What happens to the current if we: 1. add a magnetic field, 2. have an oscillating E field (e.g. light), 3. have a thermal gradient H Add a magnetic field H - - + Last time : dp/dt = -p(t)/ + f(t) 2 H field apply force to whole wire or just moving carriers? Hall Effect In a current carrying wire when in a perpendicular magnetic field, the current should be drawn to one side of the wire. As a result, the resistance will increase and a transverse voltage develops. Lorentz force = -e v x H/c H -----------------++++++++++++++++ - Last time: dp/dt = -p(t)/ + f(t) Group If a constant current flows (of velocity vD) in the positive x direction and a uniform magnetic field is applied in the positive z-direction, determine the magnitude and direction of the resulting Hall field. Lorentz force = • • • • -ev/c x H Ey= -vxHy/c =- jxHy/nec Hall coefficient RH= Ey/jxH = -1/nec, Big or small? R is very small for metals as n is very large. Useful for determining carrier density and type Hz -----------------++++++++++++++++ - x y The Hall coefficient What would happen if we had both holes and electrons? Ohm’s law contains e2 But for RH the sign of e is important. A hole is the lack of an electron. It has the opposite charge so +e. Application: For a 100-m thick Cu film, in a 1.0 T magnetic field and through which I = 0.5 A is passing, the Hall voltage is 0.737 V. H H nec Not yet prepared to discuss other quantum versions of the Hall effect With strong magnetic fields: The integer quantum Hall effect is observed in 2D electron systems at low temperature, in which the Hall conductance undergoes quantum Hall transitions to take on quantized values from D.C. Tsui, RMP (1999) and from H.L. Stormer, RMP (1999) The fractional quantum Hall effect: Hall conductance of 2D electrons shows precisely quantized plateaus at fractional values of e2/h What happens to the conduction electrons if we have an oscillating E field (e.g. light)? plasmon: charge density oscillations (Start with simple picture) d What will happen to d over the oscillation? Longitudinal Plasma Oscillations d Displacement of the entire electron gas a distance d with respect to the positive ion background. This creates surface charges s = nde & thus an electric field E = 4nde. Equation of Motion: F = ma = -eE d Nm 2 NeE Ne( 4nde) t 2 Oscillations at the d 2 Plasma Frequency d 0 p 2 t 2 4ne 2 p 2 m plasmon: charge density oscillations values for the plasma energy Pretty reasonable agreement What do you think happens if we try to oscillate really fast? Why are metals shiny? Drude’s theory gives an explanation of why metals do Where is this approximation accurate? not transmit light and rather reflect it. Continuum limit: Where the wavelength is bigger than the spacing between atoms. Otherwise diffraction effects dominate. (Future topic) AC Electrical Conductivity of a Metal Drude’s Equation of Motion dp( , t ) p( , t ) eE( , t ) for the momentum of one dt electron in a time dependent E( , t ) E( )e it electric field. Look for a steady p( , t ) p( )e it state solution of the form: eE( ) p( ) j( ) s ( )E( ) AC conductivity DC conductivity s0 s ( ) 1 i ne 2 s0 1 / i nep( ) ( ne 2 / m )E( ) j( ) m 1 / i m Works great for the continuum limit when can treat the force on each electron the same. When we have a current density, we can write Maxwell equations as: -2E = i/c (4sE/c -i E/c) -2E = 2/c2 (1 +4 si / )E ( ) =1 + 4 si / Usual wave: -2E = 2 ()E/c2 x(xE) = -2E x(xE) = x = -i H(,t) 14 = x i H(,t) /c = i /c x H(,t) x(xE) = -2E = i /c ( j = sE ) From continuum limit s0 ne s ( ) ,s 0 1 i m From Maxwell’s equations 2 ( ) =1 + 4 si / Plugging s into : ( ) =1 + 4 s0i / (1-i ) =1 + 4 s0i / ( -i 2) Plugging in s0: ( ) =1 + 4 ne2i / m( -i 2) For high frequencies 1can ignore first term in denominator Ignoring: ( ) =1 - 4 ne2/m2 p is known as the plasma frequency 15 What is a plasma? Application to Propagation of Electromagnetic Radiation in a Metal -2E = 2 ()E/c2 The electromagnetic wave equation in a nonmagnetic isotropic medium. E exp( it iK r ) Look for a solution with the dispersion 2 2 2 K ( , K ) / c relation for electromagnetic waves (1) real & > 0 → K is real & the transverse electromagnetic wave propagates with the phase velocity vph= c/1/2 E/t= -i E(,t) 2E/t2= 2 E(,t) 2E = -2 ()E/c2 =2 E Transverse optical modes in a plasma Dispersion relation for (, K ) 2 c 2 K 2 electromagnetic waves 2 p 2 c2 K 2 (1) real & positive, no damping (2) (p>) real & < 0 → K is ( ) imaginary & the wave is damped with a characteristic length 1/|K| (Why metals are shiny) E e p Kr (3) complex → K is complex & the wave is damped in space (4) = 0 longitudinally polarized waves are possible / p (2) E&M waves are totally reflected from the medium when is negative (1) E&M waves propagate with no damping when is positive & real Ultraviolet Transparency of Metals Plasma Frequency p & Free Space Wavelength lp = 2c/p Range Metals n, cm-3 1022 p, Hz 5.7×1015 lp, cm 3.3×10-5 spectral range UV Semiconductors 1018 5.7×1013 3.3×10-3 IF Ionosphere 1010 5.7×109 33 radio The Electron Gas is Transparent when > p i.e. l < lp p 2 4ne m 2 The reflection of light from a metal is similar to the reflection of radio waves from the Ionosphere! Plasma Frequency Ionosphere Semiconductors Metals What happens when you heat a metal? What do we know from basic thermo 19 th (0 law)? Drude’s Best Success: Explanation of the Wiedemann-Franz law for metals (1853) Thermal conductivity Temperature Electrical conductivity • • • 20 Wiedemann and Franz observed that the ratio of thermal and electrical conductivity for ALL METALS is constant at a given temperature (for room temperature and above). Later it was found by L. Lorenz that this constant is proportional to the temperature. Let’s try to reproduce the linear behavior and to calculate the slope Thermal conductivity A material's ability to conduct heat. Electric current density (je = I/A) Heat current density Fourier's Law for heat conduction. E Thermal current density jt sec area jt vn = Energy per particle v = velocity n = N/V 2l Many open questions: • • Why does the Drude model work so relatively well when many of its assumptions seem so wrong? In particular, the electrons don’t seem to be scattered by each other. Why? Why is the actual heat capacity of metals much smaller than predicted? From Wikipedia: "The simple classical Drude model provides a very good explanation of DC and AC conductivity in metals, the Hall effect, and thermal conductivity (due to electrons) in metals. The model also explains the Wiedemann-Franz law of 1853. "However, the Drude model greatly overestimates the electronic heat capacities of metals. In reality, metals and insulators have roughly the same heat capacity at room temperature.“ It also does not explain the positive charge carriers from the Hall effect. FYI: measurable quantity – Hall resistance H RH H Vy H More detail about H Ix n2 D ec Hall resistance E j j sE in the presence of magnetic field the resistivity and conductivity becomes tensors xx xy E x xx xy jx for 2D: j E yy yy y yx y yx s 0 E x cj y jx s 0 E y cjx j y c Ex jx jy s0 s0 c 1 Ey jx jy s0 s0 1 c s 0 1 s0 c s 0 1 s 0 c s 0 jx Ex 1 s 0 E j s 1 s 0 0 y y c 1 m xx 2 s 0 ne c H xy s 0 nec for 3D systems n2 D nLz for 2D systems n2D=n jx s xx s xy Ex j s E s yy y y yx s xx s xy xx s s yx s yy yx xy yy s0 s xx s yy 1 (c ) 2 s 0c s xy s yx 1 (c ) 2 xx s xx 2 xx xy 2 xy s xy 2 xx xy 2 1