Lecture contents. Metals: Drude model Conductivity frequency dependence Plasma waves Difficulties of classical free electron model

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1 Lecture contents Metals: Drude model Conductivity frequency deendence Plasma waves Difficulties of classical free electron model Paul Karl Ludwig Drude (German: [ˈdʀuːdə]; July, 863 July 5, 96)

2 Phenomenology of electron transort: relaxation time In conductors, valence electrons are treated as free electrons: free article swarm (Drude model) Electron motion in the field: Electrons exerience collisions similar to gas molecules in the kinetic theory of gasses Extra average velocity due to electric field: Equivalent to a friction force on a free electron : Thermal velocity is much higher than drift velocity v th 3kBT m Motion in real sace = thermal motion + drift + scattering 7 cm v s d dv m qe dt v d dv dt q E m d m qe m Relaxation time vd

3 Phenomenology of electron transort: mobility 3 Current density is roortional to drift velocity of carriers Concentration n is taken as a density of valence electrons In the steady state drift velocity is roortional to the field (m drift mobility): J qnv d q vd E m E m - with mobility m [cm /V-s] introduced in metals and semiconductors m q m And current density (s conductivity) gives Ohm s law: J qnv qnme se s d q qnm n m

4 Resistivity of metals 4 Drude theory was successful to describe basic tendencies of metal conductivity: Room temerature resistivity of single crystalline metals Deendence on Crystalline quality Temerature Alloying Frequency

5 Frequency deendence of conductivity of metals 5 Let s use simle microscoic icture: (relaxation time aroximation) and find how the conductivity deends on frequency of EM field Assuming the electric field along x-direction: E x mx qe x B i t E e x Drift velocity which can change in time and sace Smaller than electric We get the dislacement x q m i E The resonse of the material can be described with olarization (not at DC) P qnx E And dielectric function: where s static conductivity) qn s m i i we used q s n m

6 Dielectric function: At low frequency, we return to static conductivity Plasma frequency qn s m i i s s i Comare with general henomenological disersion relation: i s 6 At high frequency With introduced lasma frequency q n q n m m qn s m becomes real no attenuation! does not deend on! Let s estimate lasma frequency in metals: f qn. m 5 Hz 36 nm Dee UV

7 Dielectric function: With lasma frequency Drude otical roerties of metals s i id qn m and daming frequency At low frequency We can write otical constants (refraction and extinction indexes) d n d d d n n 7 From Hummel,

8 Drude otical roerties of metals 8 Plasma frequency qn m To imrove accuracy of Drude model, effective number of free electron is usually introduced N eff (observed) (calculated) Daming frequency d Daming frequency (scattering time) generally correlates with conductivity but not accurately From Hummel,

9 Otical roerties of metals and dielectrics 9 Reflectivity R n n n n In metals, the major feature is lasma edge, also some interband transitions aear at higher (UV) frequencies Above lasma frequency there is no difference between metals and dielectrics In dielectrics, there are vibration-related features in IR and band features in UV as in metals

10 Plasma waves Plasma frequency can be considered as maximum frequency of lasma resonse. It corresonds to internal electrostatic oscillations of lasma The electric field ulls the electrons back towards equilibrium, where they exactly neutralize the ion charge, but the kinetic energy gained in this rocess causes the electrons to overshoot to a new dislacement on the other side. Let s consider the simlest mode of lasma oscillations D oscillations, resulting in B= The nd Maxwell equation is Assuming that oscillations are faster than scattering time, the motion equation of carriers is Current density is as usual D H J t E B t E J t v m qe t J qnv substitute We get equation for oscillator with frequency : In this simlest case the wavevector does not deend on frequency More sohisticated analysis considering thermal motion gives weak deendence on wavevector v t nq v m qn m 3kvth

11 Difficulties of classical free electron model: electric roerties Mass of electron obtained for cyclotron resonance may differ significantly from free electron mass Hall effect may show ositive sign of carriers transorting current Does not exlain temerature deendence of conductivity in metals Exerimentally Since kinetic energy of electron and scattering time We can exect And conclude that -ad hoc assumtion of the model s T q s n v T m l f T E th mv th 3 kt l v f th Effect of temerature on resistivity of metals Crystal structure effects are ignored Periodicity of crystal is ignored Anisotroy of conductivity in some non-cubic metals Predicts two orders higher aramagnetic suscetibility than measured in exeriment

12 Difficulties of classical free electron mode: thermal roerties Problem even with the best triumh of the free-electron K model: Wiedemann-Franz law relationshi between C WF T thermal conductivity K and electron conductivity s : s Exact statistical analysis according to kinetic theory of gases shows that 3 8 n T. T K de k W s dt e SK Wiedemann-Franz constant is about times smaller than exerimental value Problem with the secific heat: Average energy of electron is E 3 kt Secific heat = change of average energy er unit volume with temerature The lattice with atom density N will contribute at room temerature But tyically secific heat density of metals is not higher than that of dielectrics de 3 Cv n nk dt C v, latt 3 Nk

13 Quantum theory of solids 3 Quantum mechanical treatment of carriers: wave functions, bands Periodic otential Bloch formalism: symmetry oints Fermi statistics Note: Classical free electron model is extremely useful in semiconductors