1. Quantum Statistical Physics

2. Bathtub principle for electrons
Electrons repel because of antisymmetric wavefunctions
Fill bands from bottom until you run out of electrons
Finite temperature, the surface gets fuzzy

3. Clumping principle for photons, phonons…
Bosons attract because of symmetric wavefunction
They have an incentive to agglomerate !

4. Energy distribution of particles
Classical
Particle
Fermion
Boson E E E m m ∞ f f f 1 1 1 1
e-E/kT
e(E-m)/kT + 1
e(E-m)/kT - 1

5. Finding the distribution
Pauli Exclusion Principle – each allowed state can accommodate only one electron
The total number of electrons is fixed N=Ni
The total energy is fixed ETOT =  EiNi
Maximize entropy, ie, # of possible microstates

6. E = 12, N = 5 4 2 E f
1/2 2 4
2/7
2/5

7. Why did we get this distribution?
Most likely Configuration is one with most choices as it will
be visited often during a random walk in phase space
Most choices available when we put
least # of particles on levels with
most ‘rooms’

8. What the environment looks like
For classical noninteracting systems (the ‘environment’),
higher energies have more ‘rooms’ (ie, density of states)
TransAmerica Pyramid, SF
EarthScraper, Mexico City

9. But why more rooms higher up?
# states ~ 4pp2dp ~ √E dE
In fact,
1/T = dS/dE > 0
Where S is the # of rooms (“Entropy”)

10. Derive this for “Bosons” Symmetric wavefunction, any # of particles in each ‘room’

11. For Bosons E f 2 4
1/7
4/5

12. Energy distribution of particles
Classical
Particle
Fermion
Boson E E E m m ∞ f f f 1 1 1 1
e-E/kT
e(E-m)/kT + 1
e(E-m)/kT - 1

13. Energy distribution of particles
Fermion
Boson E E m m ∞ f 1 f
Pay a penalty
m to add electrons
Can make m go to zero !!

15. Origin of irreversibility and Boltzmann
Environment has a big phase space and settles into its
equilibrium quickly. In the process, information from
system gets reset (‘erased’).
S = kBlnW
Composite system,
= W1W2
S = S1 + S2 ~ scales with size

16. Origin of irreversibility and Boltzmann
Environment has a big phase space and settles into its
equilibrium quickly. In the process, information from
system gets reset (‘erased’).
S = kBlnW
P2/P1 = W1/W2 = exp[(S1–S2)/kB]
= exp[–DE(dS/dE)/kB] = exp[–DE/kBT]
1/T = dS/dE > 0

17. Origin of irreversibility
S H A N N O N
Erasure of information gives rise to irreversibility

18. How do spins interact with
their surroundings?

19. Zeeman effect: Flip spins along magnetic field
(Origin of Stern-Gerlach) B
H = - m.B = -gmBBSz
mB =qħ/2m = 9.27 x J/T ≈ 60 meV/T
‘g’ factor ~ 2 for electrons

20. 1 0 0 -1 Magnetic field splits the energy levels B = 0 B ≠ 0
H = -gmBBSz = -gmBBħ/2
B = 0
B ≠ 0

21. D Ferromagnet: Internal B field can split levels E EF H = - JS1.S2k
H = - JS1.S2
J is the exchange
parameter EF E k
Internal field
B ~ J<S> D

22. 1 0 0 -1 Can we transition between the spins? H = -gmBBSz = -gmBBħ/2
Need (i) an off-diagonal term coupling the states for transitions  E.g., a field along the x-axis
(ii) a resonant AC field to provide the transition energy

23. 1 0 0 -1 0 1 1 0 Electron Spin Resonance (ESR) B B1coswt
H = -gmBBSz = -gmBBħ/2
H1 = -gmBB1(t)Sx = -(gmBB1ħcoswt)/2

24. Electron Spin Resonance (ESR)B
B1coswt
iħ/t = [H + H1(t)] y y
Solve analytically using some approximations
or numerically

25. Electron Spin Resonance (ESR)B
B1coswt
P(t)
So we can transition between spins with a suitable field

26. What about internal fields?
Exchange fields in metallic magnets
Spin-orbit fields in semiconductors

27. Spin-Orbit coupling +
Electron orbiting in electric field of nucleus

28. + What the electron sees B ~ v x E
Nucleus orbiting, creating a net current
and thus  DRUMROLL…. a magnetic field !!

29. + What the electron sees The Zeeman coupling of the electron spin to
this motional field is the Spin-Orbit effect
(within a factor of 2)
H ~ -S.B ~ S.(p x E) ~ S.(p x r)dU/dr
H ~ -S.L(dU/dr)

30. S-O coupling: Various manifestations
Atom: Gives rise to Hund’s Rule
Solid: Split-off states in valence band
Gated transistor: Rashba coupling
H ~ S.(p x r)E
~ r.(S x p)E
~ (sxky-sykx)Ez

31. Spintronic Devices

32. Read: GMR, TMR, spin valves
(Memory, Sensors)

33. Write: MRAMs Rotate with field Write: STTRAMs Rotate with current
Write: STTRAMs
Rotate with current
Also  Rotate with strain (multiferroics)

34. Computing: Datta-Das “FET”
H = aR
(sxky-sykx)Ez
Use Rashba field to
rotate spins in a
modulator with a
gate

35. Computing with 104 spins NkTln(pon/poff) for N charges
~kTln(pon/poff) for N spins !!

36. Using spin for computing
All spin Logic
Memristors
MQCA

37. Summary: Spin is a new
variable. It can be used
for energy-efficient
Computing