Oral Qualifying Exam
Two dimensional systems
Spin lifetime
Outline
- Motivation
- Model and solution
- Hanle curve fitting
- Regimes and results
E. Sosenko, H. Wei, and V. Aji, Phys. Rev. B 89, 245436 (2014).
Fits
Tunneling contacts
- L=2.1µm
- P=0.19
- RC=2.03×107kΩ
- τ=514.3ps
- D=0.02m2s−1
Tunneling contacts
- L=5.5µm
- P=0.1
- RC=6.7×106kΩ
- τ=451.84ps
- D=0.01m2s−1
W. Han, K. Pi, K. M. McCreary, Y. Li, J. J. I. Wong, A. G. Swartz, and R. K. Kawakami, Phys. Rev. Lett. 105, 167202 (2010).
Fits
Tunneling contacts
- L=2.1µm
- P=0.19
- RC=2.03×107kΩ
- τ=514.3ps
- D=0.02m2s−1
Tunneling contacts
- L=5.5µm
- P=0.1
- RC=6.7×106kΩ
- τ=451.84ps
- D=0.01m2s−1
W. Han, K. Pi, K. M. McCreary, Y. Li, J. J. I. Wong, A. G. Swartz, and R. K. Kawakami, Phys. Rev. Lett. 105, 167202 (2010).
Fits
Pinhole contacts
- L=3.0µm
- P=0.23
- RC=1.31×107kΩ
- τ=132.28ps
- D=0.02m2s−1
Transparent contacts
- L=3.0µm
- P=0.01
- RC=2.94kΩ
- τ=130.36ps
- D=0.04m2s−1
W. Han, K. Pi, K. M. McCreary, Y. Li, J. J. I. Wong, A. G. Swartz, and R. K. Kawakami, Phys. Rev. Lett. 105, 167202 (2010).
Fits: τ Independent Limit
Transparent contacts
- L=3.0µm
- P=0.02
- RC=0.27kΩ
- τ=9.97×109ps
- D=0.02m2s−1
Transparent contacts
- L=3.0µm
- P=0.02
- RC=0.28kΩ
- τ=9.95×1013ps
- D=0.02m2s−1
W. Han, K. Pi, K. M. McCreary, Y. Li, J. J. I. Wong, A. G. Swartz, and R. K. Kawakami, Phys. Rev. Lett. 105, 167202 (2010).
Energy Bands
- Δ—band splitting
- λ—spin splitting
- τ—valley index
- σ—spin index
MoS2
- at=3.15Å eV
- Δ=1.66eV
- 2λ=0.15eV
- μ=−0.83eV
Enτσ(k)=12(λτσ+n√(2at)2|k|2+(Δ−λτσ)2)
Optical Transitions
Pτσ(k)=m0ħ⟨u+|∇kHτσ0(k)|u−⟩
Pτσ±(k)=Pτσx±iPτσy
- Right circular polarization strongly couples to τ=+ valley transitions
- Left circular polarization strongly couples to τ=− valley transitions
D. Xiao, G.-B. Liu, W. Feng, X. Xu, and W. Yao, Phys. Rev. Lett. 108, 196802 (2012).
Vally Hall effect
Semiclassical equations
˙r=∇kE(k)−˙k×Ω(k)
˙k=−eE−e˙r×B
Anomalous velocity v=eΩ(k)×E
Berry curvature
Ωnτσ(k)=∇k×⟨unτσ(k)|i∇k|unτσ(k)⟩
Broken inversion symmetry
Ωnτ,σ(k)=−Ωn−τ,σ(k)≠0
BCS Superconductivity
Mean-field Hamiltonian
H−μN=∑∑kσξkc†kσckσ−∑∑k(ˉΔkc−k↓ck↑+Δkc†k↑c†−k↓)
Number of available parings maximized when center-of-mass momentum is zero
Quasiparticle operators
bkσ=σcosθkckσ+sinθkc†−k,−σ
Diagonalized
∑∑kσλkb†kσbkσ
Induced Superconductivity
Intervalley pairing
- aντσ—orbital operators
- bα—quasiparticle operators
- BCS pairs in opposite valleys
- Reduces to standard BCS Hamiltonian where α=τ=σ plays the role of the spin index
- Not a singlet ground state: mixture of singlet and triplet states
HV=−∑∑′k∑∑ν,τΔνaν−τ↓†(−k)aντ↑†(k)+h.c.
H−μN=∑∑′k∑∑αλαkb†kαbkα+∑∑′k(ξk↓+λ−k).
Optical Transitions
Pτσ(k)=m0ħ⟨u+|∇kHτσ0(k)|u−⟩
Pτσ±(k)=Pτσx±iPτσy
- Right circular polarization strongly couples to τ=+ valley transitions
- Left circular polarization strongly couples to τ=− valley transitions
D. Xiao, G.-B. Liu, W. Feng, X. Xu, and W. Yao, Phys. Rev. Lett. 108, 196802 (2012).
SC Optical Excitations
P(k)=m0ħ⟨Ωf|∇kHτσ(k)|Ω⟩
P±(k)=Px±iPy
|Ω⟩=∏kbk↑b−k↓|0⟩
|Ωf⟩={c+α†(k)b−α(−k)|Ω⟩k>kμc+α†(k)b†−α(−k)|Ω⟩k<kμ
SC Optical Excitations
Compare to normal transitions
- Upper band excitations are now paired with lower band quasiparticle excitations
- Valley-polarization coupling is retained even in the superconducting case
- Contrast is reduced in an region around the chemical potential