Hybrid-Monte-Carlo simulations at the van Hove singularity in monolayer graphene

Transcription

1 Hybrid-Monte-Carlo simulations at the van Hove singularity in monolayer graphene Michael Körner Institut für Theoretische Physik, Justus Liebig Universität Gießen Lunch Club Seminar January 24, 2018

2 Outline Introduction Hybrid Monte Carlo Results Summary and Outlook 1 / 24

3 Introduction Graphene - Overview Graphene - Overview History: First calculation of the band structure (Wallace et. al.) First experimental realization (Boehm et. al.) Measurement of the electron properties (Novoselov, Geim) Nobel Prize for Novoselov and Geim since at least D Materials as a independent field of research Graphene: 2-dimensional hexagonal lattice of Carbon Atoms Basic building block for Graphite, Fullerene, etc. Castro Neto, Guinea, Peres, Novoselov, Geim., Rev. Mod. Phys., Vol. 81, No. 1, 2 / 24

4 Introduction Graphene - Overview Graphene - Overview Remarkable physical properties: Tensile strength: (Structural steel: N mm 2 N mm ) 2 Thermal conductivity: 5000 W mk (Copper: 400 W mk ) Electrical conductivity: Ωcm (Copper: Ωcm ) Nearly transparent: Absorption 2.3% Focus on electrons: 3 electrons in sp 2 -orbitals build σ-band (responsible for structure) 1 electron in p z -orbital build π-band (valence band) π-band electrons are screened by σ-band electrons 3 / 24

5 Introduction Structure Structure Diatomic base: ( ) 3 a = a 0 b = a 2 ( ) 3 3 c = ( ) 0 a with the lattice constant a 1.42 Å All lattice points can be described by: R 1 = n 1 a + n 2 b R2 = n 1 a + n 2 b + c with n 1, n 2 Z. Two sublattices! 4 / 24

6 Introduction Tight binding theory of Graphene Tight binding theory of Graphene H T B = κ ( a x,s a y,s + h.c. ) x,y,s M K' K Linear Dispersion around K-points Van Hove Singularity at the M-point 5 / 24

7 Introduction Density of States Density of States Important effects are also in the DOS: ds 1 ρ(e) = S(E) (2π) 2 k ɛ( k) VHS characterized by a divergency in the DOS Full tight-binding DOS can be observed in analog experiments with photonic crystals (microwave billiards) Logarithmic divergency in finite volumes ρ max = 3 [ ln Nc 2 ln π O(Nc 1 ) ] πκ Neck-disrupting Lifshitz transition ρ(ϵ) 3κ κ 2κ 2κ κ κ ϵ Dietz, Iachello, Miski-Oglu, Pietralla, Richter, v. Smekal, Wambach Phys. Rev. B 88, (2013) 6 / 24

8 Hybrid-Monte-Carlo simulations at the van Hove singularity in monolayer graphene Introduction Lifshitz Transition Lifshitz Transition Shifting the Fermi level by a chemical potential µ the shape of the fermi surface changes The topology of the fermi surface changes when crossing the VHS! 7 / 24

9 Introduction Real Graphene Real Graphene Strongly coupled system α = e 2 4πɛ 0 v F 2.2 non-local exchange effects are not negligible α can be changed experimentally Experimental observations: Reshaping of bands smaller bandwidth S. Ulstrup et al., Phys. Rev. B, 94, (2016). Extended VHS J. L. McChesney et. al., PRL 104, (2010) Add a long-range potential and a chemical potential to the Hamiltonian: H = κ x,y,s(a x,sa y,s + h.c.) + λ q x V xy q y µ a x,sa x,s x,y x,s with q x = a x,1 a x,1 a x, 1 a x, 1 Goal: Use the HMC-Algorithm to study the influence of the long-range interaction on the DOS especially at the VHS. 8 / 24

10 Hybrid Monte Carlo Hybrid Monte Carlo HMC: Based on path-integral representation of partition function Z = Tr e βh = Dφ det [ M(φ)M (φ) ] { exp 2 Interactions are transmitted by scalar Hubbard field φ N t 1 t=0 Effect of dynamical Fermions encoded in Fermion matrix M(φ) x,y } φ x,t Vxy 1 φ y,t M (x,t)(y,t )(φ) =δ xy (δ tt e i β N φ x,t t δ t 1,t ) κ β β δ y,x+ n δ t 1,t + m s δ xy δ t 1,t N t N t Path-integral approximated by artificial Hamiltonian evolution of φ n 9 / 24

11 Hybrid Monte Carlo Potential Potential 10 8 standard Coulomb potential crpa (Wehling et. al.) partially screened potential V(r) in ev r/a 10 / 24

12 Hybrid Monte Carlo Sign problem Sign problem Through a chemical potential in the Hamiltonian the Fermion matrix M respectively M, turns into M M µ β N t 1 = M µ M M + µ β N t 1 M µ Measure is no longer real and positive definite. Interpretation as a probability impossible! Circumvention by introducing a spin-dependent chemical potential: µ = +µ s s = +1 µ = µ s s = 1 Then the condition is fulfilled and we can write det [ M(φ)M (φ) ] again. 11 / 24

13 Hybrid Monte Carlo Observable Observable Effects in DOS perserved in Thomas-Fermi-susceptibility χ(µ) = A c lim lim Π(ω, p, µ, T ) p 0 ω 0 d 2 k Π(ω, p, µ, T ) = 2π 2 f BZ s,s =±1 ( s E( k+ p) µ T ) f T 0 ρ(µ) ( ) 1 + ss Φ k Φ k+ p Φ k Φ k+ p ) ( se( k) µ T s E( k + p) se( k) ω iɛ The Thomas-Fermi susceptibility is given by the second derivative of the grand canonical potential χ(µ) = 1 ( d 2 ) [ Φ N c dµ 2 = 1 1 d 2 Z N c β Z dµ 2 1 ( ) ] 2 dz Z 2 dµ 12 / 24

14 Hybrid Monte Carlo Observable Observable Expectation values of susceptibility in the thermal ensemble: 1 d n Z Z dµ n = 1 [ d n Dφ Z dµ n det ( MM ) ] e S(φ) Using these relations we can write the susceptibility as χ = χ con + χ dis, with χ con = 2 ( ) 1 dm 1 dm ReTr M M N c β dµ dµ { [ χ dis = 4 ( )] 2 1 dm ReTr M N c β dµ ( ) } 2 1 dm ReTr M dµ The brackets on the right-hand sides are understood as averages over a representative set of field configurations. 13 / 24

15 Hybrid Monte Carlo Relevant parameters Relevant parameters - An overview Temperature T Volume N Staggered mass m s Discretization of the time axis N t Interaction strength α spin-dependent chemical potential µ 14 / 24

16 Hybrid Monte Carlo Influence of V and T in non-interacting theory Influence of V and T in non-interacting theory Finite volumina and non-zero temperature χ(µ) = 1 [ ( ) µ E(m, n) 2T N 2 sech 2 2T n,m +sech 2 ( µ + E(m, n) 2T )] Behavior at the VHS χ max = 3 { ( ) } 8κ π 2 ln + γ E + O(T ) κ πt 15 / 24

17 Hybrid-Monte-Carlo simulations at the van Hove singularity in monolayer graphene Results Influence of the time discretization Influence of the time discretization χcon(μ)inev -1 χ(μ)inev -1 δ 112eV 1 δ 124eV 1 δ 148eV 1 δ 196eV 1 δ 0 analytical, λ μ δ 16eV 1 δ 112eV 1 δ 130eV 1 δ 148eV 1 δ 0 χdis(μ)inev -1 χ(μ)inev -1 δ 16eV 1 δ 112eV 1 δ 130eV 1 δ 148eV 1 δ 0 analytical, λ μ δ 16eV 1 δ 112eV 1 δ 130eV 1 δ 148eV 1 δ 0 μ discretization generates a constant shift in χ χ dis is not affected by the discrete time axis μ 16 / 24

18 Results Influence of the volume Influence of the volume χpeak in ev -1 λ = Approximation N λ = Interactions only shift the peak to higher values N 17 / 24

19 Hybrid-Monte-Carlo simulations at the van Hove singularity in monolayer graphene Results Influence of the staggered mass Influence of the staggered mass χ(μ) in ev -1 m s = 0.5 ev m s = 0.3 ev m s = 0.1 ev - χ(μ) in ev -1 0 κ 2κ 3κ m s = 0.5 ev m s = 0.3 ev m s = 0.1 ev - χ(μ) in ev -1 μ 0 κ 2κ 3κ m s = 0.5 ev m s = 0.3 ev s m = 0.1 ev μ staggered mass influences χ only at low µ χ dis not affected by m s - μ 18 / 24

20 Hybrid-Monte-Carlo simulations at the van Hove singularity in monolayer graphene Results Influence of the long-range interaction Influence of the long-range interaction χ(μ)inev λ λ 0.1 λ λ 0.8 λ 1.0 χcon(μ)inev λ λ 0.1 λ λ 0.8 λ 1.0 χdis(μ)inev μ 0.7 λ λ 0.1 λ λ 0.8 λ μ 0.1 μ Simulation agrees qualitativly with the experimental result (bandwidth shrinking) Long-range interaction enlarge the VH-peak in finite volumina 19 / 24

21 Hybrid-Monte-Carlo simulations at the van Hove singularity in monolayer graphene Results Influence ot the Temperature Influence ot the Temperature β 2eV 1 β 3eV 1 β 2eV 1 β 3eV 1 χ(μ)inev -1 β 4eV 1 χcon(μ)inev -1 β 4eV 1 μ μ χdis(μ)inev -1 β 2eV 1 β 3eV 1 β 4eV 1 Lowering temperature enlarge the VH-peak in finite volumina Enlargement is mostly driven by χ dis in interacting systems μ 20 / 24

22 Results Influence of the Temperature at the VHS Influence of the Temperature at the VHS χ(t) 2 χ con (T) χ dis (T) 1.5 f 1(T ) = 3 π 2 ln ( κ T ) + b c T κ b [ev 1 ] 0.519(3) c [ev 1 ] 72(8) χpeak κ T κ / 24

23 Hybrid-Monte-Carlo simulations at the van Hove singularity in monolayer graphene Results Influence of the Temperature at the VHS Influence of the Temperature at the VHS χ(t) 2 χ con (T) χ dis (T) 1.5 f 1(T ) = 3 π 2 ln ( κ T ) + b c T κ b [ev 1 ] 0.519(3) c [ev 1 ] 72(8) χpeak κ Use volume dependency! T κ χpeak in ev λ = Approximation N λ = N 21 / 24

24 Hybrid-Monte-Carlo simulations at the van Hove singularity in monolayer graphene Results Influence of the Temperature at the VHS Influence of the Temperature at the VHS χ(t) 2 χ con (T) χ dis (T) 1.5 f 1(T ) = 3 π 2 ln ( κ T ) + b c T κ b [ev 1 ] 0.519(3) c [ev 1 ] 72(8) χpeak κ Use volume dependency! T κ χpeak in ev λ = Approximation N λ = N 21 / 24

25 Results Influence of the Temperature at the VHS Influence of the Temperature at the VHS χpeak κ χ(t) 2 χ con (T) χ dis (T) T κ f 1(T ) = 3 π 2 ln ( κ T ) + b c T κ b [ev 1 ] 0.519(3) c [ev 1 ] 72(8) f 2(T ) = k T Tc T c γ β c [ev 1 ] 6.1(5) γ 0.52(6) k [ev 1 ] 0.12(1) 21 / 24

26 Results Influence of the Temperature at the VHS Influence of the Temperature at the VHS χpeak κ χ(t) 2 χ con (T) χ dis (T) T κ f 1(T ) = 3 π 2 ln ( κ T ) + b c T κ b [ev 1 ] 0.519(3) c [ev 1 ] 72(8) f 2(T ) = k T Tc T c γ β c [ev 1 ] 6.1(5) γ 0.52(6) k [ev 1 ] 0.12(1) Evidence for extended van-hove singularity! 21 / 24

27 Summary and Outlook Summary Graphene System in which the coupling can be changed experimentally Method Tight-binding description/approximations as a baseline for simulations Use spin-dependent chemical potential to avoid the sign problem ( phase quenched theory ) All influences of the simulation parameters are understood HMC-simulations agrees with experiments qualitatively Bandwith shrinking Extended van-hove singularity More Information: MK, Smith, Buividovich, Ulybyshev, von Smekal, Phys. Rev. B 96, / 24

28 Summary and Outlook Outlook Differences to a pure Hubbard model Simulations with only on-site interactions Improved action Implementation of exponential action (with Maksim Ulybyshev) Sign problem Use LLR algorithm to simulate with real chemical potential (with Kurt Langfeld) Second peak in χ con Study and explain the second peak in the disconnected part of the susceptibilty 23 / 24

29 Summary and Outlook Thank you! 24 / 24