Spin and Valley Physics in Two Dimensional Systems Graphene and Superconducting Transition Metal Dichalcogenides

Doctoral Thesis Defense

evansosenko.com/deck-doctoral-thesis

August 2, 2016
TMD crystal

Background

Graphene
By AlexanderAlUS (Own work) [CC BY-SA 3.0 or GFDL], via Wikimedia Commons

2D Materials promise novel application

  • Graphene discovered 2004
  • 2D hexagonal carbon lattice, high electron mobility
  • Relativistic low-energy model: Dirac cones, linear dispersion
  • Strong, flexible & transparent
  • Monolayer TMDs: also hexagonal lattice
  • Gapped with strong spin-orbit coupling
  • Both excellent candidates for spintronic devices

Spintronics

  • Nobel Prize for GMR in 2007 (Albert Fert, Peter Grünberg)
  • A future with devices built on spin-current
  • Smaller and lower power
  • Graphene? Predicted excellent conductor for spin-current
  • TMDs? No spin degeneracy and spin couples to light polarization

Overview for Part I

Spin lifetime

  • Injection, transmission, detection of spin signals
  • Spin signals degrade via internal scattering
  • Spin lifetime measures how quickly this happens
  • Critical to find materials with long spin lifetimes

Motivation

  1. Theoretical predictions longer than measured: \( \text{ms} \) vs. \( \text{ps} \)
  2. Finite contact resistance mismatch: a potential candidate
  3. Unified analytic solution for fitting data in all limits
Spintronic device

Outline

  1. Model and solution
  2. Hanle curve fitting
  3. Regimes and results
E. Sosenko, H. Wei, and V. Aji, Phys. Rev. B 89, 245436 (2014).

Device geometry

  • \( L \) : contact spacing
  • \( D \) : diffusion constant
  • \( \tau \) : spin lifetime
  • \( \lambda = \sqrt{D \tau} \)
  • \( \omega = g \mu_B B / \hbar \)
Nonlocal spin valve
  • \( \mu_s = \frac{1}{2} \left( \mu_{\uparrow} - \mu_{\downarrow} \right) \)
  • \( J_{\uparrow \downarrow} = \sigma_{\uparrow \downarrow} \nabla \mu_{\uparrow \downarrow} \)
  • \( J_{\uparrow \downarrow}^C = \Sigma_{\uparrow \downarrow} \left( \mu^N_{\uparrow \downarrow} - \mu^F_{\uparrow \downarrow} \right)_c \)
  • \( J = J_{\uparrow} + J_{\downarrow} \)
  • \( J_s = J_{\uparrow} - J_{\downarrow} \)

$$D \nabla^2 \mu_s - \frac{\mu_s}{\tau} + \omega \times \mu_s = 0$$

$$V \propto \mu_s^N(x = L)$$

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

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

$$ R_{\text{NL}} \propto \text{Re} {\frac{e^{- \left( L / \lambda \right) \sqrt{1 + i \omega \tau}}} {2 \sqrt{1 + i \omega \tau}}} $$

$$ R_{\text{NL}} \propto \int_0^{\infty} \frac{1}{\sqrt{4 \pi D t}} \exp{\left[ - \frac{L^2}{4 D t} \right]} e^{-t / \tau} \cos{\omega t} \: dt $$

Our approach

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

Computation

  • Drift diffusion equation in contacts and semiconductor
  • Continuity of current and spin current at boundary
  • Conservation of current and spin
  • Solution gives nonlocal resistance

Non-local resistance

$$\Delta R_{\text{NL}} = 2 P^2 R_N \left\lvert f \right\rvert$$

  • \( r = \frac{R_F + R_C}{R_{\text{SQ}}} W \)
  • \( R_{\text{SQ}} = W / \sigma^N \)
  • \( R_N = \frac{\lambda}{W L} \frac{1}{\sigma^N} \)

$$ f = \text{Re} \left\{ \left( 2 \left[ \sqrt{1 + i \omega \tau} + (\lambda / r) \right] e^{\left( L / \lambda \right) \sqrt{1 + i \omega \tau}} + (\lambda / r)^2 \frac{ \sinh{\left[ \left( L / \lambda \right) \sqrt{1 + i \omega \tau} \right]}} {\sqrt{1 + i \omega \tau}} \right)^{-1} \right\} $$

Only scales that appear in \( f \)

  • \( L / \lambda \)
  • \( \lambda / r \)
  • \( \omega \tau \)
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.
  • \( L = 2.1 \: \mu \text{m} \)
  • \( P = 0.19 \: \)
  • \( R_\text{C} = 2.03 \times 10^7 \: \text{k} \Omega \)
  • \( \tau = 514.3 \: \text{ps} \)
  • \( D = 0.02 \: \text{m}^2 \text{s}^{-1} \)

Tunneling contacts

Fit to parallel field data from Fig. 4b of W. Han, et al.
  • \( L = 5.5 \: \mu \text{m} \)
  • \( P = 0.1 \: \)
  • \( R_\text{C} = 6.70 \times 10^6 \: \text{k} \Omega \)
  • \( \tau = 451.84 \: \text{ps} \)
  • \( D = 0.01 \: \text{m}^2 \text{s}^{-1} \)
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

Fit to parallel field data from Fig. 4c of W. Han, et al.
  • \( L = 3 \: \mu \text{m} \)
  • \( P = 0.23 \: \)
  • \( R_\text{C} = 2.31 \times 10^7 \: \text{k} \Omega \)
  • \( \tau = 132.28 \: \text{ps} \)
  • \( D = 0.02 \: \text{m}^2 \text{s}^{-1} \)

Transparent contacts

Fit to parallel field data from Fig. 4d of W. Han, et al.
  • \( L = 3 \: \mu \text{m} \)
  • \( P = 0.01 \: \)
  • \( R_\text{C} = 2.94 \: \text{k} \Omega \)
  • \( \tau = 130.36 \: \text{ps} \)
  • \( D = 0.04 \: \text{m}^2 \text{s}^{-1} \)
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

$$ \Delta R_{\text{NL}} = \left( P_{\Sigma}^L \right)^2 R_N e^{- L / \lambda} $$

Tunneling contacts

$$ f^{\infty} = \text{Re} {\frac{e^{- \left( L / \lambda \right) \sqrt{1 + i \omega \tau}}} {2 \sqrt{1 + i \omega \tau}}} $$

Fits independent of lifetime

\( \lambda / r \gg \sqrt{\omega \tau} \gg 1 \)

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

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

Fits: \( \tau \) Independent Limit

Transparent contacts

Fit to parallel field data from Fig. 4d of W. Han, et al.
  • \( L = 3 \: \mu \text{m} \)
  • \( P = 0.02 \: \)
  • \( R_\text{C} = 0.27 \: \text{k} \Omega \)
  • \( \tau = 9.97 \times 10^9 \: \text{ps} \)
  • \( D = 0.02 \: \text{m}^2 \text{s}^{-1} \)

Transparent contacts

Fit to parallel field data from Fig. 4d of W. Han, et al.
  • \( L = 3 \: \mu \text{m} \)
  • \( P = 0.02 \: \)
  • \( R_\text{C} = 0.28 \: \text{k} \Omega \)
  • \( \tau = 9.95 \times 10^{13} \: \text{ps} \)
  • \( D = 0.02 \: \text{m}^2 \text{s}^{-1} \)
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).

Software

Data Fitting with SciPy

Independent Python package for general curve fitting

github.com/razor-x/scipy-data_fitting

Fitalyzer

Browser based fitting visualizer

github.com/razor-x/fitalyzer

Transition metal dichalcogenides

  1. TMD overview and tight-biding, low-energy, two-valley model
  2. Optical valley selection rules and Berry curvature
  3. Intrinsic and induced superconductivity
  4. Intervalley and intravalley paring channels
  5. Superconducting optoelectronics and Berry curvature
TMD crystal (top view)
TMD crystal (side view)

TMD monolayers

  • Direct band gap semiconductors: \( \mathrm{MoS_2} \), \( \mathrm{WS_2} \), \( \mathrm{MoSe_2} \), \( \mathrm{WSe_2} \)
  • Break inversion symmetry ⇒ gapped out spectrum
  • Two inequivalent valleys ⇒ new degree of freedom
  • Strong spin-orbit coupling ⇒ large valance band spin-splitting
  • Opposite valley Berry curvature
  • Optical valley probe and valley Hall effect
  • Two state tight binding model: \( d_{z^2} \), and \( d_{xy} \pm i d_{x^2 - y^2} \)
TMD energy bands

$$ H_{\tau}^0 \left( \mathbf{k} \right) = a t \left(\tau k_x \sigma_x + k_y \sigma_y \right) \otimes I_2 + \frac{\Delta}{2} \sigma_z \otimes I_2 - \lambda \tau \left(\sigma_z - 1 \right) \otimes S_z $$

D. Xiao, G.-B. Liu, W. Feng, X. Xu, and W. Yao, Phys. Rev. Lett. 108, 196802 (2012).

Energy bands

Energy bands
  • \( \Delta \)—band splitting
  • \( \lambda \)—spin splitting
  • \( \tau \)—valley index
  • \( s \)—spin index
  • \( a t = 3.15 \) Å eV
  • \( \Delta = 1.66 \) eV
  • \( 2 \lambda = 0.15 \) eV
  • \( \mu = -0.83 \) eV
$$ E_{\tau \sigma}^n \left( \mathbf{k} \right) = \frac{1}{2} \left( \lambda \tau \sigma + n \sqrt{(2 a t)^2 {\left\lvert \mathbf{k} \right\rvert}^2 + {\left( \Delta - \lambda \tau \sigma \right)}^2} \right) $$

Optical excitations

Optical transitions
Band transitions
$$ H^A = \sum_{\mathbf{k}, \tau, s, n, n'} \frac{e A_0}{m_0} \mathbf{\epsilon} \cdot \mathbf{P}_{\tau s}^{n n'} \left( \mathbf{k} \right) {c_{\tau s}^n}^{\dagger} \left( \mathbf{k} \right) c_{\tau s}^{n'} \left( \mathbf{k} \right) $$

Optoelectronic coupling

  • Right circularly polarized light couples to right valley \( P_+ \)
  • Left circularly polarized light couples to left valley \( P_- \)

Valley Hall effect

Band transitions

Semiclassical equations

\( \dot{\mathbf{r}} = \nabla_{\mathbf{k}} E \left( \mathbf{k} \right) - \dot{\mathbf{k}} \times \mathbf{\Omega} \left( \mathbf{k} \right) \)

\( \dot{\mathbf{k}} = -e \mathbf{E} - e \dot{\mathbf{r}} \times \mathbf{B} \)

Anomalous velocity

\( \mathbf{v} = e \mathbf{\Omega} \left( \mathbf{k} \right) \times \mathbf{E} \)

Berry curvature

\( \mathbf{\Omega}_{\tau \sigma}^n \left( \mathbf{k} \right) = \nabla_{\mathbf{k}} \times \left\langle u_{\tau \sigma}^n \left( \mathbf{k} \right) \right\rvert i \nabla_{\mathbf{k}} \left\lvert u_{\tau \sigma}^n \left( \mathbf{k} \right) \right\rangle \)

Broken inversion symmetry

\( \mathbf{\Omega}_{\tau, \sigma}^n \left( \mathbf{k} \right) = - \mathbf{\Omega}_{-\tau, \sigma}^n \left( \mathbf{k} \right) \neq 0 \)

Superconducting sources

Substrate induced

$$ V = - \sum_{\mathbf{k} \nu \tau} B_{\nu} {d^{\nu}_{\tau}}^{\dagger} \left( \mathbf{k} \right) d^{\nu}_{-\tau} \left( \mathbf{k} \right) $$

Intrinsic (density-density interaction)

$$ V = \frac{1}{2} \sum_{\mathbf{R} \mathbf{R'}} v_{\mathbf{R} \mathbf{R'}} :n_{\mathbf{R}} n_{\mathbf{R'}}: $$

Assume even pairing interaction

BCS Hamiltonian

$$ H - \mu N = \sum_{\mathbf{k} \sigma} \xi_{\mathbf{k}} c_{\mathbf{k} \sigma}^{\dagger} c_{\mathbf{k} \sigma} - \sum_{\mathbf{k}} \left( \bar{\Delta}_{\mathbf{k}} c_{-\mathbf{k} \downarrow} c_{\mathbf{k} \uparrow} + \Delta_{\mathbf{k}} c_{\mathbf{k} \uparrow}^{\dagger} c_{-\mathbf{k} \downarrow}^{\dagger} \right) = \sum_{\mathbf{k} \sigma} \lambda_{\mathbf{k}} \gamma_{\mathbf{k} \sigma}^{\dagger} \gamma_{\mathbf{k} \sigma} $$

Superconducting channels

Induced

$$ \Delta_{\mathbf{k}} = \frac{1}{2} \left( B_+ + B_- \right) - \frac{1}{2} \left( B_+ - B_- \right) \cos{\theta_{\mathbf{k}}} $$

Intrinsic

$$ \Delta_{\mathbf{k}} = \chi_0 \cdot f_{\mathbf{k}} $$

Gap equation

$$ \chi_0 = \frac{v_0}{2} \sum_{\mathbf{k}} \frac{f_{\mathbf{k}} \cdot \chi_0}{\lambda_{\mathbf{k}}} \bar{f}_{\mathbf{k}} $$

Optical transition rate
  • Project into upper-valance bands (\( \tau = s \))
  • Mean field theory BCS-type solutions: gap function \( \Delta_{\mathbf{k}} \)
  • Different \( f_{\mathbf{k}} \) per channel
  • \( m = 0 \) intervalley channel always dominates
  • Both classes give constant gap function for dominant paring

Optical coupling

Optical transition rate
  • \( \mathbf{A} / A_0 = \mathbf{\epsilon} e^{i \omega t} + \mathbf{\epsilon}^* e^{-i \omega t} \), \( \mathbf{k} \rightarrow \mathbf{k} + e \mathbf{A} \)
  • Harmonic perturbation ⇒ correlated phase's optical transition rates
  • Valley selection remains coupled to circularly polarized light
$$ H^A = \sum_{\mathbf{k}, \tau, s, n, n'} \frac{e A_0}{m_0} \mathbf{\epsilon} \cdot \mathbf{P}_{\tau s}^{n n'} \left( \mathbf{k} \right) {c_{\tau s}^n}^{\dagger} \left( \mathbf{k} \right) c_{\tau s}^{n'} \left( \mathbf{k} \right) $$

Berry curvature

Optical transition rate
  • BCS ground state has zero net curvature
  • Optical pair-breaking ⇒ selectively excite carriers in single valley
  • Berry curvature retains relative sign relations

Conclusion

Spin lifetime measurements

  • Analytical solution modeling a nonlocal spin value which includes the ferromagnet contacts
  • Developed software to refit data with new result
  • Analyzed various regimes to understand the limits of the model

Superconducting states in TMDs

  • Characterized superconducting phases in regime where with locked spin and valley indexes
  • The correlated state inherits the valley contrasting phenomena
  • Pair-breaking produces quasiparticles that have the same Berry curvature (same anomalous velocity)