Sub-diffusive electronic states in octagonal tiling

We study the quantum diffusion of charge carriers in octagonal tilings. Our numerical results show a power law decay of the wave-packet spreading, $L(t) \propto t^{\beta}$, characteristic of critical states in quasicrystals at large time $t$. For many energies states are sub-diffusive, i.e. $\beta<0.5$, and thus conductivity increases when the amount of defects (static defects and/or temperature) increases.

Experimental investigations have indicated that the conduction properties of many stable quasicrystals (AlCuFe, AlPdMn, AlPdRe, ...) are unusual and differ strongly from those of simple inter-metallic alloys [1][2][3]. In particular their conductivity increases with static defects density and when temperature increases. It appears also that the medium range order and the chemical order -over one or a few nanometers-have a decisive influence [4][5][6][7][8][9]. There is now strong evidence that these non standard properties result from a new type of break-down of the semi-classical Bloch-Boltzmann theory of conduction [10][11][12][13][14]. On the other hand, the specific role of long range quasiperiodic order in electronic properties is still an open question in spite of a large number of studies (Refs.  and Refs. therein). Many studies support the existence of critical states, which are neither extended nor localised, but are characterised by a power law decay of the wave-function envelope at large distances. In the presence of critical states, the diffusion of charge carrier at sufficiently large time t follows a power law and then the spatial extension L of wave-packets should be, [34][35][36][37] where β, 0 ≤ β ≤ 1, is an exponent depending on energy E and on the Hamiltonian model. Note that in usual metallic crystals without static defects, β = 1 and the propagation is ballistic. In strongly disordered systems, β = 0.5 for a large time range and propagation is diffusive. For a localised state one has β = 0. When disorder is introduced in the perfect approximant or perfect quasicrystal in the form of static defects (elastic scatterers) and/or inelastic scattering (temperature, magnetic field...), the defects induce scattering and we expect that there is an associated time τ above which the propagation of the wave-packet is diffusive. The diffusivity D of charge carrier at energy E can be estimated by, D(E, τ ) L(E, t = τ ) 2 /τ ∝ τ 2β(E)−1 , and the conductivity σ at zero frequency is given by the Einstein formula: where n is the density of states and E F the Fermi energy. The case 0.5 < β < 1, called super-diffusive regime, leads to transport properties similar to metal, since the conductivity decreases when disorder increases -i.e. when τ decreases-. Conversely, for 0 < β < 0.5, the regime is sub-diffusive and the conductivity increases when disorder increases like in real quasicrystals. Many authors consider [18][19][20]29,[34][35][36][37] that critical states could lead to β < 0.5 but it has not yet been shown in 2D or 3D quasiperiodic structures (except for some very specific energies).
Model Hamiltonian.-The octagonal, or Ammann-Beenker, tiling [38,39] is a quasiperiodic tiling analogous to the notorious Penrose tiling. This tiling has been often used to understand the influence of quasiperiodicity on electronic transport [18][19][20][21][22][23][24][25]. A sequence of periodic approximants X 0 , X 1 , . . . , X k , . . . can be generated [40]. In approximants of order k ≥ 1, the 6 local configurations around vertexes are the same as in the octagonal quasiperiodic tiling. They have, respectively, coordination number η = 3, 4, 5, 6, 7 and 8. We consider the simple Hamiltonian, where i indexes s orbitals located on vertexes, and γ is the strength of the hopping between orbitals. i, j are the nearest-neighbours at tile edge distance a. To simulate schematically a possible effect of the presence of different chemical elements, the onesite energy i is proportional to the coordinance η i of the site i: i = η i γ. To obtain realistic time values, we use γ = 1 eV which is the order of magnitude of the hopping parameter in real inter-metallic compounds. The total density of states (total DOS) of X 7 approximant is shown figure 1(a). Quantum diffusion.-In the framework of Kubo-Greenwood approach for calculation of the conductivity, we use the polynomial expansion method developed by Mayou, Khanna, Roche and Triozon [41,42,31,27,28] to compute the mean square spreading of the wave-packet at time t and energy E: L 2 (E, t) = (X(t) −X(0)) 2 E , whereX is the position operator in the x-direction. The diffusion coefficient D(E, t) = L(E, t) 2 /t is shown in figure 1(b) for X 7 approximant at some energies. The ballistic regime due to the periodicity of the approximant is reached at very large t, when L(t) > L k where L k is the approximant cell size; then L(t) = V B t, where V B is the Boltzmann velocity, i.e. the intra-band velocity, V B = ∂E n (k)/∂k x E / , where E n (k) is the band dispersion relation [12]. For X 7 in time range shown figure 1, this Boltzmann term is negligible and, for all purposes of this discussion, the X 7 approximant is equivalent to the quasiperiodic system. Depending on the t values, three different regimes are observed at each energy: • At very small time, typically when L(t) < a, the mean spreading grows linearly with t, L(t) = V 0 t (ballistic behaviour), where V 0 > V B [10,12]. • For times, corresponding to L(t) a few a, the propagation seems to become diffusive as the diffusion coefficient is almost constant, D(t) D dif . Therefore L 1 , defined by L 1 = D dif /V 0 a few a, is a kind of effective elastic scattering length but it is not due to static scattering events because we consider perfect tilings. The corresponding effective elastic scattering time is t 1 = L 1 /V 0 . Roughly speaking, it seems that when L(t) L 1 , i.e. t t 1 , the wave-packet feels a random tiling.
• An other distance L 2 (respectively an other time t 2 , L(t 2 ) = L 2 ) appears. For L(t) > L 2 a few 10a, a new regime appears and D(t) follows a power law. It is thus characteristic of the medium and long range quasiperiodic order. Figure  1(b) shows that the β value can switch from a sub-diffusive regime (β < 0.5) to a super-diffusive regime (β > 0.5) over a small variation of energy. The t 2 values, t 2 10 −13 -10 −14 s, have the order of magnitude of the scattering time above which measurements show unusual transport properties in quasicrystals [1].
Both distances L 1 a few a, L 2 a few 10a, and the exponent β at time t > t 2 , depend a lot on the energy value E. L 1 < L 2 , but at some energy it even seems that L 1 L 2 . Further analysis are necessary to understand the energy dependence.
To summarise, we have presented quantum diffusion in a large approximant of the octagonal tiling. The charge carrier propagation is determined by the wave-packet spreading in the quasiperiodic lattice. From numerical calculation, two length scales seem to characterise this quasiperiodic spreading. L 1 , typically L 1 = a few a, above which the propagation is almost diffusive in spite of the absence of static defects. L 2 , typically L 2 = a few 10a, above which specific quasiperiodic symmetries lead to a power law dependence of the root mean square spreading, L(t) ∝ t β . For some energies states are super-diffusive or diffusive, i.e. β ≥ 0.5, whereas for other energies, a subdiffusive regime, i.e. β < 0.5, sets in as expected for critical states characteristic of quasiperiodicity. This sub-diffusive regime is the generalisation to quasicrystal of the