Landau levels and SdH oscillations in monolayer transition metal dichalcogenide semiconductors

Transcription

1 Landau levels and SdH oscillations in monolayer transition metal dichalcogenide semiconductors MTA-BME CONDENSED MATTER RESEARCH GROUP, BUDAPEST UNIVERSITY OF TECHNOLOGY AND ECONOMICS Collaborators: Andor Kormányos, Guido Burkard DEPARTMENT OF PHYSICS, UNIVERSITY OF KONSTANZ, D KONSTANZ, GERMANY Kormányos et. al., New Journal of Physics 17, (2015). 1

2 Transition metal dichalcogenides semiconductors Top view: three-fold symmetry No inversion symmetry. Atomically thin material with direct band gap. 1 2eV. DFT, LSDA for MoS 2 : (Nano-TCAD Group) Kormányos et. al, Phys. Rev. B 88, (2013). 2D workshop page 2

3 Two-band continuum model for valleys τ = ±K seven-band model ˆq = p+ e A two-ban model by Löwdin partitioning H τ eff = H 0 +H τ so +H τ k p H 0 = 2 ˆq +ˆq + ˆq ˆq + 2m e g eµ B B z s z Hso τ,s = ( τ vb s z ) 0 0 τ cb s z H τ,s k p = Hτ,s D +Hτ,s as +H τ,s 3w +Hτ,s cub, H τ,s D = ( εvb H τ,s 3w = ( ) τ γ τ,sˆq τ τ γτ,sˆq + τ ε cb 0 κ τ,s (ˆq τ + )2 κ τ,s (ˆqτ )2 0 ) H τ,s as = ( ατ,sˆq τ +ˆqτ 0 0 β τ,sˆq τ ˆq τ +, H τ,s cub,1 =... ) 2D workshop page 3

4 Landau Levels (LLs) harmonic oscillator eigenfunctions as basis states:π = 2 l B a,π + = 2 l B a,l B /eb. matrix representation ofh τ eff on a finite subspace spanned by the basis functions. Landau Levels: eigenvalues ofh τ eff. LLs in the conductance band: spin, K-valley spin, K -valley spin, K-valley spin, K -valley E [ev] b) B z [T] 2D workshop page 4

5 Approximation of the LLs Neglecting the trigonal warping and cubic terms (H τ,s 3w,Hτ,s cub,1 ). Another Löwdin-partitioning to obtain effective single-band Hamiltonian separately for conduction and valence bands. Leading to a harmonic oscillator-like Hamiltonians: E τ,s n,vb = ετ s vb + ω (τ s) vb E τ,s n,cb = ετ s cb + ω (τ s) cb ( n+ 1 ) 2 ( n+ 1 ) 2 valley splitting linearly depends onb z g eµ B B z s+ 1 2 g(s) vl,vb µ BB z τ, g eµ B B z s+ 1 2 g(s) vl,cb µ BB z τ. 2D workshop page 5

6 Approximation vs full quantum valley K, spin a) valence band b) conduction band E [ev] E [ev] B z [T] B z [T] The trigonal warping is relevant for higher magnetic fields. At higher magnetic fields the valley splitting is not linear inb. 2D workshop page 6

7 Calculation of the SdH oscillations Calculation of the conductivityσ xx : neglecting the intra-valley scattering between the spin-split bands: due to the specific form of the intrinsic SOC neglecting the inter-valley scattering in the absence of magnetic impurity: large momentum change, requires simultaneous spin-flip Considering the intra-valley, intra-band scatterings and short range scatterers. 2D workshop page 7

8 Self-consistent Born approximation random disorder potential V(r) with short range correlations V(r)V(r ) = λ sc δ(r r ) self-energyσ τ,s R = Στ,s r +iσ τ,s i Σ τ,s r +iσ τ,s i = λ sc 2πl 2 B n=0 1 E En τ,s (Σ τ,s r +iσ τ,s i ) E τ,s n are the approximated or exact LL energies. λ sc scattering rate calculated by the Born-approximation in zero magnetic field. Ando T, J. Phys. Soc. Jpn. 37, 1233 (1974). 2D workshop page 8

9 conductivityσ xx Kubo-formalism: σ τ,s xx = e2 π 2 de ( f(e) ) σ τ,s E xx(e) σ τ,s xx(e) ( ω (i) c ) 2 = n=0 (n+1)re[g τ,s A (n,e)gτ,s R (n+1,e) Gτ,s A (n,e)gτ,s A (n,e)] G τ,s R (n,e) andgτ,s A (n,e) are the retarded and advanced Greens-functions: G τ,s R,A (n,e) = [E Eτ,s n Σ τ,s R,A ] 1 2D workshop page 9

10 Calculated SdH oscillations for p-doped WSe 2 The amplitudes of the oscillations are not captured well by the approximated LLs. numerical SdH, and approximated SdH a) Reason: few LLs under the Fermi level. σ/σ ω c τ sc 2D workshop page 10

11 Calculated SdH oscillations for n-doped MoS 2 two bands contribute to σ xx = σ xx (1) +σ xx (2) parameters from GW calculations numerical SdH, and approximated SdH complex oscillation pattern due to valley splitting (different oscillation periods in the two valleys) σosc /σ d) ω c τ sc 2D workshop page 11

12 Comparing to experiments on MoS 2 Fitting the analytical expression to the experimental data of X. Cui et. al., 2015 advance online publication in Nature Nanotechnology (arxiv: ) σ osc (B)/σ(0) experimental SdH, and approximated SdH B z [T] 2D workshop page 12