Oral Qualifying Exam

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$$\newcommand{\re}{\operatorname{Re}} \newcommand{\rNL}{R_{\text{NL}}} \newcommand{\rSQ}{R_{\text{SQ}}} \newcommand{\vect}[1]{\mathbf{#1}} \newcommand{\vK}{\mathbf{k}} \newcommand{\exOfK}{\left( \vK \right)} \newcommand{\exOfMK}{\left( - \vK \right)} \newcommand{\hc}{\text{h.c.}} \newcommand{\exInteration}[2]{ {c^-_{#1}}^† \left( \vK' \right) {c^-_{#2}}^† \left( - \vK' \right) c^-_{#2} \left( - \vK \right) c^-_{#1} \left( \vK \right) } \newcommand{\exThetaMPU}[1]{\frac{θ \left( #1 \right)}{2}} \newcommand{\exExp}{e^{-i \left( ϕ_{\vK'} - ϕ_{\vK} \right)}} \newcommand{\exSin}{\sin{\exThetaMPU{k}} \sin{\exThetaMPU{k'}}} \newcommand{\exCos}{\cos{\exThetaMPU{k}} \cos{\exThetaMPU{k'}}} $$

Two dimensional systems

Spin lifetime and Hanle curve fitting

Superconducting phases of monolayer transition-metal dichalcogenides

Spin lifetime

Outline

Motivation
Model and solution
Hanle curve fitting
Regimes and results

Motivation

Theoretical lifetime predictions longer than measured values: $ \text{ms} $ vs. $ \text{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

Fit to parallel field data from Fig. 4a of W. Han, et al.

$ L = 5.5 \: \text{µm} $
$ P = 0.1 $
$ R_\text{C} = 6.7 × 10^{ 6 } \: \text{kΩ} $
$ τ = 451.84 \: \text{ps} $
$ D = 0.01 \: \text{m}^2 \text{s}^{-1} $

Tunneling contacts

Fit to parallel field data from Fig. 4b of W. Han, et al.

$ L = 2.1 \: \text{µm} $
$ P = 0.19 $
$ R_\text{C} = 2.03 × 10^{ 7 } \: \text{kΩ} $
$ τ = 514.3 \: \text{ps} $
$ D = 0.02 \: \text{m}^2 \text{s}^{-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 = \sqrt{D τ} $
$ ω = g μ_B B / ħ $

$ μ_s = \frac{1}{2} \left( μ_↑ - μ_↓ \right) $
$ J_{↑↓} = σ_{↑↓} ∇μ_{↑↓} $
$ J_{↑↓}^C = Σ_{↑↓} \left( μ^N_{↑↓} - μ^F_{↑↓} \right)_c $
$ J = J_↑ + J_↓ $
$ J_s = J_↑ - J_↓ $

$$D ∇^2 μ_s - \frac{μ_s}{τ} + ω × μ_s = 0$$

$$V ∝ μ_s^N(x = L)$$

$$R_\text{NL} = V / I$$

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

$$\rNL ∝ \re{\frac{e^{- \left( L / λ \right) \sqrt{1 + i ω τ}}}{2 \sqrt{1 + i ω τ}}}$$

$$\rNL ∝ \int_0^∞ \frac{1}{\sqrt{4 π D t}} \exp{\left[ - \frac{L^2}{4 D t} \right]} e^{-t / τ} \cos{ω t} \: dt$$

Our approach

Finite contact resistance
Exact analytic expression
Matches previous approaches in appropriate limits

Non-local resistance

$$Δ \rNL = 2 P^2 R_N \left\lvert f \right\rvert$$

$ r = \frac{R_F + R_C}{\rSQ} W $
$ \rSQ = W / σ^N $
$ R_N = \frac{λ}{W L} \frac{1}{σ^N} $

$$ \begin{multline} f = \re \left\{ \left( \vphantom{ \frac{ \sinh{ \left[ \left( L / λ \right) \sqrt{1 + i ω τ} \right] } }{\sqrt{1 + i ω τ}} } 2 \left[ \sqrt{1 + i ω τ} + (λ / r) \right] e^{\left( L / λ \right) \sqrt{1 + i ω τ}} \right. \right. \\ \left. \left. + (λ / r)^2 \frac{ \sinh{ \left[ \left( L / λ \right) \sqrt{1 + i ω τ} \right] } }{\sqrt{1 + i ω τ}} \right)^{-1} \right\} . \end{multline} $$

Only scales that appear in $ f $

$ L / λ $
$ λ / r $
$ ω τ $

E. Sosenko, H. Wei, and V. Aji, Phys. Rev. B 89, 245436 (2014).

Fits

Tunneling contacts

$ L = 5.5 \: \text{µm} $
$ P = 0.1 $
$ R_\text{C} = 6.7 × 10^{ 6 } \: \text{kΩ} $
$ τ = 451.84 \: \text{ps} $
$ D = 0.01 \: \text{m}^2 \text{s}^{-1} $

Tunneling contacts

$ L = 2.1 \: \text{µm} $
$ P = 0.19 $
$ R_\text{C} = 2.03 × 10^{ 7 } \: \text{kΩ} $
$ τ = 514.3 \: \text{ps} $
$ D = 0.02 \: \text{m}^2 \text{s}^{-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

Fit to parallel field data from Fig. 4c of W. Han, et al.

$ L = 3.0 \: \text{µm} $
$ P = 0.23 $
$ R_\text{C} = 1.31 × 10^{ 7 } \: \text{kΩ} $
$ τ = 132.28 \: \text{ps} $
$ D = 0.02 \: \text{m}^2 \text{s}^{-1} $

Transparent contacts

Fit to parallel field data from Fig. 4d of W. Han, et al.

$ L = 3.0 \: \text{µm} $
$ P = 0.01 $
$ R_\text{C} = 2.94 \: \text{kΩ} $
$ τ = 130.36 \: \text{ps} $
$ D = 0.04 \: \text{m}^2 \text{s}^{-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

$$Δ \rNL = \left( P_Σ^L \right)^2 R_N e^{- L / λ}$$

Tunneling contacts

$$f^∞ = \re{\frac{e^{- \left( L / λ \right) \sqrt{1 + i ω τ}}}{2 \sqrt{1 + i ω τ}}}$$

Fits independent of lifetime

$ λ / r ≫ \sqrt{ω τ} ≫ 1 $

Zeros determined by $$L \sqrt{\frac{D}{2 ω}} + \frac{π}{4} = \frac{n π}{2}$$

E. Sosenko, H. Wei, and V. Aji, Phys. Rev. B 89, 245436 (2014).

Fits: $ τ $ Independent Limit

Transparent contacts

$ L = 3.0 \: \text{µm} $
$ P = 0.02 $
$ R_\text{C} = 0.27 \: \text{kΩ} $
$ τ = 9.97 × 10^{ 9 } \: \text{ps} $
$ D = 0.02 \: \text{m}^2 \text{s}^{-1} $

Transparent contacts

$ L = 3.0 \: \text{µm} $
$ P = 0.02 $
$ R_\text{C} = 0.28 \: \text{kΩ} $
$ τ = 9.95 × 10^{ 13 } \: \text{ps} $
$ D = 0.02 \: \text{m}^2 \text{s}^{-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

$ \mathrm{MoS_2} $, $ \mathrm{WS_2} $, $ \mathrm{MoSe_2} $, $ \mathrm{WSe_2} $
Similar to monolayer graphene: two inequivalent valleys: $ \vect{K} $, $ \vect{K}' $
Strong spin-orbit coupling and inversion symmetry breaking
Leads to opposite valley Berry curvature
Two state tight binding model: $ d_{z^2} $, and $ d_{xy} $, $ d_{x^2 - y^2} $

$$ H_0^{τ σ} \exOfK = a t \left(τ k_x σ_x + k_y σ_y \right) ⊗ I_2 + \frac{Δ}{2} σ_z ⊗ I_2 - λ τ \left(σ_z - 1 \right) ⊗ S_z $$

$$ H_0^{τ σ} \exOfK = \left[ \begin{matrix} \dfrac{Δ}{2} & a t \left( τ k_x - i k_y \right) \\ a t \left( τ k_x + i k_y \right) & λ τ σ - \dfrac{Δ}{2} \end{matrix} \right] $$

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

$ \mathrm{MoS_2} $

$ a t = 3.15 \: \text{Å eV} $
$ Δ = 1.66 \: \text{eV} $
$ 2 λ = 0.15 \: \text{eV} $
$ μ = -0.83 \: \text{eV} $

$$ E_{τ σ}^n \exOfK = \frac{1}{2} \left( λ τ σ + n \sqrt{ (2 a t)^2 \left\lvert \vect{k} \right\rvert^2 + \left( Δ - λ τ σ \right)^2 } \right) $$

Optical Transitions

$ \vect{P}^{τ σ} \exOfK = \frac{m_0}{ħ} \left\langle u_+ \right\rvert ∇_{\vK} H_0^{τ σ} \exOfK \left\lvert u_- \right\rangle $

$ P_±^{τ σ} \exOfK = P_x^{τ σ} ± i P_y^{τ σ} $

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

$ \dot{\vect{r}} = ∇_{\vK} E \exOfK - \dot{\vect{k}} × \vect{Ω} \exOfK $

$ \dot{\vect{k}} = - e \vect{E} - e \dot{\vect{r}} × \vect{B} $

Anomalous velocity $ \vect{v} = e \vect{Ω} \exOfK × \vect{E} $

Berry curvature

$ \vect{Ω}_{τ σ}^n \exOfK = ∇_{\vK} × \left\langle u_{τ σ}^n \exOfK \right\rvert i ∇_{\vK} \left\lvert u_{τ σ}^n \exOfK \right\rangle $

Broken inversion symmetry

$ \vect{Ω}_{τ, σ}^n \exOfK = - \vect{Ω}_{-τ, σ}^n \exOfK ≠ 0 $

BCS Superconductivity

Mean-field Hamiltonian

$$ H - μ N = \sideset{}{}∑_{\vK σ} ξ_{\vK} c_{\vK σ}^† c_{\vK σ} - \sideset{}{}∑_{\vK} \left( \bar{Δ}_{\vK} c_{-\vK ↓} c_{\vK ↑} + Δ_{\vK} c_{\vK ↑}^† c_{-\vK ↓}^† \right) $$

Region of allowed parings for fixed center-of-mass momentum.

Number of available parings maximized when center-of-mass momentum is zero

Quasiparticle operators

$ b_{\vK σ} = σ \cos{θ_{\vK}} c_{\vK σ} + \sin{θ_{\vK}} c_{-\vK, -σ}^† $

Diagonalized

$ \sideset{}{}∑_{\vK σ} λ_{\vK} b_{\vK σ}^† b_{\vK σ} $

Induced Superconductivity

Intervalley pairing

BCS pairs for induced superconducting states.

$ a^ν_{τ σ} $—orbital operators
$ b_α $—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

$$ \begin{equation} H_V = - \sideset{}{'}∑_{\vK} \sideset{}{}∑_{ν, τ} Δ_ν {a^ν_{-τ ↓}}^† \exOfMK {a^ν_{τ ↑}}^† \exOfK + \hc \end{equation} $$

$$ \begin{equation} H - μ N = \sideset{}{'}∑_{\vK} \sideset{}{}∑_α λ_{\vK}^α b_{\vK α}^† b_{\vK α} + \sideset{}{'}∑_{\vK} \left(ξ_{\vK ↓} + λ_{\vK}^- \right) . \end{equation} $$

Optical Transitions

$ \vect{P}^{τ σ} \exOfK = \frac{m_0}{ħ} \left\langle u_+ \right\rvert ∇_{\vK} H_0^{τ σ} \exOfK \left\lvert u_- \right\rangle $

$ P_±^{τ σ} \exOfK = P_x^{τ σ} ± i P_y^{τ σ} $

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

Induced superconducting optical transition rates for $ τ = + $.

$ \vect{P} \exOfK = \frac{m_0}{ħ} \left\langle Ω_f \right\rvert ∇_{\vK} H^{τ σ} \exOfK \left\lvert Ω \right\rangle $

$ P_± \exOfK = P_x ± i P_y $

$ \left\lvert Ω \right\rangle = ∏_{\vK} b_{\vK ↑} b_{-\vK ↓} \left\lvert 0 \right\rangle $

$ \left\lvert Ω_f \right\rangle = \begin{cases} {c^+_α}^† \exOfK b_{-α} \exOfMK \left\lvert Ω \right\rangle & k > k_μ \\ {c^+_α}^† \exOfK b_{-α}^† \exOfMK \left\lvert Ω \right\rangle & k < k_μ \end{cases} $

SC Optical Excitations

Compare to normal transitions

Strong induced superconducting optical transition rates. Dashed lines are $ H_0 $ transitions.

Weak induced superconducting optical transition rates. Dashed lines are $ H_0 $ 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 $ \mathrm{MoS_2} $
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

$$ H_V = - \sideset{}{'}∑_{\vK, \vK'} v \left( \vK' - \vK \right) A \left( \vK, \vK' \right) $$

$$ \begin{align} A \left( \vK, \vK' \right) = & B^2 \left( \vK, \vK' \right) \exInteration{+ ↑}{+ ↑} \\ + & B^2 \left( \vK', \vK\right) \exInteration{- ↓}{- ↓} \\ + & \left\lvert B \left( \vK, \vK' \right) \right\rvert^2 \left[ \exInteration{+ ↑}{- ↓} \\ + \exInteration{- ↓}{+ ↑} \right] \end{align} $$

Two dimensional systems

Spin lifetime and Hanle curve fitting

Superconducting phases of monolayer transition-metal dichalcogenides

Spin lifetime

Outline

Motivation

Fits

Tunneling contacts

Tunneling contacts

Device geometry

Motivation for solution

Existing results

Our approach

Non-local resistance

Fits

Tunneling contacts

Tunneling contacts

Fits

Pinhole contacts

Transparent contacts

Regimes

Zero field

Tunneling contacts

Fits independent of lifetime

Fits: \( τ \) Independent Limit

Transparent contacts

Transparent contacts

Dichalcogenides

Outline

Motivation

Effective Hamiltonian

Energy Bands

\( \mathrm{MoS_2} \)

Optical Transitions

Vally Hall effect

Semiclassical equations

Berry curvature

BCS Superconductivity

Mean-field Hamiltonian

Quasiparticle operators

Diagonalized

Induced Superconductivity

Intervalley pairing

Optical Transitions

SC Optical Excitations

SC Optical Excitations

Compare to normal transitions

Future Work

Induced superconductivity

Intrinsic superconductivity