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: \( \text{ms} \) 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
- \( L = 2.1 \: \text{µm} \)
- \( P = 0.19 \)
- \( R_\text{C} = 2.03 × 10^{ 7 } \: \text{kΩ} \)
- \( τ = 514.3 \: \text{ps} \)
- \( D = 0.02 \: \text{m}^2 \text{s}^{-1} \)
Tunneling contacts
- \( L = 5.5 \: \text{µm} \)
- \( P = 0.1 \)
- \( R_\text{C} = 6.7 × 10^{ 6 } \: \text{kΩ} \)
- \( τ = 451.84 \: \text{ps} \)
- \( D = 0.01 \: \text{m}^2 \text{s}^{-1} \)
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).
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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
- \( L = 2.1 \: \text{µm} \)
- \( P = 0.19 \)
- \( R_\text{C} = 2.03 × 10^{ 7 } \: \text{kΩ} \)
- \( τ = 514.3 \: \text{ps} \)
- \( D = 0.02 \: \text{m}^2 \text{s}^{-1} \)
Tunneling contacts
- \( L = 5.5 \: \text{µm} \)
- \( P = 0.1 \)
- \( R_\text{C} = 6.7 × 10^{ 6 } \: \text{kΩ} \)
- \( τ = 451.84 \: \text{ps} \)
- \( D = 0.01 \: \text{m}^2 \text{s}^{-1} \)
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
- \( L = 3.0 \: \text{µm} \)
- \( P = 0.23 \)
- \( R_\text{C} = 1.31 × 10^{ 7 } \: \text{kΩ} \)
- \( τ = 132.28 \: \text{ps} \)
- \( D = 0.02 \: \text{m}^2 \text{s}^{-1} \)
Transparent contacts
- \( L = 3.0 \: \text{µm} \)
- \( P = 0.01 \)
- \( R_\text{C} = 2.94 \: \text{kΩ} \)
- \( τ = 130.36 \: \text{ps} \)
- \( D = 0.04 \: \text{m}^2 \text{s}^{-1} \)
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
$$ H - μ N = \sideset{}{}∑_{\vK σ} ξ_{\vK} c_{\vK σ}^† c_{\vK σ} - \sideset{}{}∑_{\vK} \left( \bar{Δ}_{\vK} c_{-\vK ↓} c_{\vK ↑} + Δ_{\vK} c_{\vK ↑}^† c_{-\vK ↓}^† \right) $$
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
- \( a^ν_{τ σ} \)—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
\( \vect{P} \exOfK = \frac{m_0}{ħ} \left\langle Ω_f \right\rvert ∇_{\vK} H^{τ σ} \exOfK \left\lvert Ω \right\rangle \)
\( P_± \exOfK = P_x ± i P_y \)
\( \left\lvert Ω \right\rangle = ∏_{\vK} b_{\vK ↑} b_{-\vK ↓} \left\lvert 0 \right\rangle \)
\( \left\lvert Ω_f \right\rangle = \begin{cases} {c^+_α}^† \exOfK b_{-α} \exOfMK \left\lvert Ω \right\rangle & k > k_μ \\ {c^+_α}^† \exOfK b_{-α}^† \exOfMK \left\lvert Ω \right\rangle & k < k_μ \end{cases} \)
SC Optical Excitations
Compare to normal transitions
- 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
$$ H_V = - \sideset{}{'}∑_{\vK, \vK'} v \left( \vK' - \vK \right) A \left( \vK, \vK' \right) $$
$$ \begin{align} A \left( \vK, \vK' \right) = & B^2 \left( \vK, \vK' \right) \exInteration{+ ↑}{+ ↑} \\ + & B^2 \left( \vK', \vK\right) \exInteration{- ↓}{- ↓} \\ + & \left\lvert B \left( \vK, \vK' \right) \right\rvert^2 \left[ \exInteration{+ ↑}{- ↓} \\ + \exInteration{- ↓}{+ ↑} \right] \end{align} $$