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

2 Last time : Add a magnetic field H dp/dt = -p(t)/  + f(t) 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

Group If a constant current flows (of velocity v D ) 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. E y = -v x H y /c =- j x H y /nec Hall coefficient R H = E y /j x H = -1/nec, Big or small? R is very small for metals as n is very large. Useful for determining carrier density and type Lorentz force = -ev/c x H HzHz x y Last time: dp/dt = -p(t)/  + f(t)

The Hall coefficient Ohm’s law contains e 2 But for R H the sign of e is important. A hole is the lack of an electron. It has the opposite charge so +e. What would happen if we had both holes and electrons?

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  V.

from D.C. Tsui, RMP (1999) and from H.L. Stormer, RMP (1999) weak magnetic fields 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 The fractional quantum Hall effect: Hall conductance of 2D electrons shows precisely quantized plateaus at fractional values of e 2 /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 Oscillations at the Plasma Frequency Equation of Motion: F = ma = -eE Displacement of the entire electron gas a distance d with respect to the positive ion background. This creates surface charges  = nde & thus an electric field E = 4  nde. d

plasmon: charge density oscillations values for the plasma energy What do you think happens if we try to oscillate really fast? Pretty reasonable agreement

Why are metals shiny? Drude’s theory gives an explanation of why metals do 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) Where is this approximation accurate?

AC Electrical Conductivity of a Metal Drude’s Equation of Motion for the momentum of one electron in a time dependent electric field. Look for a steady state solution of the form: AC conductivity DC conductivity Works great for the continuum limit when can treat the force on each electron the same. -

14 When we have a current density, we can write Maxwell equations as: j =  E  x(  xE) =  x = -i  H( ,t) =  x i  H( ,t) /c = i  /c  x H( ,t)  x(  xE) = -  2 E  x(  xE) = -  2 E = i  /c ( ) -  2 E = i  /c (4  E/c -i  E/c) -  2 E =  2 /c 2 (1 +4  i /  )E  (  ) =1 + 4  i /  Usual wave: -  2 E =  2  (  )E/c 2

15  (  ) =1 + 4  i /  From continuum limitFrom Maxwell’s equations Plugging  into  :  (  ) =1 + 4  0 i /  (1-i  ) =1 + 4  0 i / (  -i  2  ) Plugging in  0 :  (  ) =1 + 4  ne 2  i / m(  -i  2  ) For high frequencies can ignore first term in denominator Ignoring:  (  ) =1 - 4  ne 2 /m  2  p is known as the plasma frequency What is a plasma?

The electromagnetic wave equation in a nonmagnetic isotropic medium. Look for a solution with the dispersion relation for electromagnetic waves Application to Propagation of Electromagnetic Radiation in a Metal -  2 E =  2  (  )E/c 2  E/  t= -i  E( ,t)  2 E/  t 2 =  2 E( ,t)  2 E = -  2  (  )E/c 2 =  2 E  real & > 0 → K is real & the transverse electromagnetic wave propagates with the phase velocity v ph = c/  

 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) (3)  complex → K is complex & the wave is damped in space (4)  = 0 longitudinally polarized waves are possible Transverse optical modes in a plasma Dispersion relation for electromagnetic waves (2) (1) /p/p E&M waves are totally reflected from the medium when  is negative E&M waves propagate with no damping when  is positive & real  p

Ultraviolet Transparency of Metals Plasma Frequency  p & Free Space Wavelength p = 2  c/  p Range MetalsSemiconductorsIonosphere n, cm  p, Hz5.7× × ×10 9 p, cm3.3× × spectral rangeUV IF radio The Electron Gas is Transparent when  >  p i.e. < p Plasma Frequency Ionosphere Semiconductors Metals The reflection of light from a metal is similar to the reflection of radio waves from the Ionosphere!

What happens when you heat a metal? What do we know from basic thermo (0 th law)? 19

20 Drude’s Best Success: Explanation of the Wiedemann-Franz law for metals (1853) 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 20 Thermal conductivity Electrical conductivity  Temperature

Thermal conductivity A material's ability to conduct heat. Thermal current density  = Energy per particle v = velocity n = N/V Electric current density Heat current density Fourier's Law for heat conduction.  (j e = I/A) 2l2l

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.Wikipedia "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 for 3D systems for 2D systems n 2D =n in the presence of magnetic field the resistivity and conductivity becomes tensors for 2D: More detail about Hall resistance