Thermoelectric and thermospin properties of gapped Dirac materials - PowerPoint PPT Presentation

Thermoelectric and thermospin properties of gapped Dirac materials Sergei G. Sharapov ICTP, Trieste, Italy, 11-15 March, 2019 Bogolyubov Institute for Theoretical Physics of the National Academy of Sciences of Ukraine Conference on Modern Concepts and New Materials for Thermoelectricity In collaboration with: V.P. Gusynin, A.A. Varlamov

Outline 1. Thermoelectric power in gapped graphene 2. Nernst - Ettingshausen effect in graphene 3. Spin Nernst (SN) effect in silicene 4. Modified Kubo formula 1

Thermoelectric power in gapped graphene

Thermoelectric power in gapped monolayer graphene V = ( S B − S A )( T 2 − T 1 ) S = − ∆ V E ∆ T = ∇ T Nowdays we call this Seebeck effect. See a review C. Goupil, H. Ouerdane, K. Zabrocki, W. Seifert, N.F. Hinsche, and E. M¨ uller, “Thermodynamics and Thermoelectricity”: Aepinus (1762), Galvani, Volta (1786) ⇒ Ritter (1801), Schweigger (1810), Seebeck (1821) ⇒ Peltier (1834), Thomson (1851) 2

Large thermoelectric effect in graphene Wei et al. , PRL 102 , 166808 (09). Wang, Shi, PRB 83 , 113403 (11). 3

Heat and electric transport equations Electric field E and temperature gradient ∇ T result in electric and heat currents.  � It is easier to control j σ E − ˆ  = ˆ β ∇ T , j  ˆ E = ˆ ρ j + S ∇ T , rather than E , express γ E + ˆ q = ˆ ζ ∇ T ,  ˆ  via j . q = Π j − ˆ κ ∇ T , γ = ˆ Onsager relation: ˆ β T Only the diagonal transport is considered in the first part! Seebeck coefficient: Peltier coefficient: S ≡ S xx = − β σ ≡ − β xx Π = γ σ = ST σ xx Approximate Mott’s formula: β = π 2 k 2 S = − π 2 k 2 ∂σ ∂ ln σ B T B T = ⇒ 3 ∂µ 3 ∂µ e e Notice that k B / e ≈ 86 µ V / K close to observed in graphene which is much larger than in metals. 4

Odd- and evenness of transport coefficients � ∞ σ = e 2 F ν ( µ + ǫ ) τ ( µ + ǫ ) ≈ e 2 d ǫ [ − n ′ F ( ǫ )] v 2 3 [ v 2 F ν ( µ ) τ ( µ )] 3 −∞ � ∞ e d ǫ ǫ [ − n ′ F ( ǫ )] v 2 β = F ν ( µ + ǫ ) τ ( µ + ǫ ) 3 T −∞ 1.0 If the product v 2 F ν ( µ + ǫ ) τ ( µ + ǫ ) is a smooth 0.8 � n F ’ � Ε � function of ǫ , one can expand it: 0.6 0.4 v 2 F ν ( µ + ǫ ) τ ( µ + ǫ ) ≈ v 2 F ν ( µ ) τ ( µ ) + ǫ d d µ [ v 2 F ν ( µ ) τ ( µ )] ’ � Ε � Ε � n F 0.2 1st term = 0 due to oddness, and contributes 2nd. 0.0 � 0.2 � 0.4 � 4 � 2 0 2 4 Ε Arrive at Mott’s formula and in normal metals S = − π 2 k B k B T ∼ 10 − 2 T [ K ] µ V 3 e µ K much smaller than observed in graphene. 5

Band structure of graphene Low-energy excitations at two inequivalent K + , K − points have a linear dispersion E p = ± � v F | p | − µ with v F ≈ 10 6 m/s and µ being the chemical potential. The excitations are described by a pair of two-component spinors � � ψ T K ,σ = ψ KA σ , ψ KB σ , which are composed of the Bloch states residing ( A , B ) sublattices with momenta near the two inequivalent points ( K + , K − ) of the Brillouin zone. The low-energy Hamiltonian � � � � d 2 p 0 � v F ( p x − ip y ) ( 2 π ) 2 ψ † H K + = ψ K + σ , K + σ � v F ( p x + ip y ) 0 σ = ± 1 where the momentum p = ( p x , p y ) is given in a local coordinate system. 6 Semenoff, PRL 53 , 2449 (1984)

Making sublattices inequivalent and graphene gapped � � � � d 2 p ∆ � v F ( p x − ip y ) ( 2 π ) 2 ψ † H K + = ψ K + σ . K + σ � v F ( p x + ip y ) − ∆ σ = ± 1 The presence of ∆ � = 0 breaks parity P : [ x → − x , y → − y , A ⇆ B ] and � F p 2 + ∆ 2 with the mass ∆ . � 2 v 2 makes the spectrum E ( p ) = ± Graphene on top of hexagonal boron nitride (h-BN) – lattice is 1.7% percent larger. Mass (sublattice asymmetry gap) can be induced by interaction with substrate: 2 ∆ ∼ 350 K. 7

Making sublattices inequivalent and graphene gapped � � � � d 2 p ∆ � v F ( p x − ip y ) ( 2 π ) 2 ψ † H K + = ψ K + σ . K + σ � v F ( p x + ip y ) − ∆ σ = ± 1 The presence of ∆ � = 0 breaks parity P : [ x → − x , y → − y , A ⇆ B ] and � F p 2 + ∆ 2 with the mass ∆ . � 2 v 2 makes the spectrum E ( p ) = ± Graphene on top of hexagonal boron nitride (h-BN) – lattice is 1.7% percent larger. Mass (sublattice asymmetry gap) can be induced by interaction with substrate: 2 ∆ ∼ 350 K. How the gap affects thermopower? 7

Quasiparticle scattering near ETT (a) Scattering processes which do not involve the small valley. (b) Scattering processes where electron gets to the small void, but then gets back to the continuous part of the Fermi surface. In vicinity of the critical point µ = µ c , when the Fermi surface connectivity changes, the quasiparticle relaxation rate τ − 1 ( ε ) ≡ Γ( ε ) also acquires the contribution strongly Possible types of electron scattering depending on energy, that generates kinks for a double valley Fermi surface. in conductivity and peaks in thermopower. A.A. Varlamov, V.S. Egorov, and A.V. Pantsulaya, Adv. in Phys. 38 , 469 (1989). 8

Scattering in gapped graphene Zero mass, ∆ = 0 Gapped, ∆ � = 0 Self-consistent equation � a � � b � for self-energy: B � 0 B � 0 E � p � Μ K � E � p � �� � v F p K � Μ K � E � p � �� �������������������������������� � 2 � � 2 � v F2 � p 2 K � E � p � p Use relatively long-range p (a) Linear dispersion, potential � (b) A possible V ( q ) , i.e. ignore µ = 0 as in compensated modification of the scattering between K ± , graphene. but assume � spectrum by the finite gap V ( q ) to be ∆ . µ is shifted from zero momentum independent by the gate voltage. for the intra-valley scattering. Control parameter: | ∆ | < ?? > | µ | 9

Quasiparticle scattering in graphene Σ( p , ε n ) = � 3 The self-energy � i = 0 σ i ( p , ε n ) � τ i Since σ 1 , 2 = 0, arrive at the system � � ε + µ − σ R 0 ( ε ) � � ξ d ξ � W ∆ + σ R 3 ( ε ) σ R 0 ( ε ) 4 � = 3 ( ε )] 2 , � � 2 − ξ 2 − [∆ + σ R σ R πτ 0 | µ | 3 ( ε ) ε + µ − σ R 0 ( ε ) 0 A new feature , in addition to the usually considered Eq. for σ 0 we also consider Eq. for σ 3 in the gap channel. Then approximately include both channels together: 1 ∆ τ ( ε ) ≡ Γ( ε ) = − Im σ R ε + µ Im σ R 0 ( ε ) − 3 ( ε ) � | ε + µ | � ∆ 2 � ( ε + µ ) 2 − ∆ 2 � =Γ 0 + θ . | µ | | ε + µ || µ | � ( ε + µ ) 2 − ∆ 2 � The relaxation rate acquires the θ contribution. 10

Transport coefficients in graphene Using Kubo formula: � � � � � ∞ = e 2 σ d ε A ( ε, Γ( ε ) , ∆) 1 , 2 T cosh 2 ε β � ε/ ( eT ) −∞ 2 T where the function � 1 + ( µ + ε ) 2 − ∆ 2 + Γ 2 ( ε ) 1 A ( ε, Γ( ε ) , ∆) = 2 π 2 2 | µ + ε | Γ( ε ) � π �� 2 − arctan ∆ 2 + Γ 2 ( ε ) − ( µ + ε ) 2 × . 2 | µ + ε | Γ( ε ) We use regularized scattering rate: Γ full ( ε ) = Γ( ε ) + γ 0 . 11

Results Conductivity σ ( µ ) Tm.-el. coefficient β ( µ ) Thermopower S ( µ ) 4 � a � � 1 �� con, �� 0 � 1 0.10 T � 1 K 1.0 � 0.2 �� con, �� 50 K 3 � 1 0.05 � 0 � 20 K � � Ω � , �� 50 K 0.5 � 1 Σ � Σ 0 � 0.2 2 Β � Β 0 S � k B � e � 0.00 0.0 1 � 0.05 � 0.5 � 0.10 � 1.0 0 � 100 � 50 0 50 100 � 100 � 50 0 50 100 � 100 � 50 0 50 100 Μ � K � Μ � K � Μ � K � In units σ 0 = 2 e 2 β 0 = k B e / � , T = 1 K π 2 � S 0 = k B / e , T = 5 K —– ∆ = 0, Γ( ε ) = const - reference case: restore normal metal case, S = − ( π 2 / 3 e ) T /µ in the limit | µ | ≫ T , Γ 0 . —– ∆ = 50 K, Γ( ε ) = const: E. Gorbar et al. , PRB 66 , 045108 (02). —– ∆ = 50 K, Γ( ε ) - S.G. Sh. and A.A. Varlamov, Phys. Rev. B 86 , 035430 (2012). Thin lines – from Mott formula. 12

Conclusions: • Opening a gap results in appearance of a fingerprint bump of the Seebeck signal when the chemical potential approaches the gap edge. • Magnitude of the bump can be up 10 times higher than already large value of S ∼ 50 µ V / K at room temperatures observed in graphene. • Effect is related to a new channel of quasi-particle scattering from impurities with the relaxation time strongly dependent on the energy. • One can exploit the predicted giant peak of the Seebeck signal as a signature of the gap opening in monolayer graphene. • Similar phenomenon already observed in bilayer graphene, C.-R. Wang, et al. , PRL 107 , 186602 (11). 13

Nernst - Ettingshausen effect in graphene