Quantum Fluid Models for Electron Transport in Graphene. Nicola Zamponi

Transcription

1 Quantum Fluid Models for Electron Transport in Graphene Mathématique pour le Graphéne Grenoble, October 2013 Nicola Zamponi Institut für Analysis und Scientific Computing, TU Wien

2 Graphene: the new frontier of nanoelectronics Graphene is a new semiconductor material created in the first decade of this century by Geim and Novoselov. It has remarkable electronic properties which make it a candidate for the construction of new electronic devices with strongly increased performances with respect to the usual silicon semiconductors.

3 Graphene: the new frontier of nanoelectronics Physically speaking, graphene is a single layer of carbon atoms disposed as an honeycomb lattice, that is, a single sheet of graphite.

4 Graphene: the new frontier of nanoelectronics Features of charge carriers in graphene: Graphene is a zero-gap semiconductor, that is, the valence band of the energy spectrum intersects the conduction band in some isolated points, named Dirac points; around such points the energy of electrons is approximately proportional to the modulus of momentum: E = ±v F p. Relativistic massless quasiparticles!

5 Graphene: the new frontier of nanoelectronics Graphene cristal lattice is split into two nonequivalent sublattices. Charge carriers have a discrete degree of freedom, called pseudospin. Different from electron spin!

6 Graphene: the new frontier of nanoelectronics Hamiltonian (low-energy approximation, zero potential): ( ) H 0 = Op [v F (p 1 σ 1 + p 2 σ 2 )] = i v F σ 1 + σ 2 ; x 1 x 2 v F 10 6 m/s is the Fermi speed; denotes the reduced Planck constant; σ 1, σ 2 are Pauli matrices: ( ) ( ) σ 0 =, σ =, σ = ( ) ( ) 0 i 1 0, σ i 0 3 = ; 0 1 Op is the Weyl quantization: given a symbol γ = γ(x, p), ( ) x + y (Op (γ)ψ)(x) = (2π ) R 2 γ, p ψ(y)e i(x y) p/ dydp, 2 R 2 2 for all ψ L 2 (R 2, C).

7 A kinetic model for graphene Goal: Derive and study several fluid models for quantum transport of electrons in graphene. Fluid models derived from kinetic models Wigner equations. w = w(x, p, t) system Wigner function. Spinorial system w is not a scalar function! w(x, p, t) is, for all (x, p, t), a complex hermitian 2 2 matrix. Serious computational difficulties! However, we can write it in the Pauli basis: w = 3 s=0 w sσ s, with w s (x, p, t) suitable real scalar functions.

8 Collisionless Wigner equations for graphene Wigner equations for quantum transport in graphene, derivated from the Von Neumann equation with the one-particle Hamiltonian H 0 + V : 1 t w 0 + v F w + Θ (V )w 0 =0, t w + v F [ w w p ] + Θ (V ) w =0, (W0) where w (w 1, w 2, w 3 ) and: (Θ (V )w)(x, p) = i (2π) 2 R 2 R 2 [ V (x + 2 ) ξ V (x 2 )] ξ w(x, p )e i(p p ) ξ dξdp. 1 For the derivation of (W0) see: N. Zamponi and L. Barletti, Quantum electronic transport in graphene: A kinetic and fluid-dynamic approach; N. Zamponi, Some fluid-dynamic models for quantum electron transport in graphene via entropy minimization.

9 Non-statistical closure: the pure state case A first fluid-dynamic model can be derived from the Wigner equations under the hypothesis of pure state: ρ ij (x, y) = ψ i (x)ψ j (y) (i, j = 1, 2), where ρ is the system density matrix, while ψ is the wavefunction.

10 Non-statistical closure: the pure state case Let us consider the following moments, for k = 1, 2, s = 1, 2, 3: n 0 = w 0 dp charge density, n s = w s dp pseudospin density, J k = p k w 0 dp pseudomomentum current, t sk = p k w s dp pseudospin currents. By taking moments of the Wigner equations it is easy to find the following system of not-closed fluid equations: t n 0 +v F j n j = 0, t n s +v F s n 0 + 2v F η sijt ij = 0 (s = 1, 2, 3), t J k +v F s t sk + n 0 k V = 0 (k = 1, 2).

11 Non-statistical closure: the pure state case From the pure state hypothesis it follows: n 0 t sk =n s J k 2 η sαβn α k n β (k = 1, 2, s = 1, 2, 3), n 0 = n = n1 2 + n2 2 + n2 3. So we found the following pure-state fluid model ( Madelung equations for a quantum particle described by the Hamiltonian H 0 ): t n 0 +v F n = 0, t n+v F n0 + 2v F t J+vF ( J n n 0 The first equation in redundant! n J + v F ( n 0 n n n ) n = 0, ) 0 v ( F 1 2 s η sij n i nj n 0 ) + n 0 V = 0.

12 Beyond the pure state assumption Goal: Derive several models not based on the pure state hypothesis. Statistical closure: close the fluid equations by means of an equilibrium distribution obtained as a minimizer of a suitable quantum entropy functional. Problem: Which statistics should we choose? Since the energy spectrum of H 0 is not bounded from below, Fermi-Dirac statistics would be more adequate to describe quantum electron transport in this material, rather than Maxwell-Boltzmann s one; neverthless, we used in our work the Maxwell-Boltzmann statistics, for the sake of simplicity and to obtain explicit models, at the price of a modification of the hamiltonian operator H 0 : [ ] H = Op v F (p 1 σ 1 + p 2 σ 2 )+ p 2 2m σ 2 0 = H 0 σ 0 2m, with m > 0 parameter (with the dimensions of a mass).

13 Collisional Wigner equations Wigner equations for quantum transport in graphene, derivated from the Von Neumann equation with the one-particle Hamiltonian H + V, with a collisional term of BGK type: [ ] p t w 0 + m w 0 + v F w + Θ (V )w 0 = g 0 w 0, τ [ ] c p t w + m w + v F [ w0 + 2 ] (W) w p g w + Θ (V ) w =. τ c Here: g is the thermal equilibrium distribution; τ c is the relaxation time.

14 Different scalings of the Wigner equations Isothermal system: thermal equilibrium with phonon bath at constant temperature T. Two different scalings of the collisional Wigner equations (W): a diffusive scaling; an hydrodynamic scaling.

15 Diffusive scaling of the Wigner equations Diffusive scaling: with ˆx, ˆt, ˆp, ˆV satisfying: x ˆxx, t ˆtt, p ˆpp, V ˆV V, 2v F ˆp = ˆV ˆxˆp, 2ˆpv F τ c = 2ˆpv Fˆt, ˆp = mk B T ; we define the semiclassical parameter ɛ, the diffusive parameter τ and the scaled Fermi speed c as: ɛ = ˆxˆp, τ = 2ˆpv F τ c mvf 2, c = k B T. Finally let γ = c/ɛ.

16 Different scalings of the Wigner equations Collisional Wigner equations under diffusive scaling: p τ t w 0 + 2γ w 0 + ɛ 2 w + Θ ɛ [V ]w 0 = g 0 w 0, τ p τ t w + 2γ w + ɛ w 2 g w 0 + Θ ɛ [V ] w + w p =, τ (WD) where: (Θ ɛ (V )w)(x, p) = i ɛ (2π) 2 δṽ (x, ξ)w(x, p )e i(p p ) ξ dξdp, R 2 R 2 δṽ (x, ξ) =V (x + ɛ2 ) ξ V (x ɛ2 ) ξ.

17 Hydrodynamic scaling of the Wigner equations Hydrodynamic scaling: with ˆx, ˆt, ˆp, ˆV satisfying: x ˆxx, t ˆtt, p ˆpp, V ˆV V, 1 ˆt = 2v F ˆp = ˆV ˆxˆp, ˆp = mk B T ; we define the semiclassical parameter ɛ, the hydrodynamic parameter τ and the scaled Fermi speed as: ɛ = ˆxˆp, τ = τ c ˆt, c = mvf 2 k B T. Again let γ = c/ɛ.

18 Hydrodynamic scaling of the Wigner equations Collisional Wigner equations under hydrodynamic scaling: p t w 0 + 2γ w 0 + ɛ 2 w + Θ ɛ [V ]w 0 = g 0 w 0, τ p t w + 2γ w + ɛ w 2 g w 0 + w p + Θ ɛ [V ] w =. τ (WH) where (again): (Θ ɛ (V )w)(x, p) = i ɛ (2π) 2 δṽ (x, ξ)w(x, p )e i(p p ) ξ dξdp, R 2 R 2 δṽ (x, ξ) =V (x + ɛ2 ) ξ V (x ɛ2 ) ξ.

19 Different scalings of the Wigner equations Two main assumptions: 1 The semiclassical hypothesis: ɛ 1 ; 2 The Low Scaled Fermi Speed (LSFS): c ɛ. As a consequence: γ = c/ɛ 1.

20 Equilibrium distribution through MEP Minimum Entropy Principle (MEP): Given a quantum system, we define the equilibrium distribution associated to the system as the minimizer of a suitable quantum entropy functional under the constraints of given fluid-dynamic moments. Quantum Entropy Functional (actually the free energy): A (S) = Tr[S log S S + H/k B T ], defined for S D(A ) suitable subset of the set of the density operators associated to the system.

21 Equilibrium distribution through MEP Let now: { } µ (k) 0 (p) k=1...n {, µ (k) s } (p) s=1,2,3, k=1...n { M (k) (x) } k=1...n real functions of x R2 ; µ (k) µ (k) 0 (p)σ 0 + µ (k) s σ s, for k = 1... N. real functions of p R 2 ; We define the equilibrium distribution at thermal equilibrium g associated to the moments { M (k)} as the Wigner transform k=1...n g WG of the solution of the constrained minimization problem: { A (G) = min A (S) : S = Op(w) D(A ), tr µ (k) (p)w(x, p) dp = M (k) (x), k = 1... N, x R 2 }.

22 Equilibrium distribution through MEP This problem can be solved formally by means of Lagrange multipliers. Solution as a density operator: G = exp( H + Op(µ (k) (p)ˆξ (k) (x))). Solution as a Wigner function: g = Exp( ĥ[ˆξ]), ĥ[ˆξ] = Op 1 H µ (k) (p)ˆξ (k) (x). Here Exp is the so-called quantum exponential, defined by: Exp(w) Op 1 (exp(op(w))), w Wigner function.

23 Semiclassical expansion of the quantum exponential Goal: find an explicit approximation of the quantum exponential of an arbitrary classical symbol with linear ɛ-dependence: g ɛ (β) = Exp ɛ (β(a + ɛb)), β R, with a = a 0 σ 0 + a σ, b = b 0 σ 0 + b σ arbitrary classical symbols. Moyal product between arbitrary classical symbols f 1, f 2 : f 1 # ɛ f 2 = Op 1 ɛ (Op ɛ (f 1 )Op ɛ (f 2 )).

24 Semiclassical expansion of the quantum exponential Semiclassical expansion of the Moyal product: # ɛ = ɛ n # (n), n=0 f 1 # (0) f 2 = f 1 f 2, f 1 # (1) f 2 = i 2 ( x s f 1 ps f 2 ps f 1 xs f 2 ), f 1 # (2) f 2 = 1 ) ( 2 xj xs f 1 2 pj ps f xj ps f 1 2 pj xs f pj ps f 1 2 xj xs f 2, 8...

25 Semiclassical expansion of the quantum exponential Let us differentiate with respect to β the function g ɛ (β): β g ɛ (β) = 1 2 ((a + ɛb)# ɛg ɛ (β) + g ɛ (β)# ɛ (a + ɛb)), g ɛ (0) =σ 0. Expansion in powers of ɛ: β g (0) (β) = 1 2 (g (0) (β)a + ag (0) (β)), g (0) (0) =σ 0, β g (1) (β) = 1 2 (g (1) (β)a + ag (1) (β)) (g (0) (β)b + bg (0) (β)) (g (0) (β)# (1) a + a# (1) g (0) (β)), g (1) (0) =0. Hierarchy of ODE: first solve eq. for g (0) (β), then eq. for g (1) (β).

26 Semiclassical expansion of the quantum exponential g (0) 0 (β) = eβa0 cosh(β a ), g (0) (β) = e βa0 sinh(β a ) a a, ( g (1) 0 (β) = a b βeβa0 cosh(β a )b 0 + sinh (β a ) a sinh(β a ) β a cosh(β a ) βa0 + βe 4β a 3 η jks {a j, a k }a s, ( g (1) (β) = βe [sinh(β a ) βa0 b 0 η ) jks{a j, a k }a s 4 a 2 + cosh (β a ) ( sinh(β a ) βa0 + βe β a + βe + βe a b a ) a a + sinh(β a ) βa0 2 a 2 a j {a j, a} a a β a cosh(β a ) sinh(β a ) βa0 2β a 2 {a 0, a} a a. ) ] a b a a a

27 Fluid models Fluid models for quantum electron transport in graphene: two-band models: first order two-band hydrodynamic model; first order two-band diffusive model; second order two-band diffusive model; spinorial models: first order spinorial hydrodynamic model; second order spinorial hydrodynamic model; first order spinorial diffusive model; first order spinorial diffusive model with pseudo-magnetic field.

28 Two-band models We will consider moments of this type: ( m k = µ k (p) w 0 (r, p) ± p ) w(r, p) dp k = 1... n, p with {µ k given. : R 2 R k = 1... n} suitable functions of p, and n N The functions: w ± (r, p) w 0 (r, p) ± p w(r, p) p are the distribution functions of the two bands (w + is related to the conduction band, w is related to the valence band).

29 Two-band hydrodynamic model Now we present a hydrodynamic model for quantum transport of electrons in graphene with two-band structure. This model will be obtained by computing moments of the Wigner equations (WH) and making a semiclassical expansion of the equilibrium distribution g appearing in Eqs. (WD). Moments: n ± (x) = w ± (x, p) dp, J±(x) k = p k w ± (x, p) dp (k = 1, 2). The moments n ± are the so-called band densities, and measure the contribution of each band to the charge density n 0 = (n + + n )/2. The moments J 1 ±, J 2 ± are the cartesian components of the band currents: they measure the contribution of each band to the current J = (J J 1, J J 2 ).

30 Two-band hydrodynamic model The (scaled) equilibrium distribution has the following form: g[n +, n, J +, J ] =Exp( h ξ ), ( ) p 2 h ξ = 2 + A 0 + A 1 p 1 + A 2 p 2 σ 0 + (c p + B 0 + B 1 p 1 + B 2 p 2 ) p p σ, where A j = A j (x), B j = B j (x) (j = 0, 1, 2) have to be determined in such a way that: g ± [n +, n, J +, J ](x, p) dp =n ± (x) x R 2, p k g ± [n +, n, J +, J ](x, p) dp =J k ±(x) x R 2, k = 1, 2.

31 Fully quantum two-band hydrodynamic model. The following proposition holds: Proposition Let n±, τ J± τ the moments of a solution w τ of Eqs. (WH), and let g = g[n+, τ n, τ J+, τ J ]. τ If n± τ n ±, J± τ J ± as τ 0 for suitable functions n ±, J ±, then the limit moments n ±, J ± satisfy: { 1 t n ± + k 2γ Jk ± + ɛ g k ± p } k 2 p g pk 0 dp ± p Θ ɛg k dp = 0, { 1 t J±+ i k p i p k g ± dp + ɛ ( p i g k ± p ) } k 2γ 2 p g 0 dp pi p k +n 0 i V ± p Θ ɛg k dp = 0, (i = 1, 2).

32 Semiclassical expansion of the equilibrium In order to obtain an explicit model, we make first-order approximation of the previous fluid equations with respect to the semiclassical parameter ɛ. So we perform a semiclassical expansion of the equilibrium distribution g through this strategy: 1 we compute a first order expansion of the quantum exponential in the spinorial case: g = g (0) + ɛg (1) ; 2 we impose that the approximation we found satisfies the contraints: (g (0) ± +ɛg (1) ± ) dp = n ± +O(ɛ 2 ), p i (g (0) ± +ɛg (1) ± ) dp = J±+O(ɛ i 2 ).

33 Semiclassical expansion of the equilibrium g ± = n ± 2π F (u) [ ( 1 ± ɛ F (u ± ) p + (p k u±) k F )] (u ± ) e p u± 2 /2 + O(ɛ 2 ), u k p e p u 2 /2 dp 2π u R 2, u k ± J k ±/n ± (k = 1, 2) ; g g p 2 ( g p) p = ɛ p 2 Λ p + O(ɛ 2 ), [ ( n+ n Λ(x, p) exp 1 2 u + u 2π p u + + u 2)] Ψ(x, p), 2 [ Ψ(x, p) sinh Φ + 1 cosh Φ ] [ ] sinh Φ x Φ + Φ Φ cosh Φ x Ξ, Ξ(x, p) u u 2 1 ( 4 2 log n+ n ) 4π 2 u + + u p, 2 Φ(x, p) u + 2 u 2 1 ( ) 4 2 log n+ u + u p. n 2

34 First-order two-band hydrodynamic model First-order two-band hydrodynamic model: t n ± + 1 2γ kj± k ± ɛ ( ) 2 F k n ± (u ± ) u k ±ɛ x V t J i ±+ 1 2γ k [ Λ(x, p) Λ(x, p)] p 2 p 3 dp = 0, { n ± (1 ± ɛf (u ± ))(δ ik + u i ±u k ±) ±ɛn ± [(δ ik u±u i ±)F k (u ± ) + u± i F u± k (u ± ) ]} 2 F u± u i ± k (u ± ) δ ik u± s F u± s (u ± ) ± ɛ [ ( 2 )] 2 F k n ± (u ± ) u k F ± (u ± ) u i u k u i +n ± i V ± ɛ x V p i Λ(x, p) p p 3 dp = 0 (i = 1, 2).

35 Two-band diffusive models Now we present two diffusive models for quantum transport of electrons in graphene with two-band structure. These two models will be based on a Chapman-Enskog expansion of the Wigner distribution w and a semiclassical expansion of the equilibrium distribution g that appear in Eqs. (WD). The moments we choose are the band densities: n ± = w ± dp, w ± = w 0 ± p p w.

36 Two-band diffusive models The (scaled) equilibrium distribution has the following form: ( ) p 2 g[n +, n ] = Exp ɛ ( h ξ ), h ξ = 2 + A σ 0 + (c p + B) p p σ, where A = A(x), B = B(x) have to be determined in such a way that: g ± [n +, n ](x, p) dp = n ± (x), x R 2.

37 Two-band diffusive models The following (formal) result holds: Theorem Let n+, τ n τ the moments of a solution w = w τ of (WD), and let g = g[n+, τ n ]. τ Let us suppose that: n± τ n ± as τ 0 for suitable functions n +, n ; then the limit moments n +, n satisfy: t n ± = (TTg[n +, n ]) ± dp, where: Tw =σ 0 T 0 w + σ T w, p T 0 w = 2γ w 0 + ɛ 2 w + Θ ɛ [V ]w 0, p T w = 2γ w + ɛ w Θ ɛ [V ] w + w p.

38 First order semiclassical expansion of equilibrium First order semiclassical expansion of the equilibrium distribution g: g 0 [n +, n ] = e p 2 /2 2π g[n +, n ] = e p 2 /2 with: 2π { ( π n 0 + ɛγ { [ n σ + ɛγ ) } 2 p n σ + O(ɛ 2 ), ( ) π 2 p ] p n 0 p + ɛ F p } p 2 + O(ɛ 2 ), n (n + + n ) n σ 1 2 (n + n ) charge density, pseudo-spin polarization, [ ] F 1 2 n σ x n 0 + n 2 0 nσ 2 n x 0 n σ x n 0 [log(n 0 + n σ ) log(n 0 n σ )] n0 2. n2 σ

39 First order two-band diffusive model First order two-band diffusive model: t n 0 = 1 ( 4γ 2 n 0 + ɛγ ) π 2 2 n σ + 1 2γ t n σ = 1 ( 4γ 2 n σ + ɛγ ) π 2 2 n γ ɛ π V 2γ 2 [ ] F + γf + ɛ 4 Γ + V [( n σ + ɛγ e ρ2 /2 ρ log ρ dρ > 0. ) π 2 n 0 Γ + ɛγ (n 0 V ) + O(ɛ 2 ), [( n σ + ɛγ ) π 2 2 n 0 π n 2 0 V ] π 2 n 0 + O(ɛ 2 ), ] V

40 Second order two-band diffusive model: assumptions We are going to derive another diffusive model for quantum transport in graphene. Exploit the Wigner equations in diffusive scaling (WD). Same fluid-dynamic moments n ± of the Wigner distribution w. Same equilibrium distribution. Stronger assumptions than (LSFS): consider also O(ɛ 2 ) terms in the fluid equations.

41 Second order two-band diffusive model: assumptions Assumptions: semiclassical hypothesis ɛ 1; Strongly Mixed State hypothesis (SMS): c ɛ, B = O(ɛ). Remember that: ( ) p 2 g[n +, n ] = Exp( h ξ ), h ξ = 2 + A σ 0 + (c p + B) p p σ, g ± [n +, n ](x, p) dp = n ± (x) x R 2. These further assumptions are necessary to overcome the computational difficulties arising from the spinorial nature of the problem: without these hypothesis, it would be hard to compute the second order expansion of the equilibrium distribution.

42 Second order two-band diffusive model: assumptions Consequence on the choice of moments: n + n n + + n = n σ n 0 = O(ɛ). Decoupling of h ξ in a scalar part of order 1 and a spinorial perturbation of order ɛ; this fact will be very useful in computations.

43 Second order two-band diffusive model: semiclassical expansion of equilibrium Now let us define, for an arbitrary positive scalar function n(x): M ɛ [n] n 2 2π e p /2 [1 + ɛ2 24 ( (σ 0 p p) log n) ] ; then the equilibrium distribution has this semiclassical expansion: [ g ɛ [n +, n ] =M ɛ n 0 n ( ( 0 2 π ) )] ɛ 2 γ 2 + n2 σ σ n 0 2 2π e p /2 + n 0 2 4π e p /2 [ ɛγ [ ɛγ n 2 0 ( ) π 2 p + n ] σ p n 0 p σ ( ) π 2 p + n ] 2 σ σ 0 + O(ɛ 3 ). n 0 Exploiting this expansion and the fully quantum two-band diffusive equations, we obtain:

44 Second order two-band diffusive model Second order two-band diffusive model: t n 0 = 4γ 2 t n σ = 4γ 2 + ɛ 4 [ ( 1 + ɛ 2 γ 2 π ) n 0 + ɛγ π 4 2 [ ] ɛγ π 2 2 n 0 + n σ + 2γ π n 2 0 V V ] + 2 n σ [( n σ + ɛγ 2 2γ ( ) n 0 (V + VB ) + O(ɛ 3 ), π 2 n 0 ) ] V { π ɛγ 2 (1 + Γ)n 0 + Γn σ where V B is (up to a constant) the so-called Bohm potential: V B = 1 ɛ 2 n 0. 2γ 6 n0 } + O(ɛ 3 ).

45 Spinorial models Now we will derive two spinorial hydrodynamic models and two spinorial diffusive models for quantum electron transport in graphene following a strategy similar to that one employed in the derivation of the previous diffusive models. Spinorial models: the Pauli components of the Wigner matrix are considered separately from each other, not through a linear combination.

46 Spinorial hydrodynamic models Moments: n s = w s dp (s = 0, 1, 2, 3), J k = p k w 0 dp (k = 1, 2). n 0 is the charge density; n = (n 1, n 2, n 3 ) is the spin vector; J = (J1, J 2, 0) is the current vector. Note that the current vector has only two components because graphene s cristal lattice is a two-dimensional object.

47 Spinorial hydrodynamic models The equilibrium distribution has the following form: g[n 0, n, J] =Exp( h ξ ), ( ) p 2 h ξ = 2 + p kξ k + ξ 0 σ 0 + (ξ s + cp s )σ s, with ξ 0 (x), (ξ s (x)) s=1,2,3, (Ξ k (x)) k=1,2 Lagrange multipliers to be determined in such a way that: g 0 [n 0, n, J] (x) = n 0 (x), g[n 0, n, J] (x) = n(x), pg 0 [n 0, n, J] (x) = J(x), for x R 2.

48 Spinorial hydrodynamic models, fully quantum system The following theorem holds: Theorem Let n τ 0, nτ, J τ the moments of a solution w τ of Eqs. (WH), and let g = g[n τ 0, nτ, J τ ]. If n τ 0 n 0, n τ n, J τ J as τ 0, then the limit moments n 0, n, J satisfy: t n 0 + 2γ J + ɛ 2 n =0 t n + 2γ g p dp + ɛ n g p dp =0 ( ) t J J + 2γ J + P + ɛ 2 p g dp + n 0 V =0 n 0 where P is the so-called quantum stress tensor: P = ( p J/n 0 ) ( p J/n 0 )g 0 dp

49 First-order semiclassical expansion of the equilibrium distribution First-order semiclassical expansion of the equilibrium distribution: g 0 [n 0, n, J] = n 0 2 2π e p u /2 + O(ɛ 2 ), g[n 0, n, J] = n ( 0 2 2π e p u /2 n ɛγz( p u) n 0 ) + O(ɛ 2 ), Z ij n ( in j n 2 + ω δ ij n ) in j n 2 n0+ n n 0 n ω n /n 0 log. + 1 ω 2γ η n k iks n j ( ) ns n (i, j = 1, 2, 3), Exploiting this expansion and the fully quantum hydrodynamic spinorial equations, we obtain:

50 First-order spinorial hydrodynamic model First-order spinorial hydrodynamic model: t n 0 + ( ) 2γ J + ɛγ n = 0, t n+ 2γ ( n u ɛγn 0Z + ɛγn 0 I ) + n u + ɛ 2 n 0(1 ω)[ s s ] s = 0, t J+ 2γ [n 0 (I + u u) + ɛγ ( u n ɛγn 0 Z T )] + n 0 V = 0, where: s n n.

51 Second order spinorial hydrodynamic model: assumption We are going to derive another hydrodynamic model for quantum transport in graphene. Exploit the Wigner equations in hydrodynamic scaling (WH). Same fluid-dynamic moments n 0, n, J of the Wigner distribution w. Same equilibrium distribution. Stronger assumptions than (LSFS): consider also O(ɛ 2 ) terms in the fluid equations.

52 Second order spinorial hydrodynamic model: assumptions Assumptions: semiclassical hypothesis ɛ 1; Strongly Mixed State hypothesis (SMS): [ 3 ] 1/2 c ɛ, (ξ s ) 2 = O(ɛ). s=1 Remember that: ( ) g[n 0, n, p 2 J] = Exp( h ξ ), h ξ = 2 + p kξ k + ξ 0 σ 0 + (ξ s + cp s )σ s, g s [n 0, n, J](x, p) dp = n s (x) (s = 0, 1, 2, 3), x R 2, p k g 0 [n 0, n, J](x, p) dp = J k (x) (k = 1, 2), x R 2. These further assumptions are necessary to overcome the computational difficulties arising from the spinorial nature of the problem: without these hypothesis, it would be hard to compute the second order expansion of the equilibrium distribution.

53 Second order spinorial hydrodynamic model: assumptions Consequence on the choice of moments: n = O(ɛ). n 0 Decoupling of h ξ in a scalar part of order 1 and a spinorial perturbation of order ɛ; this fact will be very useful in computations.

54 Second order spinorial hydrodynamic model: semiclassical expansion of equilibrium Let us define, for an arbitrary positive function N (x) and an arbitrary vector function J (x) = (J 1 (x), J 2 (x), 0): M ɛ [N, J ] = N )] 2π e p U 2 /2 [1 (2 ɛ2 N 2 log N + 24 N 2 Q(N, J ), where: Q(N, J ) =3( A + p k U k + i U j j U i ) 2 i U j (p i U i )( j A + p k j U k ) ( ija 2 + p k iju 2 k )(p i U i )(p j U j ) + (A + p k U k ) 2, ( ) U = J N /N, A = log U 2 2π 2.

55 Second order spinorial hydrodynamic model: semiclassical expansion of equilibrium Second-order semiclassical expansion of the equilibrium distribution: [ ( ) ( ) ] g 0 [n 0, n, n 2 J] =M ɛ n 0 n 0 + ɛ 2 γ 2, n 2 J + ɛγ n + ɛ 2 γ 2 J + n 0 4π e p J/n 0 g[n 0, n, J] = n 0 2π e p J/n 0 2 /2 2n 2 0 2n 2 0 ( 2 /2 n ɛγ p ) J 2 + O(ɛ 3 ), n 0 n 0 ( ( n ɛγ p )) J + O(ɛ 3 ). n 0 n 0 Exploiting this expansion and the fully quantum hydrodynamic spinorial equations, we obtain:

56 Second order spinorial hydrodynamic model Second order spinorial hydrodynamic model: t n 0 + 2γ ( J + ɛγ n) =O(ɛ 3 ), t n + ( 2γ n ) J + n J =O(ɛ 3 ), n 0 n 0 ( ) t J ( J + 2γ J + ɛγ n) n + 0 2γ + n 0 (V + V B ) =O(ɛ 3 ), n 0 where V B is again (up to a constant) the so-called Bohm potential: V B = 1 ɛ 2 n 0. 2γ 6 n0

57 Spinorial diffusive models Now we will present two spinorial drift-diffusion model for quantum transport of electrons in graphene. Both first-order model: second order too much computationally demanding! Difference: a theoretical Pseudo-Magnetic external field which is supposed to interact with the charge carriers pseudo-spin and which will provide a strong coupling between the second model equations.

58 Spinorial diffusive models Moments: n 0 (x, t) = n(x, t) = w 0 (x, p, t) dp w(x, p, t) dp charge density, spin vector. The (scaled) equilibrium distribution can be written as: ( ) p 2 g[n 0, n] = Exp( h A, B ), h A, B = 2 + A σ 0 + (c p + B) σ, where A(x, t), B(x, t) = (B 1 (x, t), B 2 (x, t), B 3 (x, t)) are Lagrange multipliers to be determined in such a way that: g 0 [n 0, n](x, p, t) dp = n 0 (x, t), g[n 0, n](x, p, t) dp = n(x, t), for (x, t) R 2 R.

59 Spinorial diffusive models: assumptions We assume that the semiclassical parameter ɛ and the diffusive parameter τ are of the same order and small, so we will perform a limit τ 0 in the Wigner equations: c ɛ τ. Moreover we define: λ c τ 1.

60 Spinorial diffusive models: semiclassical expansion of equilibrium First order semiclassical expansion of equilibrium distribution: g (0) g[n 0, n] =g (0) [n 0, n] + ɛg (1) [n 0, n] + O(ɛ 2 ), 0 [n 0, n] = e p 2 /2 n 0, g (0) /2 [n 0, n] = e p 2 2π 2π n, 2 /2 g (1) 0 [n 0, n] = γ e p n p, 2π g (1) /2 [n 0, n] = γ e p 2 2π [ ( n n n 0 (1 ω) n 2 (1 ω) [( p x ) n] n 2γ n 2 ], ) + ω I p with: { ω n n 0 + n log n 0 n 0 n } 1.

61 First order spinorial drift-diffusion model: derivation From eqs. (WD) make a Chapman-Enskog expansion of the Wigner function w. Take moments of eqs. (WD). Exploit the semiclassical expansion of the equilibrium. We obtain:

62 First order spinorial drift-diffusion model. First order spinorial drift-diffusion model Quantum Spin Diffusion Equations 1 (QSDE1): t n 0 = n 0 + div(n 0 V ), t n j = s A js + F j, (j = 1, 2, 3) [ ] ) A js = (δ jl + b k η jkl s n l + n j s V nn0 2η jsl n l + b k [ n n 0 ] (δ jk n s δ js n k ), (j, s = 1, 2, 3) F j =η jkl n k l V 2n j + b s [ n n 0 ] s n j b j [ n n 0 ] s n s, (j = 1, 2, 3) where we defined, for all v R 3, 0 < v < 1: [ ] v 2 v b[ v] = λ v 2 1. log(1 + v ) log(1 v )

63 First order spinorial drift-diffusion model with pseudo-magnetic field In the model QSDE1 the charge density n 0 evolves independently from the spin vector n: we are going to modify the QSDE1 model in order to obtain a fully coupled system by adding a pseudo-magnetic field able to interact with the charge carriers pseudospin. Negulescu and Possanner, in their article 2, consider a semiconductor subject to a magnetic field interacting with the electron spin, and derive a purely semiclassical (without quantum corrections) diffusive model for the charge density n 0 and the spin vector n throught a Chapman-Enskog espansion of the Boltzmann distribution. We will follow a similar procedure to obtain our new model. 2 S. Possanner and C. Negulescu. Diffusion limit of a generalized matrix Boltzmann equation for spin-polarized transport. Kinetic and Related Models (2011).

64 First order spinorial drift-diffusion model with pseudo-magnetic field: derivation We define two quantities: ζ =ζ(x, t) ω = ω(x, t) pseudo-spin polarization of scattering rate; direction of local pseudo-magnetization. s = 1 + ζ(x, t) 1 ζ(x, t) s, where s are the scattering rates of electrons in the upper band and in the lower band; it is bounded between 0 and 1. The vector ω, being a direction, has modulus equal to 1.

65 First order spinorial drift-diffusion model with pseudo-magnetic field: derivation New Wigner equations in diffusive scaling: τ t w + τ t w 0 + p 2γ w 0 + ɛ 2 w + Θ ɛ [V ]w 0 = Q 0(w), τ p 2γ w + ɛ w 2 Q(w) 0 + w p + Θ ɛ [V ] w + τ ω w =, τ (WD2) with the collision operator Q(w) defined by: Q(w) = P 1/2 (g w)p 1/2, P = σ 0 + ζ ω σ. P is the so-called polarization matrix.

66 First order spinorial drift-diffusion model with pseudo-magnetic field: derivation From eqs. (WD2) make a Chapman-Enskog expansion of the Wigner function w. Take moments of eqs. (WD2). Exploit the semiclassical expansion of the equilibrium. We obtain:

67 First order spinorial drift-diffusion model with pseudo-magnetic field First-order spinorial drift-diffusion model with pseudo-magnetic field Quantum Spin Diffusion Equation 2 (QSDE2): t n 0 = s M 0s, t n j = s M js + η jks (M ks + n k ω s ) { [ ] } n + s b k (η jkl s n l + δ jk n s δ js n k ) n 0 [ ] [ ] n n + b s s n j b j s n s (j = 1, 2, 3), n 0 n 0 M 0s =φ 2 {(n 0 + n 0 s V ) ζω k ( s n k + n k s V + η kls n l )}, M js =φ 2 { ζω j (n 0 + n 0 s V ) + [ω j ω k + φ(δ jk ω j ω k )]( s n k + n k s V + η kls n l )}, [ 1 φ = 1 ζ 2, b[ v] = λ v v 2 2 v log(1 + v ) log(1 v ) ].

68 Analytical results Now we present some analytical results concerning the model QSDE1. Existence and uniqueness of (weak) solutions satisfying suitable L bounds. Entropy inequality. Long-time behaviour of the solutions.

69 Analytical results We considered model QSDE1 for (x, t) Ω T Ω [0, T ] with Ω R 2 bounded domain: t n 0 =div ( n 0 + n 0 V ) x Ω, t [0, T ], λ 2 D V =n 0 C(x) x Ω, t [0, T ], n 0 (x, t) =n Γ (x, t) x Ω, t [0, T ], t n =div J + F x Ω, t [0, T ], n(x, t) =0 x Ω, t [0, T ], V (x, t) =U(x, t) x Ω, t [0, T ], n 0 (x, 0) =n 0I (x) x Ω, n(x, 0) = n I (x) x Ω, F j =η jkl n k l V 2n j + b k [ n/n 0 ] k n j b j [ n/n 0 ] n, J js = ( δ jl + b k [ n/n 0 ]η jkl ) s n l + n j s V (Pb) 2η jsl n l + b k [ n/n 0 ](δ jk n s δ js n k ), (j, s = 1, 2, 3), [ { ( )} ] 1 v 1 + v b[ v] =λ v v log v R 3, 0 < v < 1. 1 v

70 Analytical results We split problem (Pb) into the following two problems: t n 0 =div ( n 0 + n 0 V ) x Ω, t [0, T ], λ 2 D V =n 0 C(x) x Ω, t [0, T ], n 0 (x, t) =n Γ (x, t) x Ω, t [0, T ], V (x, t) =U(x, t) x Ω, t [0, T ], n 0 (x, 0) =n 0I (x) x Ω, t n =div J + F x Ω, t [0, T ], n(x, t) =0 x Ω, t [0, T ], (Pb-ns) n(x, 0) = n I (x) x Ω, F j =η jkl n k l V 2n j + b k [ n/n 0 ] k n j b j [ n/n 0 ] n, J js = ( δ jl + b k [ n/n 0 ]η jkl ) s n l + n j s V (Pb-n0V) 2η jsl n l + b k [ n/n 0 ](δ jk n s δ js n k ), (j, s = 1, 2, 3), [ { ( )} ] 1 v 1 + v b[ v] =λ v v log v R 3, 0 < v < 1. 1 v

71 Analytical results: existence and regularity for first problem We studied first the existence and regularity of solutions (n 0, V ) of pb. (Pb-n0V). Conditions on the data: n Γ H 1 (0, T ; H 2 (Ω)) H 2 (0, T ; L 2 (Ω)) L (0, T ; L (Ω)), n 0I H 1 (Ω), inf Ω n 0I > 0, n 0I = n Γ (0) on Ω, inf Ω (0,T ) n Γ > 0, U L (0, T ; W 2,p (Ω)) H 1 (0, T ; H 1 (Ω)), C L (Ω), C 0 in Ω, for some p > 2.

72 Analytical results: existence, uniqueness and regularity for first problem Theorem Let T > 0. Under the previous assumptions there exists a unique solution (n 0, V ) to pb. (Pb-n0V) satisfying: n 0 L ([0, T ], H 2 (Ω)) H 1 ([0, T ], H 1 (Ω)) H 2 ([0, T ], (H 1 (Ω)) ), V L ([0, T ], W 1, (Ω)) H 1 ([0, T ], H 2 (Ω)), 0 < me µt n 0 M in Ω, t > 0, where µ = λ 2 D and M = max { { m = min sup Ω (0,T ) n Γ, sup Ω inf n Γ, inf n 0I Ω (0,T ) Ω n 0I, sup C Ω } > 0. },

73 Analytical results: existence and uniqueness for second problem Theorem Let (n 0, V ) be the solution to pb. (Pb-n0V) according to the previous theorem and n 0 H0 1 (Ω) such that: n 0 (x) sup x Ω n 0I (x) < 1 ; then pb. (Pb-ns) has a solution n such that: n L 2 ([0, T ], H 1 0 (Ω)) H 1 ([0, T ], H 1 (Ω)), sup Ω T n n 0 < 1 ; furthermore, there exists at most one weak solution with the property stated above and satisfying n L ([0, T ], W 1,4 (Ω)).

74 Entropicity of the system Let (n 0, n, V ) be a solution to pb. (Pb) according to previously stated existence theorems. We assume that the boundary data is in global equilibrium, i.e. n Γ = e U, V = U, n = 0 on Ω, where U = U(x) is time-independent. Then the macroscopic entropy: { 1 S(t) = 2 (n 0 + n ) ( log(n 0 + n ) 1 ) is nonincreasing in time. Ω (n 0 n ) ( log(n 0 n ) 1 ) } + (n 0 C(x))V λ2 D V 2 dx 2

75 Entropicity of the system Proposition The entropy dissipation ds/dt can be written as: ds dt = 1 (n 0 + n ) (log(n 0 + n ) + V ) (n 0 n ) (log(n 0 n ) + V ) ( ) [ ] n0 + n n n log G 0, 2 n 0 n n where G is defined by: G [ v] j,k ( j v k ) v curl v + 2 v 2 0 v H 1 (Ω) 3.

76 Long-time decay of the solutions Let (n 0, n, V ) be a solution to pb. (Pb) according to the existence theorems. It is possible to prove that, under suitable assumptions on the electric potential, the spin vector converges to zero as t. To prove the stated result we exploited the following: Lemma Let G as in the previous proposition: G [ v] = j,k ( j v k ) v curl v + 2 v 2 v H 1 (Ω) 3. A constant K Ω > 0 exists, depending only on Ω, such that: G [ u] K Ω u 2, u H 1 (Ω) 3.

77 Long-time decay of the solutions Theorem Let K Ω as in the previous Lemma, and let 2 < p < arbitrary. 1 A positive constant c = c(p, Ω) exists such that: if sup ΩT V < c then: n Lp (Ω)(t) n I Lp (Ω)e kt t > 0, for a suitable number k = k(p, Ω, sup ΩT V ) > 0. 2 If sup ΩT V < K Ω then: n L2 (Ω)(t) n I L 2 (Ω)e k t t > 0, with k = 2K Ω sup ΩT V > 0. 3 If sup ΩT V < 0 then: n L (Ω)(t) n I L (Ω)e k t t > 0, with k = sup ΩT V > 0.

78 Numerical simulations We solved model (QSDE2) and, as a particular case, model (QSDE1), in one space dimension, by means of Crank-Nicholson finite difference scheme. We simulated a ballistic diode to which a certain bias is applied: we chose global equilibrium initial conditions and we observed the evolution of the system towards a new equilibrium due to the applied potential.

79 Numerical simulations Boundary conditions: n 0 = C, n = 0, V = U on Ω = {0, 1}, t > 0, where U(x) = V A x/l, V A = 1 v is the applied voltage, and L = 10 7 m is the device lenght. Initial conditions: n 0 (x, 0) = q exp( V eq (x)), n(x, 0) = 0, where q = C m 2, V eq is the (scaled) equilibrium potential: λ 2 D 2 xxv eq = exp( V eq ) C/q in Ω, V eq (0) = V eq (1) = 0, and λ 2 D = 10 3 L 2. The pseudo-spin polarization and the direction of the local magnetization are defined by: ζ = 0.5, ω = (0, 0, 1).

80 Numerical simulations The doping profile corresponding to a ballistic diode: 1 C(x) ( C m 2 ) x (10 7 m) Figure: Doping profile corresponding to a ballistic diode.

81 Numerical simulations 1 Particle density 5 x 10 4 Spin vector, component 1 n 2 ( C m 2 ) n 0 ( C m 2 ) t = 0 s t = s t = s x (10 7 m) Spin vector, component t = 0 s t = s t = s n 1 ( C m 2 ) n 3 ( C m 2 ) 0 5 t = 0 s t = s t = s x (10 7 m) Spin vector, component t = 0 s 0.2 t = s t = s x (10 7 m) x (10 7 m) Figure: Model QSDE2: Particle density and components of the spin vector versus position at times t = 0 s, t = s, and t = s.

82 Numerical simulations 1 Charge density for models QSDE1, QSDE2 at time t=0.05 versus position QSDE1 QSDE n x Figure: Charge density for models QSDE1 and QSDE2 versus position at time t = s (maximized difference).

83 Numerical simulations Ratio n /n 0 t = s t = s t = s t = s 0.5 n / n x (10 7 m) Figure: Model QSDE2: Ratio n /n 0 versus position at several times.

84 Numerical simulations Particle density, deviation from equilibrium (logarithmic scale) model QSDE1 model QSDE2 log( n 0 (t) n 0 ( ) 2 / n 0 ( ) 2 ) t ( s) Figure: Relative difference n 0(t) n 0( ) / n 0(t) versus time (semilogarithmic plot) for the models QSDE1 (solid line) and QSDE2 (dashed line).

85 Numerical simulations Steady state current density at outflow point model QSDE2 model QSDE1 Current density ( C m 1 s 1 ) Applied voltage ( v) Figure: Static current-voltage characteristics for the models QSDE1 and QSDE2.

86 Conclusions The purpose of our work was the description of quantum transport of electrons in graphene by means of fluid models: we presented a kinetic model, that is, the Wigner equation, as the starting point of the derivation of fluid models; we defined the quantum equilibrium distribution by means of the quantum minimum entropy principle, computing a semiclassical expansion of the quantum exponential in the spinorial case; we derived one hydrodynamic and two diffusive two-band models, which means, models for conduction and valence band densities; we derived two hydrodynamic and two diffusive spinorial models, including all the components of the spin vector; we performed an analysis of the first diffusive spinorial model, proving existence of solutions, uniqueness of the solution under a regularity condition on the moments, entropicity for the model and long-time decay of the spin vector; we obtained some numerical simulations for the spinorial diffusive models, showing the temporal evolution of the moments.

87 Bibliography N. Zamponi, L. Barletti. Quantum electronic trasport in graphene: a kinetic and fluid-dynamic approach. Math. Meth. Appl. Sci. 2011, 34, N. Zamponi. Some fluid-dynamic models for quantum electron transport in graphene via entropy minimization. Kinetic and Related Models 2012, vol. 5, no. 1, N. Zamponi, A. Jüngel. Two spinorial drift-diffusion models for quantum electron transport in graphene. Communication in Mathematical Sciences 2013, vol. 11, no. 3, N. Zamponi. Quantum Fluid Models for Electron Transport in Graphene. PhD Thesis (2013).

88 Thank you for your attention! Thank you for your attention!