The ac conductivity of monolayer graphene

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1 The ac conductivity of monolayer graphene Sergei G. Sharapov Department of Physics and Astronomy, McMaster University Talk is based on: V.P. Gusynin, S.G. Sh., J.P. Carbotte, PRL 96, 568 (6), J. Phys.: Condens. Matter 19, 6 (7), cond-mat/6777, cond-mat/7153; V.P. Gusynin, V.A. Miransky, S.G. Sh., I.A. Shovkovy, PRB 7, 1959 (6); cond-mat/6188 January 1, 7 S.G. Sharapov (McMaster University) The ac conductivity of monolayer graphene Kavli Institute 1 / 17

2 Outline 1 AC conductivity: introduction for experts Microwave conductivity 3 Optical probe of the Dirac quasiparticles Optical conductivity sum rule 5 New quantum Hall states, σ xy =, ±e /h for B > T 6 Summary S.G. Sharapov (McMaster University) The ac conductivity of monolayer graphene Kavli Institute / 17

3 AC probe of the Dirac quasiparticles: zero magnetic field case, B = In addition to intraband transitions, there are interband transitions. S.G. Sharapov (McMaster University) The ac conductivity of monolayer graphene Kavli Institute 3 / 17

4 AC probe of the Dirac quasiparticles: zero magnetic field case, B = Σ xx h e Μ K.1 K T 1K K T 5K K T 1K 5K 1 In addition to intraband transitions, there are interband transitions. 1 3 GHz The microwave conductivity σ xx (Ω, T) in units e /h vs frequency Ω in GHz. In addition to Drude peak (intraband) there is a constant background (interband). V.P. Gusynin, S.G. Sh., J.P. Carbotte, PRL 96, 568 (6). S.G. Sharapov (McMaster University) The ac conductivity of monolayer graphene Kavli Institute 3 / 17

5 Microwave conductivity: Drude term In weak scattering limit for energy dependent scattering rate Γ(ω) conductivity in microwave range is given by σ xx (Ω, T) = σ ( dω n ) F(ω) π ω Γ(ω) ω Ω + Γ (ω), with Ω photon frequency, n F (ω) = 1/[e (ω µ)/t + 1] and σ = e /(π ). Take µ = (Dirac point) and assume the weak impurity scattering limit in Born approximation: Γ(ω) = γ + α ω, (γ small), we get a cusp-like dependence σ xx (Ω, T) πσ α [ 1 π 8α Ω T ], γ < Ω T. W. Kim, F. Marsiglio, and J. P. Carbotte, PRB 7, 655(R) (). S.G. Sharapov (McMaster University) The ac conductivity of monolayer graphene Kavli Institute / 17

6 Microwave conductivity in HTSC This cusp behavior of the microwave conductivity σ(ω) is observed in a very pure YBCO 6.5 orthoii phase: P.J. Turner et al., PRL 9, 375 (3). In HTSC it is explained by W. Kim, F. Marsiglio, and J.P. Carbotte, PRB 7, 655(R) (). For graphene found that for µ = or V g = the same [ equation ] σ xx (Ω, T) πσ α 1 π Ω 8α T, holds and predict in Born limit the same cusp like behavior. S.G. Sharapov (McMaster University) The ac conductivity of monolayer graphene Kavli Institute 5 / 17

7 Microwave conductivity in HTSC This cusp behavior of the microwave conductivity σ(ω) is observed in a very pure YBCO 6.5 orthoii phase: P.J. Turner et al., PRL 9, 375 (3). In HTSC it is explained by W. Kim, F. Marsiglio, and J.P. Carbotte, PRB 7, 655(R) (). For graphene found that for µ = or V g = the same [ equation ] σ xx (Ω, T) πσ α 1 π Ω 8α T, holds and predict in Born limit the same cusp like behavior. Unitary limit [see e.g. N.M.R. Peres, F. Guinea, A.H. Castro Neto, PRB 73, 1511 (6)] is, of course, different, but HTSC shows that the cusp is not impossible... S.G. Sharapov (McMaster University) The ac conductivity of monolayer graphene Kavli Institute 5 / 17

8 Microwave conductivity in HTSC This cusp behavior of the microwave conductivity σ(ω) is observed in a very pure YBCO 6.5 orthoii phase: P.J. Turner et al., PRL 9, 375 (3). In HTSC it is explained by W. Kim, F. Marsiglio, and J.P. Carbotte, PRB 7, 655(R) (). For graphene found that for µ = or V g = the same [ equation ] σ xx (Ω, T) πσ α 1 π Ω 8α T, holds and predict in Born limit the same cusp like behavior. Unitary limit [see e.g. N.M.R. Peres, F. Guinea, A.H. Castro Neto, PRB 73, 1511 (6)] is, of course, different, but HTSC shows that the cusp is not impossible... What happens if we vary the gate voltage V g? S.G. Sharapov (McMaster University) The ac conductivity of monolayer graphene Kavli Institute 5 / 17

9 Tunable Drude peak Away from the Dirac point µ = µ sgnv g Vg is tunable by the gate voltage as well as the width of the Drude peak: Γ(µ) σ xx (Ω, T) = σ π µ, µ T. Ω + Γ (µ) A possibility to measure the dependence Γ(ω) by changing µ and extracting Γ(µ). S.G. Sharapov (McMaster University) The ac conductivity of monolayer graphene Kavli Institute 6 / 17

10 Optical probe of the Dirac quasiparticles: finite magnetic field case, B 3 Μ 1 K K ' b 1 3 The allowed transitions between LLs n =,.... The pair of cones at K and K are combined. Left E < µ < E 1 ; middle E 1 < µ < E ; right E < µ < E 3. S.G. Sharapov (McMaster University) The ac conductivity of monolayer graphene Kavli Institute 7 / 17

11 Optical probe of the Dirac quasiparticles: finite magnetic field case, B 3 Μ 5K B.1T Μ 1 Μ 5K B 1T a T 1K Μ 51K B 1T b K K ' 6 15K Μ 66K B 1T 1 3 ReΣ xx h e K The allowed transitions between LLs n =,.... The pair of cones at K and K are combined. Left E < µ < E 1 ; middle E 1 < µ < E ; right E < µ < E cm 1 Reσ xx (Ω) in units of e /h vs Ω. red µ = 5K and B = 1 T, black µ = 5K, blue µ = 51K, green µ = 66K all three for B = 1T. S.G. Sharapov (McMaster University) The ac conductivity of monolayer graphene Kavli Institute 7 / 17

12 Optical probe: filling sweep Dependence of Hall conductivity, σ xy (ν) = e /h(1 + [ν/]) at T = ([] denotes integer part) on the filling factor ν which varies linearly with the time. The first absorption line Ω = 9 cm 1 always appears with full intensity or is entirely missing, while all other lines disappear in two steps. This indicates the Dirac nature of quasiparticles in graphene. Dependence of diagonal ac conductivity, σ xx (Ω) at T on optical frequency Ω as the function of ν. S.G. Sharapov (McMaster University) The ac conductivity of monolayer graphene Kavli Institute 8 / 17

13 Optical conductivity sum rule Partial optical spectral weight up to Ω m Μ 5K B.1T T 1K a Μ 5K B 1T Μ 51K B 1T ReΣ xx h e 6 15K K Μ 66K B 1T cm 1 Area under conductivity gives total optical spectral weight to energy Ω m : W (Ω m ) = Ω m dω Reσ xx (Ω) S.G. Sharapov (McMaster University) The ac conductivity of monolayer graphene Kavli Institute 9 / 17

14 Optical conductivity sum rule Partial optical spectral weight up to Ω m ReΣ xx h e 6 T 1K 15K K a Μ 5K B.1T Μ 5K B 1T Μ 51K B 1T Μ 66K B 1T h e W m cm Μ 5K B T Μ 5K B.T Μ 5K B.T T.5K K cm 1 Area under conductivity gives total optical spectral weight to energy Ω m : W (Ω m ) = Ω m dω Reσ xx (Ω) K m cm 1 Plateaux with the steps corresponding to the various peaks in Re σ xx (Ω). S.G. Sharapov (McMaster University) The ac conductivity of monolayer graphene Kavli Institute 9 / 17

15 Full optical conductivity sum rule Due to two atoms per unit cell and inter- as well as intra-band optical transitions sum rule is different from sum over bands: dω Reσ xx (Ω) = e d k π BZ (π) [n F (ǫ(k)) n F ( ǫ(k))] [ ( ) ] ϕ(k) ǫ(k), here ϕ : t e ikδ ǫ(k)e iϕ(k) kα k α δ S.G. Sharapov (McMaster University) The ac conductivity of monolayer graphene Kavli Institute 1 / 17

16 Full optical conductivity sum rule Due to two atoms per unit cell and inter- as well as intra-band optical transitions sum rule is different from sum over bands: dω Reσ xx (Ω) = e d k π BZ (π) [n F (ǫ(k)) n F ( ǫ(k))] [ ( ) ] ϕ(k) ǫ(k), here ϕ : t e ikδ ǫ(k)e iϕ(k) kα k α δ or in the linearized approximation π dω Re σ xx (Ω) = α e t e a 9π v F ( µ 3 + π µ T ), µ T where t is the hopping parameter, three vectors δ connect nearest neighbors, α.61, v F is the Fermi velocity, a is the lattice constant. S.G. Sharapov (McMaster University) The ac conductivity of monolayer graphene Kavli Institute 1 / 17

17 Optical Hall angle sum rule Optical Hall angle: t H (Ω) tanθ H (Ω) = σ xy(ω) σ xx (Ω) The Hall response j x (Ω) = σ xy (Ω)E y (Ω), where, because an injected current is j y (Ω) = σ xx (Ω)E y (Ω) the response function t H (Ω): j x (Ω) = t H (Ω)j y (Ω) S.G. Sharapov (McMaster University) The ac conductivity of monolayer graphene Kavli Institute 11 / 17

Optical Hall angle sum rule Optical Hall angle: t H (Ω) tanθ H (Ω) = σ xy(ω) σ xx (Ω) obeys the sum rule The Hall response j x (Ω) = σ xy (Ω)E y (Ω), where, because an injected current is j y (Ω) = σ

18 Optical Hall angle sum rule Optical Hall angle: t H (Ω) tanθ H (Ω) = σ xy(ω) σ xx (Ω) obeys the sum rule The Hall response j x (Ω) = σ xy (Ω)E y (Ω), where, because an injected current is j y (Ω) = σ xx (Ω)E y (Ω) the response function t H (Ω): j x (Ω) = t H (Ω)j y (Ω) π dωre t H (Ω) = ω H, where in D electron gas the Hall frequency ω H coincides with a bare cyclotron frequency, ω H = ω c = eb mc. H.D. Drew and P. Coleman, PRL 78, 157 (97). S.G. Sharapov (McMaster University) The ac conductivity of monolayer graphene Kavli Institute 11 / 17

19 Partial Hall angle sum rule Ret H B.T Μ K T 5K Μ 3K 1K K Μ K Μ 5K cm 1 Optical Hall angle, Ret H (Ω) Notice that while Re σ xx (Ω) and Im σ xy (Ω) peaked near Ω = Landau level energy difference, Hall angle Re t H (Ω) has a Drude -like peak. S.G. Sharapov (McMaster University) The ac conductivity of monolayer graphene Kavli Institute 1 / 17

20 Partial Hall angle sum rule Ret H B.T T 5K Μ K m 3cm 1 Μ 3K m 7cm 1 1K K Μ K m 1cm 1 3 Μ 5K m 1cm 1 5 W B cm 1 15 Μ K cm 1 Optical Hall angle, Ret H (Ω) Notice that while Re σ xx (Ω) and Im σ xy (Ω) peaked near Ω = Landau level energy difference, Hall angle Re t H (Ω) has a Drude -like peak. 1 5 T 1K 15K K B T Area under Ret H (Ω) gives total weight to energy Ω m : W (Ω m ) = Ω m dω Re t H (Ω) Observe crossover from W B for low Ω m to W B for large Ω m S.G. Sharapov (McMaster University) The ac conductivity of monolayer graphene Kavli Institute 1 / 17

21 Hall angle sum rule for graphene dω Ret H (Ω) = ω H = 1 eb π α c = 9πα ta ρa eb v F c µ sgnµ t 3, where t is the hopping parameter, α.61, v F is the Fermi velocity, a is the lattice constant, ρ is the carrier imbalance. Full spectral weight linear in B and ρ V G. Interestingly, when the spectrum is gapped (as considered by many groups): E = v F (p 1 + p ) + ρ = 1 (µ )θ(µ )sgnµ π vf The gap can be extracted from the change in ω H obtained from magneto-optical measurements as done by Drew in HTSC. S.G. Sharapov (McMaster University) The ac conductivity of monolayer graphene Kavli Institute 13 / 17

22 New quantum Hall states, σ xy =, ±e /h xy (e /h) V g (V) - R (k ) V g (V) V g (V) Y. Zhang, et al., PRL 96, (6). xy (e /h) 9T, 11.5T, 17.5T, 5T, 3T, 37T, 37T, T, 5T. The observed filling factor sequence: ν = for B > 11Tesla, ν = ±1 for B > 17Tesla. Thus the four fold (sublattice-spin) degeneracy of n = Landau level is totally resolved for B > 17Tesla. The four fold degeneracy of n = 1 level is partially resolved into ν = ± which originates from spin splitting leaving two fold degeneracy. S.G. Sharapov (McMaster University) The ac conductivity of monolayer graphene Kavli Institute 1 / 17

23 Theoretical predictions E E L L Σ L ΜBB L ΜBB ΜBB ΜBB L ΜBB L ΜBB Σ a c E E L L Σ d L ΜBB L ΜBB ΜBB ΜBB ΜBB ΜBB V.P. Gusynin, et al., PRB 7, 1959 (6). 6 b L ΜBB L ΜBB Σ Illustration of the spectrum and the Hall conductivity in the n = and n = 1 Landau levels: (a) = and E Z = (no Zeeman term). (b) and E Z =. (c) = and E Z. (d) and E Z. Thickness of the lines represents the degeneracy,, and 1 of the energy states; L = v F eb /c. Both gap, and Zeeman term, E Z are necessary to explain σ xy =, ±e /h plateaux. S.G. Sharapov (McMaster University) The ac conductivity of monolayer graphene Kavli Institute 15 / 17

24 Manifestation of the gap in optics 1 ħeb v F c b Μ 1 Μ 1 Μ 1 ħeb v F c Transitions between LLs with an excitonic gap. green µ <, blue µ =, black µ >. For µ > two different lengths for transitions = peak splits S.G. Sharapov (McMaster University) The ac conductivity of monolayer graphene Kavli Institute 16 / 17

25 Manifestation of the gap in optics 1 ħeb v F c 1 1 b Μ Μ Μ ReΣ xx h e T 1K 15K B 1T Μ 15K K Μ 15K 1K Μ 15K 15K Μ 15K K 1 ħeb v F c Transitions between LLs with an excitonic gap. green µ <, blue µ =, black µ >. For µ > two different lengths for transitions = peak splits 6 8 cm 1 Re σ xx (Ω) in units of e /h vs Ω. red = K, black = 1K, blue = 15K, green = K. S.G. Sharapov (McMaster University) The ac conductivity of monolayer graphene Kavli Institute 16 / 17

26 Summary Optical measurements can be used to establish the presence of the Dirac quasiparticles, anomaly of the first peak Optical signatures of the Dirac quasiparticles already observed in epitaxial graphite M.L. Sadowski, et al., PRL 97, 665 (6), but not quite, because the sample is multilayer. Observation in graphene APS 7 abstract of P. Kim... Theoretical work on FIR in bilayer D.S.L. Abergel, V.I. Fal ko, Preprint cond-mat/ S.G. Sharapov (McMaster University) The ac conductivity of monolayer graphene Kavli Institute 17 / 17

arxiv: v1 [cond-mat.mes-hall] 25 May 2007

arxiv: v1 [cond-mat.mes-hall] 25 May 2007 arxiv:0705.3783v [cond-mat.mes-hall] 5 May 007 Magneto-optical conductivity in Graphene V P Gusynin, S G Sharapov and J P Carbotte Bogolyubov Institute for Theoretical Physics, 4-b Metrologicheskaya Street,