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Home , Charge carrier density, Drude model, Hall effect

The

Ref. Ihn Ch. 10 Reminder: The

F = q[E + (v  B)] Lorentz Force: Review • Cyclotron Motion 2 FB = mar  qvB = (mv /r) • Radius r = [(mv)/(|q|B)] = [p/(q|B|]) • Orbit Frequency A Momentum Filter!! ω = 2πf = (|q|B)/m • Orbit Energy A Mass Measurement Method! K = (½)mv2 = (q2B2R2)/2m • Velocity Filter Undeflected trajectories in crossed E & B fields: v = (E/B) Hall Effect

Electrons move in the y direction and an component appears in the y direction, Ey. This will continue until the Lorentz force is equal and opposite to the electric force due to the buildup of – that is, a steady condition arises. B The Classical Hall Effect 1400

Hall Resistance 1200 Slope Related to RH & R (Ohms) 1000 Sample Dimensions Ly xy 800 d 600 400

200 0 Ax 024 6810 ()

Ly is the transverse width of the sample Ax is the transverse cross sectional area

The Lorentz Force tends to deflect jx. However, this sets up an E-field which balances that Lorentz Force. Balance occurs when

Ey = vxBz = Vy/Ly. But jx = nevx (or ix = nevxAx). So

Rxy = Vy / ix = RH Bz ×(Ly/Ax) where RH = 1/ned 3d-Hall coefficient Hall’s Experiment (1879):

+ + + +

Vy I ----

Vx

B Hall’s Experiment (1879):

+ + + +

Vy I ----

Vx

Roughly Explained by Drude Theory R =V / I R =V / I 1900 yx y x B xx x x

B B The Drude theory of : the free theory of metals Paul Drude (1900): theory of electrical and thermal conduction in a application of the kinetic theory of to a metal, which is considered as a of electrons mobile negatively charged electrons are confined in a metal by attraction to immobile positively charged

in a metal isolated

nucleus charge eZa Z electrons are weakly bound to the nucleus (participate in chemical reactions)

Za –Zcore 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 the valence electrons wander far away from their parent called conduction electrons or electrons The basic assumptions of the

1. between collisions the interaction of a given electron with the other electrons is neglected independent electron approximation and with the ions is neglected free electron approximation

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 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 E  j j  E j=I/A – the current  – the resistivity L R=L/A – the resistance A = 1/ the conductivity j  env v is the average electron velocity

eE  ne2  v   j   E m  m   ne2 j  E  m  m  ne2 at room resistivities of metals are typically of the order of microohm centimeters (ohm-cm) and  is typically 10-14 – 10-15 s mean free path l=v0 v0 – the average electron speed l measures the average distance an electron travels between collisions 2 7 estimate for v0 at Drude’s time1 2 mv 0  3 2 k B T → v0~10 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 average average dp(t) p(t) momentum velocity    f(t) dt  pv()tmt ()

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 Hall effect and 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×B when a current j flows across a magnetic field B

the Lorentz force FvL eB 

in equilibrium jy = 0 → the transverse field (the Hall field) Ey due to the accumulated charges balances the Lorentz force quantities of interest: Vx Ex magnetoresistance RH() Rxx resistivity ()H xx (transverse magnetoresistance) I x jx V E Hall (off-diagonal) resistance y y Ryx  Hall resistivity yx I x jx

Ey the Hall coefficient RH  jBx RH → measurement of the sign of the carrier charge RH is positive for positive charges and negative for negative charges Hall Effect

Initially, v  vxxˆ  vyyˆ  vzzˆ E  Ex xˆ B  Bz zˆ   d 1     F  m  v  e(E  v  B)  dt   m = effective mass  d 1  Fx  m  vx  e(Ex  vyB)  dt   d 1  Fy  m  vy  e(vx B)  dt   Hall Effect

Steady state condition: mvx   e(Ex  vyB) mv y  e(E  v B)  y x   e E v    x  v eB x m C y   C e E  m v   y  v  y m C x Hall Effect  e E  v   y  v  0 y m C x Cvx   Ey  m e eB E    E   E y C x m x e v   E x m x v  E  m x x e Hall Effect

The Hall coefficient is defined as: eB E Ey m x 1 RH      j B ne2 ne x E B m x Semiconductors: Density via Hall Effect • Why is the Hall Effect useful? It can determine the carrier type (electron vs. hole) & the carrier density n for a . • How? Place the semiconductor into external B field, push

current along one axis, & measure the induced Hall VH along the perpendicular axis.

n = [(IB)/(qwVH)]   Hole  FqvBB   Electron

+ charge –charge Modern Physics VERY LOW TEMPERATURE 2 D SAMPLE VERY LOW DISORDER + + + +

VH I ----

VL

Study the electrons Not the disorder

SEMICONDUCTORB PHYSICS!

Silicon Technology (i.e., Transistor) III-V MBE (GaAs) The 2D Hall Effect

The surface is sx = vxn2Dq, where n2D is the surface .

RH = 1/n2De.

Rxy = Vy / ix = RH Bz. since sx = ix /Ly & Ey = Vy /Ly. So, Rxy does NOT depend on the shape of the sample. measurable quantity – Hall resistance Vy B Rxy  I xDne2 for 3D systems n2D  nLz E  j j  E for 2D systems n2D=n in the presence of magnetic field the jEx xx xy  x resistivity and conductivity tensors become       jEyyxyyy   E  j    xx xy  x xx xy  x   for 2D:         1    E j  xx xy   xx xy   yx yy  yyxyyy                yx yy   yx yy   E jj  0 x cy x E jj   0 0 ycxy  xx  yy   2   1 ( c )  ne2 / m  0       0 c 1   xy yx  2 c 1 ( c ) Ej j Ex 1 00c jx xx y  00 E    j  yy c 001  1 c  Ejjyxy  1 m  xx   xx  2 2  00 xx 2  ne   xx  xy     0   1  0 c  0     c B xy    xy    2 2  c 0 1  0  xy 0 ne xx   xy Results of Hall measurements A measurement of the two independent components of the resistivity tensor allows us to determine the density and the mobility (scattering time) of the electron gas. 2D Hall Effect ( + 100 Years)

+ + + + V I ---- H

VL

B

CLASSICAL PREDICTION

RH=VH / I RL=VL / I

B B (Data: M. Cage) T=1.2K

Hall Bar VERY FLAT!

2 1 R  Shubnikov H e2 i deHass  RK / i

i = integer

Classical Regime ZERO! (Dissipationless)

RH=VH / I RL=VL / I

B B Nobel prize, 1985

Why Quantization? Why So Precise? Why doesn’t X destroy effect?

X = Dirt = Imperfect Sample Shape = Imperfect Contacts …. etc

Accuracy = better than 1 part in 109

Now DEFINES the Ohm

RK=25812.807449  The Integer Hall Effect

Very important: For a 2D electron system only

First observed in 1980 by Klaus von Klitzing Awarded 1985 Nobel Prize.

Hall Conductance is quantized in units of e2/h, or 2 Hall Resistance Rxy = h/ie , where i is an integer. The quantum of conductance h/e2 is now known as the "Klitzing“!! B Hall resistance   RB HH ne 1*m weak   the Drude magnetic xx  ne2  0 model fields B c the classical c  1 xy  0 ne Hall effect

   1 strong magnetic fields c  h quantization of Hall resistance  xy e2  eH  at integer and fractional   n    hc  the integer and the fractional quantum Hall effect from D.C. Tsui, RMP (1999) and from H.L. Stormer, RMP (1999) The Fractional Quantum Hall Effect The Royal Swedish Academy of Sciences awarded The 1998 Nobel Prize in Physics jointly to Robert B. Laughlin (Stanford), Horst L. Störmer (Columbia & Bell Labs) & Daniel C. Tsui, (Princeton) • The researchers were awarded the Nobel Prize for discovering that electrons acting together in strong magnetic fields can form new types of "particles", with charges that are fractions of electron charges. Citation: “For their discovery of a new form of quantum fluid with fractionally charged excitations” • Störmer & Tsui made the discovery in 1982 in an experiment using extremely powerful magnetic fields & low temperatures. Within a year of the discovery Laughlin had succeeded in explaining their result. His theory showed that electrons in a powerful magnetic field can condense to form a kind of quantum fluid related to the quantum fluids that occur in & helium. What makes these fluids particularly important is that events in a drop of quantum fluid can afford more profound insights into the general inner structure dynamics of matter. The contributions of the three laureates have thus led to yet another breakthrough in our understanding of quantum physics & to the development of new theoretical concepts of significance in many branches of modern physics.