Detection of topological phase transitions through entropy - PowerPoint PPT Presentation

Detection of topological phase transitions through entropy measurements Valeriy Gusynin Bogolyubov Institute for Theoretical Physics, Kyiv, Ukraine of the National Academy of Sciences of Ukraine Conference on Modern Concepts and New Materials for Thermoelectricity ICTP, Italy, March 14, 2019 In collaboration with: Y. Galperin, D. Grassano, A. Kavokin, A. Varlamov, O. Pulci, S. Sharapov, V. Shubnyi

Table of contents 1. Entropy per particle and its measurements 2. Quasi-2DEG: Quantization of entropy 3. Entropy spikes in gapped Dirac materials 4. Detection of topological transition 5. Entropy in transition-metal dichalcogenides

Entropy per particle Internal energy for a varying particle number: d E ( S , V , N ) = TdS − PdV + µ dN , where the entropy, dS = δ Q T with δ Q being the heat. Entropy is important not only for describing thermodynamical experiments, but also in interpreting heat transport experiments, e.g. Seebeck and Nernst - Ettingshausen effects are characterized by the entropy per particle: � ∂ S � � ∂ S � = 1 s = , ∂ N V ∂ n T T set V = 1 and use instead of carrier number N the carrier density n .

Entropy per particle via Maxwell relation System in the thermostat described by the Gibbs free energy: dG ( T , P , N ) = d ( E − TS + PV ) = − SdT + VdP + µ dN , so that � ∂ G � � ∂ G � = − S , = µ ∂ T ∂ n p , n T , p Maxwell relation, � ∂µ � ∂ S � � s = = − ∂ n ∂ T T n makes entropy per particle measurable quantity. A.Y. Kuntsevich, Y.V. Tupikov, V.M. Pudalov, I.S. Burmistrov, “Strongly correlated two-dimensional plasma explored from entropy measurements”, Nature Commun. 6 , 7298 (15). Thermodynamic method to measure the entropy per electron in gated structures. Technique appears to be three orders of magnitude superior in sensitivity to a.c. calorimetry, allowing entropy measurements with only 10 8 electrons.

Measuring entropy per particle Modulation of the sample temperature T ( t ) = T 0 + ∆ T cos( ω t ) changes the chemical potential and causes recharging of the gated structure. The derivative ∂µ/∂ T is directly determined in the experiment from the measured recharging current: Example of the measurements in i ( T ) = ∂µ Si-MOS structures. Magnetic field ∂ T ∆ T ω C sin( ω t ). Here, C stands for the capacitance applied ⊥ to the structure B = 9 T between the gate electrode and 2D electron layer, ∆ T ∼ 0 . 1 K , ω/ (2 π ) ∼ 0 . 5 Hz . ∂ S /∂ n versus electron density (10 11 cm − 2 ).

Quasi-two-dimensional electron gas Electron subbands in the quasi-2D � 2 k 2 electron gas: ε j ( k ) = E j + 2 m ∗ . � In the absence of scattering, the density of electronic states in a non-interacting 2DEG has a staircase-like shape ∞ D ( ε ) = m ∗ � θ ( ε − E j ) . π � 2 j =1 Electronic topological transition or Lifshitz transition in quasi-2DEG, δ D ( ε ) = C θ ( E − E c ). The presence of impurities results in the level broadening: γ θ ′ ( ε ) = δ ( ε ) → δ γ ( ε ) ≡ π ( ε 2 + γ 2 ) , where � /γ is a finite life-time. The steps of the DOS are smeared � � θ γ ( ε ) = 1 2 + 1 ε π arctan . γ

Quantization of entropy per particle Formal matters : � ∂µ � ∂ n + ∞ � − 1 � � � ∂ n D ( ε ) � s = − = , n ( µ, T ) = + 1 d ε. � ε − µ ∂ T ∂ T ∂µ � exp n µ T T −∞ A. Varlamov, A. Kavokin, and Y. Galperin, PRB 93 , 155404 (16). The value of the entropy per particle in the N -th maximum depends only on the size-quantization quantum number N : ln 2 s ( T → 0) | µ = E n = k B = 1 . N − 1 / 2 In the absence of scattering this result is independent of the shape of the transversal potential that confines 2DEG and of the material parameters including the electron effective mass and dielectric constant.

Exact formulas for 2DEG at finite temperature and scattering rate � ∂ n 2 m ∗ � � γ + i ( µ − E j ) � 1 2 + γ + i ( µ − E j ) � � = Re Ψ π � 2 ∂ T 2 T 2 π T µ j =1 � � 1 �� � γ 2 + γ + i ( µ − E j ) + π − 2 T − π ln Γ 2 ln(2 π ) , 2 π T m ∗ � ∂ n � � 1 + 2 � 1 2 + γ + i ( µ − E j ) �� � = π Im Ψ , 2 π � 2 ∂µ 2 π T T j =1 where Ψ( z ) is the digamma function. We have taken into account that µ ≫ T . The general expression for entropy per particle in the quasi-2DEG can be calculated from the above expressions.

Role of temperature and disorder Dependence of the entropy per particle, s , on ( µ − E N ) / 2 T ≡ δ N / 2 T for (a) N = 2 , 3; γ → 0; (b) N = 2; γ/ 2 T = 0 , 0 . 2. Asymptotic in the vicinity of the peak: � � − | δ N |  exp | δ N | T � , δ N ≪ − T ,   � − | δ N | T N − 1+exp  T s ( µ = E N + δ N ) = ln 2 N − 1 / 2 , 0 < δ N ≪ T ,   δ N  TN exp( − δ N / T ) , δ N ≫ T . The peak is suppressed by the elastic scattering of electrons: s | µ = E N = ln 2 − ( γ/π T ) . N − 1 / 2

Low-buckled Dirac materials The same hexagonal lattice as in graphene, but due to buckling there is also a strong intrinsic spin-orbit interaction H SO = i ∆ SO � c † √ i σ ( ν ν ij · σ ν σ σ ) σσ ′ c j σ ′ , 3 3 � � i , j � � σσ ′ ν z ij = ± 1, ∆ SO ≈ 4 . 2 meV in silicene, Silicene: vertical distance between ∆ SO ≈ 11 . 8 meV in germanene. sublattices 2 d ≈ 0 . 46˚ A. Lattice constant a = 3 . 87˚ A. So far grown on metallic substrates Ag, Au, Pt, Al, as well as less interactive substrates such as MoS 2 (gap ∼ 1 . 23 − 1 . 8eV ). 2D sheets of Ge, Sn and P atoms (germanene, Perpendicular to the plane electric field stanene and phosphorene). No E z opens the tunable gap ∆ el = E z d . transport measurements yet. Interplay of two gaps: ∆ SO and ∆ el .

Low-energy Hamiltonian of silicene H η = σ 0 ⊗ [ � v F ( η k x τ 1 + k y τ 2 ) + ∆ el τ 3 ] − η ∆ SO σ 3 ⊗ τ 3 , τ σ τ τ and σ σ – sublattice and spin; k is measured from the K η points. There is a spin σ = ± , and valley η = ± dependent gap (or mass ∆ ησ / v 2 F ) � � 2 v 2 k 2 + ∆ 2 ∆ ησ = ∆ el − ησ ∆ SO , ε ησ ( k ) = ± ησ . C. Liu, W. Feng, and Y. Yao, PRL 107 , 076802 (11); N. Drummond, V. Z´ olyomi, and V. Fal’ko, PRB 85 , 075423 (12); M. Ezawa, New J. Phys. 14 , 033003 (12), J. Phys. Soc. Jpn. 81 , 064705 (12). Time-reversal symmetry (TRS) is unbroken. The band structure: bulk and edge states in nanoribbons for varying ∆ el .

DOS of the Dirac materials M ε 2 − ∆ 2 � � � Generic form of the DOS: D ( ε ) = f ( ε ) θ . i i =1 M = 1: gapped graphene M = 2: silicene, germanene, etc. F k 2 + ∆ 2 � � 2 v 2 � F k 2 + ∆ 2 ε ( k ) = ± � 2 v 2 ε ησ ( k ) = ± ησ and and f ( ε ) = 2 | ε | / ( π � 2 v 2 F ) f ( ε ) = | ε | / ( π � 2 v 2 F ). (spin-valley degeneracy is i = 1 corresponds to η = σ = ± with included). ∆ 1 = | ∆ SO − ∆ z | and i = 2 corresponds to η = − σ = ± with ∆ 2 = | ∆ z + ∆ SO | . Since D ( ε ) = D ( − ε ), instead of the total density of electrons one operates with the difference between the densities of electrons and holes ( γ = 0): � ∞ n ( T , µ, ∆ 1 , ∆ 2 , . . . , ∆ M ) = 1 � tanh ε + µ − tanh ε − µ � d ε D ( ε ) . 4 2 T 2 T −∞ Clearly, n ( T , µ ) = n ( T , − µ ), so that n ( T , µ = 0) = 0. V. Tsaran, A. Kavokin, S. Sharapov, A. Varlamov, and V.G., Sci. Rep. 7 , 10271 (2017).

Quantization of entropy in Dirac materials We need ∂ n /∂ T and ∂ n /∂µ . For ∆ i < | µ | < ∆ i +1 and T → 0: = D ′ ( | µ | ) π 2 T ∂ n ( T , µ ) sign ( µ ) , ∆ i > 0 . ∂ T 3 and at the discontinuity points µ = ± ∆ N at T → 0, ∞ � ∂ n ( T , µ ) � x dx � = ± [ D (∆ N + 0) − D (∆ N − 0)] cosh 2 x = ± f (∆ N )ln 2 . � ∂ T � µ = ± ∆ N 0 If µ = ± ∆ N with N < M and T → 0, one obtains M � ∂ n ( T , µ ) � � θ (∆ 2 N − ∆ 2 = f (∆ N ) i ) = f (∆ N )( N − 1 / 2) , � ∂µ � µ = ± ∆ N i =1 The entropy per particle in Dirac materials is ln 2 s ( T → 0 , µ = ± ∆ N ) = ± N − 1 / 2 , N = 1 , 2 , . . . M .

Gapped graphene and silicene: analytics Carrier imbalance in gapped graphene: � �  µ − ∆  1 + exp 2 T 2  ∆ T � − e − µ +∆ � � µ − ∆ � n ( T , µ, ∆) = T ln � + Li 2 − Li 2 − e T T  π � 2 v 2 � − µ +∆ 1 + exp F T where Li 2 ( x ) is the dilogarithm function: Li s ( x ) = � ∞ z k k s , Li ν (1) = ζ ( ν ). k =1 E. Gorbar, V.G., V. Miransky, and I. Shovkovy, PRBB 66, 045108 (2002). The Fermi-Dirac integral � ∞ ǫ α e ǫ − µ + 1 = − α Γ( α )Li α +1 ( − e µ ) . F α ( µ ) = d ǫ 0 One can consider silicene as a superposition of two gapped graphene layers characterized by different gaps: n ( T , µ, ∆ 1 , ∆ 2 ) = 1 / 2 [ n ( T , µ, ∆ 1 ) + n ( T , µ, ∆ 2 )] .