The free electron theory of metals The Drude theory of metals Paul Drude (1900): theory of electrical and thermal conduction in a metal application of the kinetic theory of gases to a metal, which is considered as a gas of electrons mobile negatively charged electrons are confined in a metal by attraction to immobile positively charged ions isolated atom nucleus charge eZa Z valence electrons are weakly bound to the nucleus (participate in chemical reactions) Za – Z core electrons are tightly bound to the nucleus (play much less of a role in chemical reactions) in a metal – the core electrons remain bound to the nucleus to form the metallic ion the valence electrons wander far away from their parent atoms called conduction electrons or electrons in a metal

Doping of Semiconductors C, Si, Ge, are valence IV , Diamond fcc structure. Valence band is full Substitute a Si (Ge) with P. One extra electron donated to conduction band N-type semiconductor

density of conduction electrons in metals ~1022 – 1023 cm-3 rs – measure of electronic density rs is radius of a sphere whose volume is equal to the volume per electron mean inter-electron spacing in metals rs ~ 1 – 3 Å (1 Å= 10-8 cm) rs/a0 ~ 2 – 6 Å – Bohr radius ● electron densities are thousands times greater than those of a gas at normal conditions ● there are strong electron-electron and electron-ion electromagnetic interactions in spite of this the Drude theory treats the electron gas by the methods of the kinetic theory of a neutral dilute gas

The basic assumptions of the Drude model 1. between collisions the interaction of a given electron with the other electrons is neglected and with the ions is neglected 2. collisions are instantaneous events Drude considered electron scattering off the impenetrable ion cores the specific mechanism of the electron scattering is not considered below 3. an electron experiences a collision with a probability per unit time 1/τ dt/τ – probability to undergo a collision within small time dt randomly picked electron travels for a time τ before the next collision τ is known as the relaxation time, the collision time, or the mean free time τ is independent of an electron position and velocity 4. after each collision an electron emerges with a velocity that is randomly directed and with a speed appropriate to the local temperature independent electron approximation free electron approximation

DC electrical conductivity of a metal V = RI Ohm’s low the Drude model provides an estimate for the resistance introduce characteristics of the metal which are independent on the shape of the wire j=I/A – the current density r – the resistivity R=rL/A – the resistance s = 1/r - the conductivity L A v is the average electron velocity

at room temperatures resistivities of metals are typically of the order of microohm centimeters (mohm-cm) and t is typically 10-14 – 10-15 s mean free path l=v0t v0 – the average electron speed l measures the average distance an electron travels between collisions estimate for v0 at Drude’s time → v0~107 cm/s → l ~ 1 – 10 Å consistent with Drude’s view that collisions are due to electron bumping into ions at low temperatures very long mean free path can be achieved l > 1 cm ~ 108 interatomic spacings! the electrons do not simply bump off the ions! the Drude model can be applied where a precise understanding of the scattering mechanism is not required particular cases: electric conductivity in spatially uniform static magnetic field and in spatially uniform time-dependent electric field Very disordered metals and semiconductors

motion under the influence of the force f(t) due to spatially uniform electric and/or magnetic fields equation of motion for the momentum per electron electron collisions introduce a frictional damping term for the momentum per electron average momentum average velocity Derivation: fraction of electrons that does not experience scattering scattered part total loss of momentum after scattering

Hall effect and magnetoresistance Edwin Herbert Hall (1879): discovery of the Hall effect the Hall effect is the electric field developed across two faces of a conductor in the direction j×H when a current j flows across a magnetic field H the Lorentz force in equilibrium jy = 0 → the transverse field (the Hall field) Ey due to the accumulated charges balances the Lorentz force quantities of interest: magnetoresistance (transverse magnetoresistance) Hall (off-diagonal) resistance resistivity Hall resistivity the Hall coefficient RH → measurement of the sign of the carrier charge RH is positive for positive charges and negative for negative charges

force acting on electron equation of motion for the momentum per electron in the steady state px and py satisfy cyclotron frequency frequency of revolution of a free electron in the magnetic field H at H = 0.1 T multiply by the Drude model DC conductivity at H=0 the resistance does not depend on H RH → measurement of the density weak magnetic fields – electrons can complete only a small part of revolution between collisions strong magnetic fields – electrons can complete many revolutions between collisions j is at a small angle f to E f is the Hall angle tan f = wct

measurable quantity – Hall resistance for 3D systems for 2D systems n2D=n in the presence of magnetic field the resistivity and conductivity becomes tensors for 2D:

strong magnetic fields quantization of Hall resistance weak magnetic fields the Drude model the classical Hall effect strong magnetic fields quantization of Hall resistance at integer and fractional the integer quantum Hall effect and the fractional quantum Hall effect from D.C. Tsui, RMP (1999) and from H.L. Stormer, RMP (1999)

AC electrical conductivity of a metal Application to the propagation of electromagnetic radiation in a metal consider the case l >> l wavelength of the field is large compared to the electronic mean free path electrons “see” homogeneous field when moving between collisions

response to electric field mainly historically both in metals and dielectric described by used mainly for electric current j conductivity s = j/E metals polarization P polarizability c = P/E dielectrics dielectric function e =1+4pc electric field leads to e(w,0) describes the collective excitations of the electron gas – the plasmons e(0,k) describes the electrostatic screening

AC electrical conductivity of a metal equation of motion for the momentum per electron time-dependent electric field seek a steady-state solution in the form AC conductivity DC conductivity the plasma frequency a plasma is a medium with positive and negative charges, of which at least one charge type is mobile

equation of motion of a free electron even more simplified: no electron collisions (no frictional damping term, ) equation of motion of a free electron if x and E have the time dependence e-iwt the polarization as the dipole moment per unit volume

Application to the propagation of electromagnetic radiation in a metal transverse electromagnetic wave

Application to the propagation of electromagnetic radiation in a metal electromagnetic wave equation in nonmagnetic isotropic medium look for a solution with dispersion relation for electromagnetic waves (1) e real and > 0 → for w is real, K is real and the transverse electromagnetic wave propagates with the phase velocity vph= c/e1/2 (2) e real and < 0 → for w is real, K is imaginary and the wave is damped with a characteristic length 1/|K|: (3) e complex → for w is real, K is complex and the waves are damped in space (4) e =  → The system has a final response in the absence of an applied force (at E=0); the poles of e(w,K) define the frequencies of the free oscillations of the medium (5) e = 0 longitudinally polarized waves are possible

Transverse optical modes in a plasma dispersion relation for electromagnetic waves (1) for w > wp → K2 > 0, K is real, waves with w > wp propagate in the media with the dispersion relation an electron gas is transparent (2) for w < wp → K2 < 0, K is imaginary, waves with w < wp incident on the medium do not propagate, but are totally reflected w/wp (2) (1) metals are shiny due to the reflection of light w/wp w = cK electromagnetic waves are totally reflected from the medium when e is negative electromagnetic waves propagate without damping when e is positive and real forbidden frequency gap cK/wp vph > c → vph does not correspond to the velocity of the real physical propagation of any quantity

Ultraviolet transparency of metals plasma frequency wp and free space wavelength lp = 2pc/wp range metals semiconductors ionosphere n, cm-3 1022 1018 1010 wp, Hz 5.7×1015 5.7×1013 5.7×109 lp, cm 3.3×10-5 3.3×10-3 33 spectral range UV IF radio the reflection of light from a metal is similar to the reflection of radio waves from the ionosphere electron gas is transparent when w > wp i.e. l < lp plasma frequency ionosphere semiconductors metals reflects transparent for metal visible UV ionosphere radio visible

Skin effect when w < wp electromagnetic wave is reflected the wave is damped with a characteristic length d = 1/|K|: the wave penetration – the skin effect the penetration depth d – the skin depth the classical skin depth d >> l – the classical skin effect d << l – the anomalous skin effect (for pure metals at low temperatures) the ordinary theory of the electrical conductivity is no longer valid; electric field varies rapidly over l not all electrons are participating in the absorption and reflection of the electromagnetic wave only electrons that are running inside the skin depth for most of the mean free path l are capable of picking up much energy from the electric field only a fraction of the electrons d’/l are effective in the conductivity d’ l

Longitudinal plasma oscillations a charge density oscillation, or a longitudinal plasma oscillation, or plasmon nature of plasma oscillations: displacement of entire electron gas through d with respect to positive ion background creates surface charges s = nde and an electric field E = 4pnde equation of motion oscillation at the plasma frequency for longitudinal plasma oscillation wL=wp w/wp transverse electromagnetic waves longitudinal plasma oscillations forbidden frequency gap cK/wp equation of continuity for j~exp(-iwt) Gauss’s law

Thermal conductivity of a metal assumption from empirical observation - thermal current in metals is mainly carried by electrons thermal current density jq – a vector parallel to the direction of heat flow whose magnitude gives the thermal energy per unit time crossing a unite area perpendicular to the flow Fourier’s law k – thermal conductivity the thermal energy per electron 1D: to 3D: high T low T after each collision an electron emerges with a speed appropriate to the local T → electrons moving along the T gradient are less energetic the electronic specific heat Wiedemann-Franz law (1853) Lorenz number ~ 2×10-8 watt-ohm/K2 Drude: application of classical ideal gas laws success of the Drude model is due to the cancellation of two errors: at room T the actual electronic cv is 100 times smaller than the classical prediction, but v2 is 100 times larger

Thermopower Seebeck effect: a T gradient in a long, thin bar should be accompanied by an electric field directed opposite to the T gradient high T low T gradT E thermoelectric field thermopower mean electronic velocity due to T gradient: 1D: to 3D: mean electronic velocity due to electric field: in equilibrium vQ + vE = 0 → Drude: application of classical ideal gas laws no cancellation of two errors: observed metallic thermopowers at room T are 100 times smaller than the classical prediction inadequacy of classical statistical mechanics in describing the metallic electron gas

The Sommerfeld theory of metals the Drude model: electronic velocity distribution is given by the classical Maxwell-Boltzmann distribution the Sommerfeld model: electronic velocity distribution is given by the quantum Fermi-Dirac distribution Pauli exclusion principle: at most one electron can occupy any single electron level normalization condition T0

consider noninteracting electrons electron wave function associated with a level of energy E satisfies the Schrodinger equation L periodic boundary conditions 3D: 1D: a solution neglecting the boundary conditions normalization constant: probability of finding the electron somewhere in the whole volume V is unity energy momentum velocity wave vector de Broglie wavelength

apply the boundary conditions components of k must be the area per point the volume k-space apply the boundary conditions components of k must be nx, ny, nz integers a region of k-space of volume W contains states i.e. allowed values of k the number of states per unit volume of k-space, k-space density of states

the number of allowed values of k within the sphere of radius kF consider T=0 the Pauli exclusion principle postulates that only one electron can occupy a single state therefore, as electrons are added to a system, they will fill the states in a system like water fills a bucket – first the lower energy states and then the higher energy states the ground state of the N-electron system is formed by occupying all single-particle levels with k < kF state of the lowest energy volume density of states the number of allowed values of k within the sphere of radius kF kx ky kF Fermi sphere Fermi surface at energy EF to accommodate N electrons 2 electrons per k-level due to spin Fermi wave vector ~108 cm-1 Fermi energy ~1-10 eV Fermi temperature ~104-105 K Fermi momentum Fermi velocity ~108 cm/s compare to the ~ 107 cm/s at T=300K classical thermal velocity 0 at T=0

density of states total number of states with wave vector < k total number of states with energy < E the density of states – number of states per unit energy the density of states per unit volume or the density of states k-space density of states – the number of states per unit volume of k-space

Ground state energy of N electrons add up the energies of all electron states inside the Fermi sphere volume of k-space per state  smooth F(k) the energy density the energy per electron in the ground state

remarks on statistics I in quantum mechanics particles are indistinguishable systems where particles are exchanged are identical exchange of identical particles can lead to changing of the system wavefunction by a phase factor only repeated particle exchange → e2ia = 1 system of N=2 particles x1, x2 - coordinates and spins for each of the particles antisymmetric wavefunction with respect to the exchange of particles symmetric wavefunction with respect to the exchange of particles p1, p2 – single particle states fermions are particles which have half-integer spin the wavefunction which describes a collection of fermions must be antisymmetric with respect to the exchange of identical particles fermions: electron, proton, neutron bosons are particles which have integer spin the wavefunction which describes a collection of bosons must be symmetric with respect to the exchange of identical particles bosons: photon, Cooper pair, H atom, exciton if p1 = p2 y = 0 → at most one fermion can occupy any single particle state – Pauli principle unlimited number of bosons can occupy a single particle state obey Fermi-Dirac statistics obey Bose-Einstein statistics

distribution function f(E) → probability that a state at energy E will be occupied at thermal equilibrium fermions particles with half-integer spins bosons integer spins both fermions and bosons at high T when Fermi-Dirac distribution function Bose-Einstein Maxwell-Boltzmann degenerate Fermi gas fFD(k) < 1 Bose gas fBE(k) can be any classical gas fMB(k) << 1 m=m(n,T) – chemical potential

remarks on statistics II v x BE and FD distributions differ from the classical MB distribution because the particles they describe are indistinguishable. Particles are considered to be indistinguishable if their wave packets overlap significantly. Two particles can be considered to be distinguishable if their separation is large compared to their de Broglie wavelength. x y(x) vg=v superposition of many waves Dx g(k’) Dk k’ thermal de Broglie wavelength particles become indistinguishable when i.e. at temperatures below k0 A particle is represented by a wave group or wave packets of limited spatial extent, which is a superposition of many matter waves with a spread of wavelengths centered on l0=h/p The wave group moves with a speed vg – the group speed, which is identical to the classical particle speed Heisenberg uncertainty principle, 1927: If a measurement of position is made with precision Dx and a simultaneous measurement of momentum in the x direction is made with precision Dpx, then at T < TdB fBE and fFD are strongly different from fMB at T >> TdB fBE ≈ fFD ≈ fMB electron gas in metals: n = 1022 cm-3, m = me → TdB ~ 3×104 K gas of Rb atoms: n = 1015 cm-3, matom = 105me → TdB ~ 5×10-6 K excitons in GaAs QW n = 1010 cm-2, mexciton= 0.2 me → TdB ~ 1 K

T≠0 the Fermi-Dirac distribution density of states distribution function [the number of states in the energy range from E to E + dE] [the number of filled states in the energy range from E to E + dE] EF E density of filled states D(E)f(E,T) density of states D(E) per unit volume shaded area – filled states at T=0

specific heat of the degenerate electron gas, estimate T ~ 300 K for typical metallic densities T = 0 specific heat U – thermal kinetic energy f(E) at T ≠ 0 differs from f(E) at T=0 only in a region of order kBT about m because electrons just below EF have been excited to levels just above EF classical gas the observed electronic contribution at room T is usually 0.01 of this value classical gas: with increasing T all electron gain an energy ~ kBT Fermi gas: with increasing T only those electrons in states within an energy range kBT of the Fermi level gain an energy ~ kBT number of electrons which gain energy with increasing temperature ~ the total electronic thermal kinetic energy the electronic specific heat EF/kB ~ 104 – 105 K kBTroom / EF ~ 0.01

specific heat of the degenerate electron gas the way in which integrals of the form differ from their zero T values is determined by the form of H(E) near E=m replace H(E) by its Taylor expansion about E=m the Sommerfeld expansion successive terms are smaller by O(kBT/m)2 for kBT/m << 1 replace m by mT=0 = EF correctly to order T2

specific heat of the degenerate electron gas FD statistics depress cv by a factor of (1) (2)

thermal conductivity thermal current density jq – a vector parallel to the direction of heat flow whose magnitude gives the thermal energy per unit time crossing a unite area perpendicular to the flow Wiedemann-Franz law (1853) Lorenz number ~ 2×10-8 watt-ohm/K2 Drude: application of classical ideal gas laws success of the Drude model is due to the cancellation of two errors: at room T the actual electronic cv is 100 times smaller than the classical prediction, but v is 100 times larger for degenerate Fermi gas of electrons the correct at room T the correct estimate of v2 is vF2 at room T

thermopower Seebeck effect: a T gradient in a long, thin bar should be accompanied by an electric field directed opposite to the T gradient high T low T gradT E thermoelectric field thermopower Drude: application of classical ideal gas laws for degenerate Fermi gas of electrons the correct at room T Q/Qclassical ~ 0.01 at room T

Electrical conductivity and Ohm’s law equation of motion Newton’s law in the absence of collisions the Fermi sphere in k-space is displaced as a whole at a uniform rate by a constant applied electric field because of collisions the displaced Fermi sphere is maintained in a steady state in an electric field Ohm’s law ky F kavg Fermi sphere kx the mean free path l = vFt because all collisions involve only electrons near the Fermi surface vF ~ 108 cm/s for pure Cu: at T=300 K t ~ 10-14 s l ~ 10-6 cm = 100 Å at T=4 K t ~ 10-9 s l ~ 0.1 cm kavg << kF for n = 1022 cm-3 and j = 1 A/mm2 vavg = j/ne ~ 0.1 cm/s << vF ~ 108 cm/s