight, it appears that Eq. (1.1) annot be appliable sine (1) the alulation does not aount for the magneti field, B() of the EM wave, and () the derivation assumes the E() field to be spatially uniform. For point (1), the effet of B()is in fat negligible sine B()/E() ~ (()()) 1/ ~ 1/ (where is the speed of light, () the eletri permittivity, and () the magneti permeability of the metal). Therefore, the ratio of the Lorentz fore (due to B()) to the eletri fore (due to E()) is ~ evb()/e E() ~ v/ (v is the average veloity of the eletrons) ~ 0.01, hene is negligible. As for point (), we reall that Eq. (1.19) assumes the E field to exerise its influene on the eletron motion only over the distane between ollisions. On average, this distane is of the order of the mean free path, l, whih is 1 to 10 Å. Most of our disussions onern visible or UV lights, where the wavelength is 10 3 to 10 4 Å (>> l), so eqn. (1.1) is appliable. 8

9 With the above result, we examine how an EM wave with wavelength muh bigger than the eletron mean free path propagates in a metal. Using Maxwell s equations, we have: E 0, B 0, ( E 0 no net harge.) E B / t, B r j (1/ ) E /. ( r 1 for most non-magneti metals.) 0 t Substitute B = B()exp(-it), E = E()exp(-it) and j = ()E()exp(-it) in the last two equations. We obtain: E ib B ( ( ) i / ). 0 E E ( i0 ( ) / ) E. ( ( E) E E E 0) E i E. 0 1 i In the above, we have used = 1/( 0 0 ) 1/, = 0 /(1 it) from Eq. (1.1). In the limit, >> 1, and use 0 = ne /m, we have: ne / m 0 E 1 E. (1.) Writing Eq. (1.) in the form E ( ) E, (1.3) and p = ne /m 0, we have: p ( ) 1, (1.4) where p is widely known as the plasma frequeny. Appendix: Derivation of eqn. (1.3) : p E j 0 ( 1) (where p here denotes eletri polarization, not momentum) t t 9

10 E 1 B 0 0 ( 1) t E = i B = ( / )E E E t t i E If we now write E = E(,t)exp[i(k r t)], and substitute it in Eq. (1.3), we obtain the usual dispersion relation: k ( ). (1.5) Eq. (1.4) shows that for < p, is negative, and by eq. (1.5) k = ik I is purely imaginary. That means E ~ exp(ikz) ~ exp(-k I z) and so deay exponentially in spae (Here, without loss of generality we have assumed k = kz). But for > p, is positive and k = k R is real, so Eexp(ikz) ~ exp(ik R z)an propagate inside the metal. Notie that this result is valid only if the above assumption, namely >> 1, holds in the neighborhood of p. By using 1/n = 4r s 3 /3 and r s 3 =a 0, we dedue that p ~ rad s -1. (k p ~ p / ~ m -1 = /k = m, in UV.) Reall that ~ s. Hene, >> 1. Indeed Eq. (1.4) has been found to be obeyed by the alkali metals. But notie that in other metals, different ontributions to the dieletri funtion ompetes with the Drude term shown in Eq. (1.4). At = p, = 0 from Eq. (1.4). By Eq. (1.5), this auses k = 0 or = /k =. Therefore, the propagation mode with p orresponds to a uniform osillation of the eletron sea in the metal against the positively harged bakground. This mode is known as the plasma osillation. Derive p : Consider the eletron sea in a metal displaed uniformly by distane x in the +x diretion. This auses aumulation of a surfae harge density, = -nex on the right side and +nex on the left side of the metal. E = 0 E = 0. (Notie that here denotes the surfae harge, not eletrial ondutivity.) Apply Newton s law, namely (mass)(aeleration) = fore, we have md x/dt = (-e)e = (-e)(nex/ 0 ) (E = (surfae harge density)/ 0 and is in the same/opposite diretion as/to the surfae normal if the surfae harge is positive/negative.) d x/dt = -ne x/( 0 m) p = ne / 0 m) 10

11 1.4 Thermal ondutivity of a metal When a metal bar is subjet to a temperature gradient, T, heat flows from the high temperature side to the low temperature side. If T is small, the flow of thermal energy per unit area per unit time (= thermal urrent density, j q (in Wm - )) is related to T by: j q = -T, (1.6) where is the thermal ondutivity (in Wm -1 K -1 ). Note that j q is always in a diretion opposite to T beause j q is direted towards the low temperature side whereas T inreases towards the high temperature side. As a result, is always positive. Without loss of generality, we hoose T and hene the heat urrent to be along x: 1/nv x 1/nv x T x v x x x + v x (T[x v x T x (T[x v x (T T At any point x, eletrons may migrate from either the +x or x diretion with number density per unit time of nv x and nv x, respetively. Reall that the eletrons, through ollisions, aquire the loal temperature at where they reside in. At point x, the most nearby ollisions our at x v x and x v x in the x and + x diretion, respetively. On arriving at x, the eletrons bring in the thermal energy they just aquired from the last ollision, i.e., (T[x v x for those migrating from x and (T[x v x for those from +x. The thermal urrent density is thus: j q = nv x /{ (T[x v x (T[x v x } (1.7) If v x is very small, we may expand Eq. (1.7) about x: j q = nv x (d/dt)(d/dx) (1.8) In 3-dimensions (3D), v x in Eq. (1.8) would be the x-omponent of the average veloity of the eletrons, whih an have omponents along y and z. Sine <v x >=<v y >=<v z > = v /3, and n d/dt = v, the eletron speifi heat apaity (JK -1 m -3 ), we have: j q = (1/3)v v (T) (1.9) Hene, = (1/3)v v = (1/3)lv v, (1.30) where v is the mean square eletron veloity, l is the mean free path. 11

12 The most impressive suess of the Drude model at the time was its ability to explain the empirial law of Wiedemann and Franz (W-F) (1853). The law states that the ratio, for metals is diretly proportional to T. In the following, we shall derive this law by applying the Drude model. Speifially, we apply eqs. (1.5) and (1.9) and find: (1/3) v mv ne. (1.30) In the spirit of the Drude model, i.e., kineti theory of eletron gas, it is natural to assume v = (3/)nk B and (1/)mv = (3/)k B T. This leads to the result: T 3 kb 8 e W/K. (1.31) The above ratio is alled the Lorentaz number. Experimentally found values of the Lorentz number are typially ~.5e-8 J/K s, whih is in remarkable agreement with the Drude value. It turns out this agreement is a result of an inidental anellation of two quantities in eq. (1.30), namely v and v, that are ~100 times off from their atual values. Notie that Drude had only onsidered the eletroni ontribution to the thermal ondutivity. It was rationalized by the observation that the thermal ondutivity of metals was usually higher than the insulators. In the next setion, we briefly disuss the reason. However, it involves onepts that we won t over until we disuss Lattie Dynamis. Thermopower In the above derivation, we have negleted the possibility that v(x v) and v(x + v) may be different due to the temperature gradient. It turns out it is the orret to assume them to be the same. It is beause the thermal ondutivity measurement is onduted in an open iruit onfiguration, so any net flow of the eletrons to the low temperature side will ause aumulation of the eletrons on that side, produing an eletri field whih opposes the flow. At equilibrium, the effet of the eletri field, E, exatly anels that of temperature gradient, T. This is alled the Seebek effet. The thermopower, Q, relates E and T as follows: E = QT (1.3) To estimate the thermopower, note that in the above one-dimensional heat flow model the mean eletron veloity at a point x due to the temperature gradient is v Q = ½[v x (x v x v x (x v x ] = dv x d v v x dx dx. (1.33) 1

13 Write v = v x /3, we have v Q dv T 6 dt (1.34) The mean veloity due to the thermoeletri field is Requiring that v Q = v E, we have v E = -ee/m (1.35) d Q 3e dt mv k ne e 1 v B 4 V/K. (1.36) Observed values of thermopower at room temperature are of the order of V/K, whih is 100 times smaller than Drude s predition, illustrating inadequay of his model. 13