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GIANT Quantum Faraday effect in graphene systems I.V. Fialkovsky1,2, D. Vassilevitch2,3 1 Instituto de Física, Universidade de São Paulo, Brasil of Theoretical Physics, St. Petersburg State University, Russia 3 Universidade Federal ABC, São Paulo, Brasil 2 Department 2 Contents • A reminder on Dirac description of Graphene • One-loop effective theory: ▫ Polarization operator ▫ Matching conditions ▫ Light propagation • Conductivity and Faraday effect: ▫ Non-Drudeness of the conductivity ▫ Cyclotron resonance: fitting of existing experiment ▫ Predicting new results Pictures’ courtesy: • Novoselov & Geim’s group • Kuzmenko et al. Related talks • Earlier today, by V. Marachevskiy • Tomorrow by D. Vassilevich Faraday effect in Graphene, I. Fialkovksy @QFEXT’11 3 Tight-binding model Two sub-lattices, two atoms in a primitive cell. Strong covalent -bonds in the graphene plane, Weekly-intersected -orbitals perpendicular to the plane. Hamiltonian function of -electrons hopping to the nearest atom H t a † n , n , b с.с. n, i , t - hopping parameter, n - atom position, , - spin Faraday effect in Graphene, I. Fialkovksy @QFEXT’11 4 The Linearity of Spectrum In vicinity of Dirac (K-)points energy of quasy-particles has linear dispersion E(k ) vF k vF 106 m s Fermi velocity Thus the low-energy quasy-particles are quasy-relativistic Dirack fermions H 0 p ip 2p d y † x vF 2 0 DC (2 ) 0 px ip y 0 0 0 0 0 0 px ip y px ip y 0 0 0 Note 8-fold degeneracy: 2 sub-lattices * 2 valleys * 2 spins Wallace, 47, Semenoff, Apelquist,Redlich, 80-s Faraday effect in Graphene, I. Fialkovksy @QFEXT’11 5 Action functional x1 The action of the systems is straightforward (with EM field) S S EM S S d x i eAx3 0 m ... 3 x3 x2 SEM 14 d 4 x F2 l l , l 0,1,2 , 0 0 1,2 c 1, vF (300)1 vF , 1,2 2 0 1,2 2 1 With a second plate – Casimir effect, see the talk by Valeriy Marachevsky The model is well established and proved by several experiments: absorption of light, etc. Gusynin et al, 2004 Faraday effect in Graphene, I. Fialkovksy @QFEXT’11 6 QFT approach The generating functional of the theory is 2 i ( x ) i eA m AJ ( ) i F 3 Z [ J , ] D [ A ] e 4 The Casimir energy in this approach is given by QFT E i Ln Z [0] TS To calculate the generating functional we first ‘integrate out’ the fermions 2 i ( x ) S ( A) AJ i F 3 eff Z[J ] D A e 4 where the effective action is given by Seff A i Ln Det i eA m Faraday effect in Graphene, I. Fialkovksy @QFEXT’11 7 QFT approach II In the quadratic approximation the effective action is given by polarization operator SEff 1 2 3 3 lm d xd y A ( x ) ( y x ) Am ( y ) l Al Am |x3 0 x ( x 0 , x1 , x 2 ) The full line denotes the fermion propagtor of considered system 1 i m eAex Sˆ (m, , T , B,...) and the one-loop polarization operator is defined in a standard way 2 r 2 dq0 d q ˆ (q , q)%l Sˆ (q p , q p)%m ( p0 , p) ie tr S 0 0 0 (2 )3 lm Faraday effect in Graphene, I. Fialkovksy @QFEXT’11 8 Effective EM theory Thus in the lowest order approximation we had the following effective action S d 4 x 14 F2 12 x3 Al lm Am This gives the modification to the Maxwell equations F x3 A 0 the delta potential is equivalent to the following matching conditions A |z 0 A |z 0 , A A z z0 z z0 A |z0 This lets one to describe the plane waves propagation in the system: - reflection coefficients (for TE,TM modes, etc.) - rotation of polarization (if any) - absorption of light Faraday effect in Graphene, I. Fialkovksy @QFEXT’11 9 Normal Incidence For our purposes we need to consider the normal incidence only ( ) ( , p 0) when the tensorial structure of PO is quite simple 0 ( ) 0 gij evn i ij odd 0 In the language of condensed matter physics, polarization operator П corresponds to ac (and dc) conductivity ij ( ) ij ( ) dc lim ( i0) 0 Faraday effect in Graphene, I. Fialkovksy @QFEXT’11 10 Propagation of light x1 For linearly polarized waves one has A eit e x eik3 z rxx e x rxy e y e ik3 z , z 0 ik3 z t e t e e , z0 xx x xy y x3 x2 the (cumbersome) calculation of the reflection coefficient gives 2 (i evn 2 ) t xx 2 2 , 2 2 2 evn 4i evn (4 odd ) rxx t xx 1, 2 odd 2 t xy 2 2 2 evn 4i evn (4 2 odd ) 2 rxy t xy If both P-even and P-odd parts of polarization operator are real, the flux is conserved | txx |2 | txy |2 | rxx |2 | rxy |2 1 Faraday effect in Graphene, I. Fialkovksy @QFEXT’11 11 Faraday Effect The transmission coefficient is then given in the first order in T t xx t xy 2 2 Im evn 1 2 while the rotation of the angle of the polarization and elipticity of transmitted wave are defined as R Re imp 2 Im imp 2 IVF, D. Vassilevitch, 09 O( 2 ) O( 2 ) Earlier predictions on the issue: Mikhailov, Volkov 1985, O’Connell, Wallace, 1982 See also works by Morimoto, Hatsugai, Aoki Faraday effect in Graphene, I. Fialkovksy @QFEXT’11 12 Polarization Operator Just to remind you lm ( ) 2 dq d q 2 0 ˆ (q , q)%l Sˆ (q , q p)%m ie tr S 0 0 (2 )3 The full line denotes the fermion propagator of considered system i m eAex Sˆ (m, , T , B,...) 1 We will be interested in the case of zero temperature but with external magnetic field and non-vanishing chemical potential. Faraday effect in Graphene, I. Fialkovksy @QFEXT’11 13 Propagator, B>0 In presence of external magentic field perpendicular to the graphene plane one has for the propagator Sˆ (q0 , q; B) eq 2 Sn (q0 , q) (1) 2 2 ( q i sgn q ) M n 0 0 0 n /|eB| n 2q 2 2q 2 2q 2 1 Sn 2( q0 0 m) P Ln P Ln 1 4vF qLn 1 | eB | | eB | | eB | where Ln are the Laguerre polynomials, and P (1 i 1 2 B ) / 2, B sgn B, M n 2nvF2 | eB | m2 Initially the propagator in this case was calculated in the field of the heavy ions collisions. Chodos, Evering, Owen, 1990 Faraday effect in Graphene, I. Fialkovksy @QFEXT’11 14 Polarization Operator In the normal incidence case, the internal moment integration can be resolved analytically due to the orthogonality of the Laguerre polynomials 0 dx e x x a Lan ( x ) Lam ( x ) n a 1 nm n! which lead to the following expressions for P-odd and P-even components evn ( ) 2 N B vF2 | eB | dq0 I n,n 1 I n 1,n n 0 odd ( ) where I k ,n 2 B vF2 | eB | dq I n 0 0 n , n 1 I n 1,n q0 ( q0 ) m2 1 2 i (q0 i sgn(q0 ))2 M k2 (q0 i sgn(q0 )) 2 M n2 Faraday effect in Graphene, I. Fialkovksy @QFEXT’11 15 Relativistic Hall effect In the limit 0, m 0 we obtain the relativistic Hall conductivity xy lim odd 0 e2 N B 1 2n0 4 h 2 n0 2 2vF | eB | For graphene N=4, and thus Novoselov et al, 2005 Experimental: Novoselov et al, 2005; Kim et al, 2005 Theory: Zheng, Ando 2002; Gusynin, Sharapov, 2005, Peres et al., 2006 Faraday effect in Graphene, I. Fialkovksy @QFEXT’11 16 Non-Drudeness of conductivity The next step is in resolving the frequency integration which leads to odd ( ) i 8 2 B vF2 | eB | G n 0 n Gn dq I 0 n , n 1 I n 1,n g1 ( M n , ' M n 1 ) g1 ( M n , ' M n 1 ) ( ) 2i ( M n ' M n 1 ) , ' ( M n ' M n 1 ) Surprisingly, G has NO POLES in the complex plane!! Only a tricky combination of logarithmic divergences: g1 : log i M n1 i M n i M n1 i M n g2 : log i M n1 i M n i M n1 i M n However, its behavior on real axe only is almost undistinguishable from the Drude conductivity, see below! Faraday effect in Graphene, I. Fialkovksy @QFEXT’11 17 Non-Drudeness of conductivity A similar results holds for the P-even part: NO POLES evn ( ) 2 N B vF2 | eB | H n n 0 Hn dq I 0 n , n 1 I n 1,n Unlike P-odd part the P-even is neither odd, nor even as a function of chemical potential for non vanishing mass gap. Actually, polarization operator in 2+1 is power-counting divergent. It reveals itself in the even part Hn ; n 1 3/2 O ( n ) 2 4vF | eB | n However, at our level of consideration it does not effect the results since T 1 Im evn 2 Faraday effect in Graphene, I. Fialkovksy @QFEXT’11 18 Small chemical potential Small chemical potentials corresponds to suspended graphene layers. In this limit we have Gn ; 0 M 2 2 M n21 M n2 2 3i 2 2 2 n 1 2 M 2 n i 2 M 2 n 1 i 2 M n2 the maximum/minimum of rotation angle is of the order of ; 2 M 1 2 and are achieved, correspondingly, at ; M1 In idealistic case, the chemical potential as well as Г are identically zero. However, in a realistic situation there is an interplay between them due to unavoidable presence of impurities. Faraday effect in Graphene, I. Fialkovksy @QFEXT’11 19 Suspended graphene Thus, possible experimental curves should have the following form , rad T , meV , meV Just 1Å of graphene at B=7T rotates polarization of light by about 4.5 degrees and absorbs 25%! Faraday effect in Graphene, I. Fialkovksy @QFEXT’11 20 Epitaxial graphene Large Fermi energy shifts are usually invoked by the interaction with a substrate. For big chemical potential one can show 2 B vF2 | eB | n0 Re G n n0 n T 1 8 vF2 | eB | n0 Im H n n0 n i.e. in cyclotron resonance the main contribution is coming from around 2 n0 2 2 v | eB | F For numerical investigation it is sufficient consider : 0.3 n0 M 12 This holds already for : 2 No nicer formulas can be obtained. Faraday effect in Graphene, I. Fialkovksy @QFEXT’11 21 Faraday effect in Graphene, I. Fialkovksy @QFEXT’11 22 Faraday effect in Graphene, I. Fialkovksy @QFEXT’11 23 Data fitting Aims: - Yet again to check the Dirac model - Reveal possible field dependence of the parameters (mass, scattering rate, chemical potential) The fit was performed around the following initial values | | 340 meV 5 meV 0 The Diraс model (nicely) fits the experimental curves: - mass gap doesn’t influence the result (in a reasonable interval) - Gamma’s dependence on the magnetic filed is about +/- 10-15%. So, it cannot be resolved at this stage. - The fitted value of chemical potential is essentially lower then the Fermi energy shift measured at zero field, and grows with magnetic field Faraday effect in Graphene, I. Fialkovksy @QFEXT’11 24 Data fitting Faraday effect in Graphene, I. Fialkovksy @QFEXT’11 25 Data fitting Faraday effect in Graphene, I. Fialkovksy @QFEXT’11 26 Chemical potential Chemical potential (as a parameter entering the Dirac model) grows with magnetic field: while experimentally measured Fermi energy shift at zero magnetic field | | 340 meV Self-consistent perturbation theory seems to be a possible solution for the apparent discrepancy. Faraday effect in Graphene, I. Fialkovksy @QFEXT’11 27 vs Drude As promised we can compare our results with the Drude ones xx 1/ i c2 ( i / ) 2 2D xy c c2 ( i / ) 2 2D Thus, the predictions for the real frequencies are rather indistinguishable, while analytical properties in the whole complex plane are completely different! Faraday effect in Graphene, I. Fialkovksy @QFEXT’11 28 Conclusions After reminding of the Dirac description of graphene we • revealed the non-Drudeness of the conductivity • predicted a new regime to observe ‘Giant Faraday Rotation in Graphene’ • showed that model nicely describes the experimental curves in the cyclotron resonance regime Attn.: tomorrow at 18.15h “Quantum Field Theory in Graphene” by D. Vassilevich Fialkovsky I.V., Vassilevitch D.V. Parity-odd effects and polarization rotation in graphene J. Phys. A: Math. Theor. 42 (2009) 442001, arXiv: 0902.2570 [hep-th] Bordag M., Fialkovsky I. V., Gitman D. M., Vassilevich D. V., Polarization rotation and Casimir effect in suspended graphene films, arXiv:1003.3380 [hep-th] Faraday effect in Graphene, I. Fialkovksy @QFEXT’11 29 Faraday in Graphene, IVF, QFEXT'11 Thank you! 30 QFT approach II No caso mais simples i S S (m) p m o laço dos férmions foi calculado muitos anos atrás e muitas vezes j l n % % p p mn m jl jkl %k l ( p ) 2 j par ( p ) g 2 i imp ( p ) p % vF p mj diag(1, vF , vF ), % p j l j p l onde as funções tem seguinte forma explicita % ( % % 2m ) / 2 p % par ( p ) 2mp p 2 4m 2 ) arctanh( p % 2m) p % 1 imp ( p ) 2m arctanh( p Semenoff, 1984, Redlich, 1984, Appelquist, 1986, Dunne, 1996-1999, etc. Faraday effect in Graphene, I. Fialkovksy @QFEXT’11 31 QFT approach II % % %l pj% pl pj% pl % p jul u j p u jul 2 n jl % B l ( p ) 2 g 2 A 2 p 2 % vF p p pu % pu % % mn m j Onde a velocidade da media pode ser escolhida como u (1,0,0) As funções escalares A,B se expressam através dos alguns dois componentes p2 A 2 00 tr p 1 p2 % p2 B 2 2 00 tr vF p p 2 Faraday effect in Graphene, I. Fialkovksy @QFEXT’11