Introduction to Solid State Physics
Introduction to Solid State Physics PHYS 4340
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This 5 page Class Notes was uploaded by Mrs. Peter Toy on Friday October 30, 2015. The Class Notes belongs to PHYS 4340 at University of Colorado at Boulder taught by Staff in Fall. Since its upload, it has received 13 views. For similar materials see /class/232125/phys-4340-university-of-colorado-at-boulder in Physics 2 at University of Colorado at Boulder.
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Date Created: 10/30/15
NOTES ON THE DRUDE MODEL KYLE MCELROY ABSTRACT Here are some notes about the classical theory of metals and the Drude model The classical theory of electrons in metals is used to describe the basics of metallic behavior These include their electrical conductivity heat conductivity re ection of optical wavelengths etc 1 ASSUMPTIONS AND BASICS The Drude model was developed at the turn Of the 20th century by Paul Drudei It came a few years after ll Thompson discovered the electron in 1897 It predates quantum theory but still can tell us a lot about electrons in metals ackground the the model we should get tO know the electrons and how many we are dealing with We re going tO keep the valence assumptioni This assumption rests on the intuition that the core electrons will be more tightly bound tO their nuclei and hence will not be free tO wander around and contribute tO conductioni Essentially this lowers the number Of electrons from Z tO ZC where ZC is the number Of conduction electronsi SO in a sample Of metal say Sodium Na the density Of conduction electrons n is 11 n NA leat07n1gtlt105g7n3 ZQgmol where NA is AvogadrO s number pm is the density Of the metal A is the atomic number Of the element and the numbers are for Na For the actual model we re going tO eliminate all the electron ion interactions and replace them by a single parameter Z 2 602 X 1023awmsmoz 2 X 10286m3 1 We will treat collisions between els and ions are instantaneous uncorrelated even si 2 We will ignore all other interactions iiei potentials from ions or other electrons except applied elds This means that electrons travel in straight lines between scattering events 3 Probability Of an electron having a collision in a time interval dt is dtTi n 739 oes not depend on the electron position or momentumi 4 Collisions 7thermalize7 electronsi This means that after a collision the elec trons have the temperature Of the local environment 2 EQUATIONS OF MOTION The rst thing you need is tO gure out how an electron s momentum on average will evolve over time TO dO this we7ll just nd the average equation Of motion for an electron TO nd this let s start with the momentum Of an electron at time t 1 2 KYLE MCELROY 15t and nd it at time t dti If the electrons had a collision it would on average have no momentum 1300 dt 0 at time t dt and by the third assumption above this has the probability PC dtTi This means that the probability of no collision is PM l 7 dtT this is because PC PM 1 If there were no collision the electrons would have evolved normally and the electrons momentum becomes 17m t dt F tdti This makes the new momentum 21gt 5m dtgt PC ctdt PM 4M dtgt lt17 mm aw Using this to nd the derivative take 50 1W dt 50 50 d dt 7 dt 7 739 Fa And you have the equation of motion averaged over electronsi Of note there are a few regimes and solutions to consider o If 0 the solution to this homogeneous equation is 0e which is why 739 is called the relaxation time If you impart momentum to the electrons on average they will relax back to no momentum exponentially with a time constant 739 0 With a constant you can show that the solution to the momentum 15390equotT FT 0 After along time t gt 739 the exponential term becomes negligible leaving 13t FT 22 47 3 OHM7s LAW You all know Ohm s law 31 V IR with V is the Voltage applied to a metal I is the resulting current and R is the proportionality constant The main empirical fact here being the proportionality of current to the applied Voltagei This is what we7re going to try to predict But let s recast it in a form that is not dependent on the geometry of the experiment 32 i US where a is the conductivity of the metal and is the current density Since is the current density it is the number of electrons passing a given point or j lnev where n is still the electron density 6 is still the electron charge and v is the average drift velocity of the electrons In an applied electric eld E the EoM for long times t gt 739 gives us 13t GET GET 17 t 7 0 m Plugging this into the expression for the current density gives us 2 33 i KE m which is Ohmls law with a ne2Tm There are a couple of implications of this For a metal like Na with a resistivity pNa 10 50 nQm the relaxation time is about 10 s 4 AC CONDUCTIVITY AND THE SHINY ln HW1 you proved that a complex form of Ohm s law still holds for a metal in the Drude model with an oscillating electric eld incident on a metal iiei Et ReE weiWi With an E field of this form Ohmls law takes the form a ne2739 l a J w 7 M where the w dependent conductivity is given by 2 l 0wne 739 m inl Since the electric eld is oscillating electric we have an electromagnetic wave So we should look at Maxwellls equations which with no net charge look like vE0 v30 a a E a a a a VXE E VXB0jM060 Since we have the complex Ohmls law we can replace the current density in the last equation and simplify it to be 6 X E Meow in060E Lets take the curl of the third equation and use the first one and what we just found to simplify it to d X E a T WWOUW ZWM060E After some rearranging this can be written as wwmjm 7W gm 71 c wee Now notice that if you take the limit where wT gtgt 17 00 A aoin leaving us with 2 2 a2 a w me a V E 7 62 m60w2 1E Now if you assume that the electric eld has a plane wave spatial structure ie Et ReE weiweik39F then the above equation simpli es to the dispersion relation 16252 02 7 mg with mp nEQmeo being the plasma frequency This dispersion relationship tells us that k is imaginary for w lt mp which is the result for an exponentially decaying electric eld This means that the EM wave doesn7t propagate into a metal at these lower frequencies and instead is reflected On the other hand for w gt mp this gives a real k which is the form for a travelling wave This implies that for frequencies higher than the plasma frequency metals become transparenti Which experiments verify see gure For metals the plasma frequency is in the ultra violet and for Na it is about mp 1015s 1 which corresponds to a wavelength of light at A 200nmi 4 KYLE MCELROY Note We disregard the direct magnetic eld effects because they are much smaller than the electric eld forces 5 JOULE HEATING See the solution set for HW1 6i HALL EFFECT We worked out what happened with a constant electric eld put onto a metal But what if you use a Voltage to drive current through a metal and put a perpen dicular magnet eld on it 7 THERMAL CONDUCTIVITY The electrons in a metal aren t just good conductors of electricity7 they are also good at conducting heati So we re asking about the amount of heat that travels along a sample of metal as heat is applied There is a simple phenomenological model that is analogous to Ohm s law7 it is called Fourier7s law and it looks like i4 ian The minus sign is because heat ows to lower temperature7 and q is the heat current density AQTimeA7 heat per time per area in units of Wattsm2 and H is the thermal conductivity which has units of WattsmKli So what does the Drude model predict A couple of assumptions are important here First we need to remember the last assumption of the Drude model This assumption means that after a collision an electron carries the thermal energy of the local environment Also we need to assume that T varies a little over 1 the mean free path electrons travel before scatteringi To calculate the heat delivered to a point we need to simplify this to a 1D problem and then we need to calculate the heat it gets from the left and subtract the heat it gets from the right Let7s write the thermal energy an electron at temperature T has as Ti Then at a point z the average electron coming from the left brings with it an energy 5Tz 7117 the average electron from the right delivers 5Tz 117 So for electrons of density n and average velocity v remembering that half will travel towards the point z and half away this leaves us with q nv5Tz 7 v7 7 5Tz UT a 1 d8 dT q 7 7 7 2nvdT dz m39 a d8 dT 39q 2 i if J m 7dT d1 CV or the speci c heat per volume Also when we generalize to 3D we note that lt v gtlt vy gtlt 1 gt lt v gt i Putting all this together we have Now note a few things n Hg lt R H 2 TTCVEVT So the prefactor up there is equal to the thermal conductivity There is another way to look at the speci c heat Its called the Wiedemann Franz ratioi Which is given by n m lt v gt2 TcV E 7 362 Now in a classical gas using the equipartition theorem the total thermal energy per volume is given by E nngT so the speci c heat per volume the derivative of the energy With respect to energy is CV nngi And also note that the total thermal energy is the kinetic energy is given by m lt 1 gt2 nngT Which we can use to get rid of the m lt v gt2 in the WF ratioi All together this gives H 3163 UT This number only has fundamental constants in it and it is 12 the real value you get experimentally But is really close for such a simple model 2 1i11X10 8WQK2
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