University of Kurdistan logoNanoelectronics: A New Perspective on TransportWeek 9 · Smart contacts
University of Kurdistan logo
WEEK 9 · SMART CONTACTS

Nanoelectronics:
A New Perspective on Transport

p-n junctions, solar cells, spin valves and spin potentials
Prof. Mohammad RazaghiUniversity of Kurdistan · Graduate course in Electronics Engineering
Based on S. Datta, Lessons from Nanoelectronics, Part A: Basic Concepts, Chapter 12
One fixed session. Next week (Week 10) brings thermoelectricity and circular session C5, which includes a lab on this week's p-n and spin-valve models.

RecapWhere we are

Weeks 7–8 gave us

  • µ⁺ ≠ µ⁻: δµ = (µ₁ − µ₂)/(1 + L/λ) (Eq. 12.1)
  • Probes read weighted averages: αµ⁺ + (1 − α)µ⁻
  • Contacts drive groups of states apart; scattering pulls them together

Today's question

What if a contact could choose which states it talks to: the conduction band but not the valence band, up spins more than down spins? Datta: the most interesting devices are built on exactly this.

Week 9Today's goals

  1. Explain why a device may need several quasi-Fermi levels, one per group of states.
  2. Derive the diode law I ∝ eqV/kT − 1 from a two-level model with inelastic transitions.
  3. Get the solar-cell Isc and VOC = ħω(1 − T/T₀)/q, and see why the contacts must be asymmetric.
  4. Compute RP, RAP and the magnetoresistance of a spin valve.
  5. Measure a spin potential with a magnetic probe; explain spin-momentum locking.

Physics refresherRefresher: the p-n junction and band bending

  • Equilibrium: µ is flat, so the bands bend across a depletion region.
  • Built-in potential: Vbi = (kT/q)ln(NAND/ni²) ≈ 0.71 V for 10¹⁶ cm⁻³ Si.
  • Forward bias: one µ splits into Fn (conduction band, from the n contact) and Fp (valence band, from the p contact).
Shockley: I = I₀(eqV/kT − 1). Today Datta derives it with no holes and no drift–diffusion.

Physics refresherRefresher: magnetization and spin

  • Each state comes in two spins; normally we just multiply by g = 2.
  • In Fe, Co, Ni the exchange interaction shifts the up- and down-spin bands.
  • At EF: D↑ ≠ D↓. Majority and minority spins see different interface resistances.
  • Reverse the magnetization: up and down swap roles.
Polarization P = (D↑ − D↓)/(D↑ + D↓); typical transition metals: P ≈ 0.3–0.5.

Datta Sec. 12.1Many quasi-Fermi levels

In/A = −(σ0,n/q) dµn/dzEq. 12.3 · one µ per group n
dIn/dz = Σm(RGm→n − RGn→m)Eq. 12.4 · groups coupled by R-G
  • Ideal junction (Fig. 12.3): states near µ₁ see only contact 1, near µ₂ only contact 2.
  • An elastic resistor would carry no current; electrons must change energy.
  • The R-G step limits the current: I = I₀(eqV/νkT − 1) (Eq. 12.5).
Datta: a p-n junction is a totally different device from a FET: the FET is about σ controlled by U; the junction is about transfer between two groups.

Datta Sec. 12.1.1The two-level model

I ∼ DBAfA(1 − fB) − DABfB(1 − fA)Eq. 12.6 · transition from 2nd index to 1st
DAB/DBA = e−ħω/kT₀Eq. 12.7 · absorbing ħω is harder
I ∼ DABfB(1 − fA)(X − 1)
X = eħω/kT₀ − ħω/kTe(µA − µB)/kTEqs. 12.8, 12.11

T₀ = T, µA − µB = qV ⇒ I ∝ eqV/kT − 1: the ideal diode, ν = 1.

(1 − f)/f = e^{(ε − µ)/kT} (Eq. 12.10) turns Eq. 12.9 into Eq. 12.11. Equilibrium (V = 0, T₀ = T) gives X = 1, I = 0.

Datta Sec. 12.1.1Energy conversion: a heat engine with two contacts

Isc ∼ exp[(ħω/k)(1/T₀ − 1/T)] − 1Eq. 12.12 · current with no voltage
qVOC/ħω = 1 − T/T₀X = 1 at open circuit: the Carnot efficiency
  • Sun T₀ ≈ 6000 K, ħω = 1.1 eV ⇒ VOC = 1.05 V.
  • T₀ < T: Isc reverses (Feynman's ratchet).
  • Sunlight fills a tiny solid angle (δ ≈ 2 × 10⁻⁵): −kT|lnδ| ≈ −0.28 eV, ≈ 0.77 V.
Datta credits his colleague Ashraf for the remark that the small angular spectrum of sunlight lowers the ideal efficiency. The diluted-source formula is our simple version of that.

Datta Sec. 12.1.2Contacts are fundamental

  • (a) Asymmetric contacts: each electron raised from B to A leaves to the left; B is refilled from the right. One photon, one electron in the circuit.
  • (b) Contacts that talk equally to A and B: why should electrons go left rather than right? No output, by symmetry.
  • This discrimination, not the materials, makes p-n junctions and solar cells different from FETs.
"Smart contacts" engineer the electrochemical potentials, as gates engineer the electrostatic one.

InteractiveExplore: a p-n junction and a solar cell from two levels

A p-n junction from two levels and two smart contacts  (Datta Ch. 12, Figs. 12.5–12.6)

Level A (energy εA) talks only to contact 1, level B (εB = εA − ℏω) only to contact 2. No elastic path connects the contacts: current flows only through inelastic jumps A ↔ B caused by scatterers at temperature T0 (Eqs. 12.6–12.7). The device is at T = 300 K. Set T0 = T for a dark diode, T0 = 6000 K for "sunlight". The model is solved exactly in steady state (rates in units of the contact coupling γ).

Short-circuit current Isc (units qγ)–
Open-circuit voltage VOC · Carnot ℏω(1 − T/T0)/q + (kT/q)lnδ–
Dark slope (Eq. 12.5 with ν = 1: 59.5 mV/dec)–
At the set V: (μA − μB)/q and I–

Try this

  1. Dark diode: read the slope on the log plot. Is it 59.5 mV/decade?
  2. Press Sunlight, Datta's ideal: compare VOC with ħω(1 − T/T₀).
  3. Press Real sunlight: how much voltage does dilution cost?
  4. Tick symmetric contacts: where did Isc go?

Scroll inside the panel for all controls. Swipe or use the arrow buttons outside it to change slide.

Rates in units of the contact coupling γ. The model is Datta Eqs. 12.6–12.11 solved exactly in steady state.

Datta Sec. 12.2.1The spin valve

RP = [1/(2r + Rch) + 1/(2R + Rch)]⁻¹Eq. 12.13
RAP = (r + R + Rch)/2Eq. 12.14
MR = (R − r)²/4rR = P²/(1 − P²)Eqs. 12.15–12.16, Rch = 0, P = (R − r)/(R + r)

Tunnel junctions: resistances multiply, MR = 2P²/(1 − P²) (Julliere-type form).

r: majority-spin interface resistance; R: minority. Spin is treated semiclassically as two kinds of electrons, red and blue.

ApplicationFrom the lab to every laptop

1988GMR discovered
2007Nobel prize
33%P = 0.5, metallic
600%MTJ, P = 0.866

Read heads (late 1990s on), then MRAM: spin-transfer torque uses non-equilibrium spins to switch nanomagnets.

InteractiveExplore: spin valve and spin potentials

Spin valve: two spin channels, two spin potentials  (Datta Sec. 12.2, Figs. 12.7–12.9)

Up spins (red) and down spins (blue) are two parallel channels. Each magnetic contact has a small interface resistance r for its majority spin and a large one R for its minority spin; the channel adds Rch to each spin. Spin flips (length λs) leak current between the two channels. Below: µup(z) and µdn(z) from the resistor network (µ₁ = 1, µ₂ = 0), and a magnetic probe that reads a weighted average (Eq. 12.17).

RP and RAP (units r) · Eqs. 12.13–12.14–
MR = RAP/RP − 1 · P²/(1 − P²) (Rch = 0)–
Spin potential µS = (µup − µdn)/2 at mid-channel–
Probe: µP(+m) − µP(−m) · 2PprobeµS (Eq. 12.20)–

Try this

  1. P = 0.5, Rch = 0: check RP = 1.5r, RAP = 2r, MR = 33%.
  2. Switch P ↔ AP and watch µup and µdn in the channel.
  3. Raise Rch, then allow spin flips with small λs/L: what happens to MR?
  4. Flip the probe magnet: does the reading change by 2PprobeµS?

Scroll inside the panel for all controls. Swipe or use the arrow buttons outside it to change slide.

The network: two spin chains with r or R at each interface, R_ch per spin, spin-flip shunts.

Datta Sec. 12.2.2Measuring the spin potential

µP = (gupµup + gdnµdn)/(gup + gdn)Eq. 12.17 · like Eq. 10.4
µ = (µup + µdn)/2,  µS = (µup − µdn)/2Eq. 12.18
µP(+m) − µP(−m) = 2PµSEq. 12.20 · P = (gup − gdn)/(gup + gdn)

With 12.18 the algebra gives µP = µ + PµS (Datta prints PµS/2 in 12.19a; 12.20 needs PµS).

Unlike µ⁺ and µ⁻, spin QFLs can be measured: just reverse the magnet of the probe.

Datta Sec. 12.2.3Spin-momentum locking

  • Ordinary: M(up±) = M(dn±). Magnet: M(up) ≠ M(dn).
  • Topological insulator surface: M(up+) = M(dn−) = M, M(dn+) = M(up−) = N.
µS = p(µ⁺ − µ⁻)/2,  p ∼ (M − N)/(M + N)Eqs. 12.22–12.23
µS = (q/2GB) p IEq. 12.25 · with I = GB(µ⁺ − µ⁻)/q

Rashba–Edelstein effect (maybe also the spin Hall effect); by reciprocity the inverse effects.

The channel polarization p belongs to the channel; the probe polarization P belongs to the magnet that measures µ_S.

CheckConcept check

One minute with a neighbour, then click to reveal.

1. Why is DAB < DBA in Eq. 12.7? (click)
B→A must absorb ħω from scatterers at T₀; that is less likely by e−ħω/kT₀ than giving it up.
2. Carnot VOC for ħω = 1.4 eV, T₀ = 6000 K, T = 300 K? (click)
1.4 × 0.95 = 1.33 V. Real cells are lower: diluted sunlight and non-radiative losses.
3. R/r = 4, Rch = 0. P and MR? (click)
P = 3/5 = 0.6; MR = 0.36/0.64 = 56%.
4. A probe with P = 0.3 changes by 12 µV on reversal. µS/q? (click)
2PµS = 12 µV ⇒ µS/q = 20 µV.

Wrap-upSummary

Take-home

  • Smart contacts drive groups of states apart; inelastic or spin-flip processes pull them together
  • p-n: I ∝ X − 1; Isc, qVOC = ħω(1 − T/T₀); asymmetric contacts
  • Spin valve: MR = P²/(1 − P²); spin potential µS measured by 2PµS
  • Spin-momentum locking: µS = (q/2GB)pI

Next: Week 10 and C5

Thermoelectricity (Datta Ch. 13–14): a temperature difference drives a current, Seebeck and Peltier effects, phonons. In C5 we optimise a thermoelectric and revisit today's two models in a Colab lab. Open week09_smart_contacts.ipynb and the Week 9 study toolkit.

Prof. Mohammad Razaghi · University of Kurdistan · after S. Datta, Lessons from Nanoelectronics