Oral Qualifying Exam
- Evan Sosenko
- with
- Vivek Aji
Two dimensional systems
Spin lifetime
Outline
- Motivation
- Model and solution
- Hanle curve fitting
- Regimes and results
Motivation
- Theoretical lifetime predictions longer than measured values: ms vs. ps
- Finite contact resistance mismatch: a potential candidate
- Unified analytic solution for fitting data in all limits
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).
Device geometry
- L : contact spacing
- D : diffusion constant
- τ : spin lifetime
- λ=√Dτ
- ω=gμBB/ħ
- μs=12(μ↑−μ↓)
- J↑↓=σ↑↓∇μ↑↓
- JC↑↓=Σ↑↓(μN↑↓−μF↑↓)c
- J=J↑+J↓
- Js=J↑−J↓
D∇2μs−μsτ+ω×μs=0
V∝μNs(x=L)
RNL=V/I
J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and I. Žutić, Acta Physica Slovaca 57, 565 (2007).
M. Johnson and R. H. Silsbee, Phys. Rev. B 37, 5312 (1988).
S. Takahashi and S. Maekawa, Phys. Rev. B 67, 052409 (2003).
M. Popinciuc, C. Józsa, P. J. Zomer, N. Tombros, A. Veligura, H. T. Jonkman, and B. J. van Wees, Phys. Rev. B 80, 214427 (2009).
Motivation for solution
Existing results
- All existing analytic expressions ignore contact resistance
- Assume infinite resistance for convenience
- Integral form widely used but difficult to fit
- Only a numeric treatment of finite contact resistance
RNL∝Ree−(L/λ)√1+iωτ2√1+iωτ
RNL∝∫∞01√4πDtexp[−L24Dt]e−t/τcosωtdt
Our approach
- Finite contact resistance
- Exact analytic expression
- Matches previous approaches in appropriate limits
Non-local resistance
ΔRNL=2P2RN|f|
- r=RF+RCRSQW
- RSQ=W/σN
- RN=λWL1σN
f=Re{(sinh[(L/λ)√1+iωτ]√1+iωτ2[√1+iωτ+(λ/r)]e(L/λ)√1+iωτ+(λ/r)2sinh[(L/λ)√1+iωτ]√1+iωτ)−1}.
Only scales that appear in f
- L/λ
- λ/r
- ωτ
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
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).
Regimes
Zero field
ΔRNL=(PLΣ)2RNe−L/λ
Tunneling contacts
f∞=Ree−(L/λ)√1+iωτ2√1+iωτ
Fits independent of lifetime
λ/r≫√ωτ≫1
Zeros determined by L√D2ω+π4=nπ2
E. Sosenko, H. Wei, and V. Aji, Phys. Rev. B 89, 245436 (2014).
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).
Dichalcogenides
Outline
- Motivation
- Valley physics
- Superconducting phase
- Future work
Motivation
- Active and emerging field
- Monolayer graphene-like system with new valley physics
- Potentially a natural spin valve material
Effective Hamiltonian
- MoS2, WS2, MoSe2, WSe2
- Similar to monolayer graphene: two inequivalent valleys: K, K′
- Strong spin-orbit coupling and inversion symmetry breaking
- Leads to opposite valley Berry curvature
- Two state tight binding model: dz2, and dxy, dx2−y2
Hτσ0(k)=at(τkxσx+kyσy)⊗I2+Δ2σz⊗I2−λτ(σz−1)⊗Sz
Hτσ0(k)=[Δ2at(τkx−iky)at(τkx+iky)λτσ−Δ2]
D. Xiao, G.-B. Liu, W. Feng, X. Xu, and W. Yao, Phys. Rev. Lett. 108, 196802 (2012).
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
Future Work
Induced superconductivity
- Only looked at MoS2
- Can tune the parameters to understand how they affect the physics
- Look at other properties of this state, e.g., magnetic susceptibility
Intrinsic superconductivity
- Derived from density-density interactions
- Already have the projected interaction term
- Apply mean-field and an analogous analysis
- Both intervalley and intravalley pairing
HV=−∑∑′k,k′v(k′−k)A(k,k′)
A(k,k′)=B2(k,k′)c−+↑†(k′)c−+↑†(−k′)c−+↑(−k)c−+↑(k)+B2(k′,k)c−−↓†(k′)c−−↓†(−k′)c−−↓(−k)c−−↓(k)+|B(k,k′)|2[c−+↑†(k′)c−−↓†(−k′)c−−↓(−k)c−+↑(k)+c−−↓†(k′)c−+↑†(−k′)c−+↑(−k)c−−↓(k)]