Diffusion of semi-Dirac excitations P. Adroguer1, D.Carpentier1, G. Montambaux2, E. Orignac1 1Laboratoire

de Physique, Ecole Normale Supérieure de Lyon, France 2Laboratoire de Physique des Solides, Univ. Paris-Sud, Orsay, France

Semi-Dirac excitations • Hexagonal lattice (graphene) qD , m∗ cx = qD /m∗ qD , cx m∗ = qD /cx m∗ , cx qD = m∗ cx p ∗ m , ∆ cx = −2∆/m∗ cx , ∆ qD = −2∆/cx 2 qD , ∆ m∗ = −qD /2∆

4

2 ∆ = −qD /2m∗ ∆ = −cx qD /2 ∆ = −m∗ c2 /2 √ qD = −2m∗ ∆ m∗ = −2∆/c2x cx = −2∆/qD

to the velocity cy , the universal Hamiltonian is described by two independent parameters (left column) parameters may be deduced

III.

PROPERTIES OF THE UNIVERSAL HAMILTONIAN

•

2 Dirac fermions, with topologic number

H = �vF �σ .�q

stomay be deduced the velocity cy , the universal Hamiltonian is described by two independent parameters (left column)

Semi-Dirac excitations

arameters may be deduced

PROPERTIES OF THE UNIVERSAL HAMILTONIAN III.

PROPERTIES OF THE UNIVERSAL HAMILTONIAN

•

�2 qx2 x )σ + �vF σ y q y Merging of the Dirac cones H = (∆ + 2m Montambaux et al., 2009

•

� q σ + �vF σ q Δ = 0 : Semi-Dirac excitation, H = 2m

2 ∆∆ 2isisvaried FIG. 1: Evolution Evolution of of the the spectrum spectrum when when the the quantity quantity variedand andchanges changesininsign signatatthethetopol top x x y y units). The low-energy low-energy spectrum spectrum stays stays linear linearininthe theqyqydirection. direction.

Without loss of of generality, generality, we we assume assume m m∗∗ >> 0.0. When When∆∆varies variesfrom fromnegative negativetotoposit pos transition from aa semi-metallic semi-metallic phase phasewith withtwo twoDirac Diraccones conesand anda aband bandinsulator insulatorwith witha agap ga the transition, the the spectrum spectrum isis hybrid, hybrid,aareminiscence reminiscenceofofthe thesaddle saddlepoint pointininthe thesemi-metall semi-meta When ∆ < 0, the the spectrum spectrum exhibits exhibitstwo twoDirac Diracpoints pointsthe theposition positionofofwhich whichalong alongthe thex xax

Also predited in VO2/TiO2 √√ heterostructures ∗ q = −2m∗ ∆

qDD = −2m ∆ he spectrum when the quantity ∆ is varied and changes in sign at the topological Pardo andtransition Pickett, 2009 (arbitrary um when the is qvaried and spectrum changes in sign these at theDirac topological (arbitrary and the linear around points istransition characterized by the velocity c along spectrum staysquantity linear in∆the y direction. spectrum around these Dirac points is characterized by the velocityxcx alon stays linear in the qy direction. !!

e temperature increases from 2 to 30 K, implying its mechanical nature. It is well known that the UCF es undergoes the average reduction when the sample ns are longer than LQ26,28. Quantitatively, following lations in Ref.28,33, the 2D UCF theory predicts dGrms, )/(2N)1/2 at T50 K which agrees to the experimental value

Diffusion • • •

while varying h due to the isotropic LQ in a 3D system. We can find dGrms maintains comparable while h is below 40u. However, when h exceeds 45u, dGrms drops abruptly, which can be explained by the contribution of a 2D conducting states. In a TI system, the UCF contributions from the electrons of the bulk states gradually predominate while h is increasing17,20. The present anisotropic

Scattering processes � pi Classically, �σ� ∝ i

Coherent regime, memory � of the phase : quantum interferences δσ ∝ a∗i aj i�=j

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Regime of diffusive transport • • •

Semi classical approach, λF � le (perturbative approach)

• •

Sample length � mean free path le

Experimental regime : high Fermi energy (good metal) �2 qx2 x Hamiltonian : H = σ + �vF σ y q y 2m �V (�r)V (r�� )� = γδ(�r − r�� ) �V (�r)� = 0 Weak disorder regime - Boltzmann equation - Diagrammatics

Boltzmann equation • Spinorial nature : Anisotropy of the scattering Pure Dirac fermions

0.6

|ψ(�k)� =

0.4

0.2

0.2

�0.2

0.4

0.6

0.8

1.0

�

1 eiθ�k

�

� 1 + cos(θ − θ ) 2 � � � |�ψ(k)|ψ(k )�| = 2

�0.4

�0.6

•

Anisotropy of the density of states

2

High density 1

�1.0

�0.5

0.5

�1

Low density

�2

1.0

Boltzmann equation •

Stronger anisotropy of the scattering for semi-Dirac excitations compared to Dirac fermions |�ψ(k�y )|ψ(k�� )�|2 Φ |�ψ(k�x )|ψ(k�� )�|2 y

0.5

0.6

Φ 0.2

0.4

0.6

0.8

0.4

1.0

Φ

0.2

x �1.0

�0.5

0.5 �0.2 �0.4 �0.6

�0.5

•

Combination of these two anisotropies : Anisotropy of the diffusion Dx �= Dy

Φ

1.0

Diagrammatics •

Direction dependant elastic mean free time 1 cos θ x −�Σ = Id + ∗ σ τe τe

• • σxx

2 diffusive modes, 1 diffuson and 1 cooperon Drude conductivity tensor

e2 2E�2 2e = h mγ a ˜

•

σyy

e2 �2 vF2 2c = h γ a ˜

σxy = σyx = 0

Weak anti-localization (Quantum interferences)

∆

2 m∗ = −qD /2∆

cx = −2∆/qD

Topological phase transition

niversal Hamiltonian is described by two independent parameters (left column) ed

IES OF THE UNIVERSAL HAMILTONIAN

•

σxx

Dependance in Δ of the conductances

e2 2E�2 2e 2 the2 topological 2 = FIG. 1: Evolution ofe�ã FIG. Evolution of the the quantity spectrum∆when the quantity ∆ is in varied changes in sign at e � v 2c transiti the 1: spectrum when is varied and changes signand at the topological transition (arbitrary F h mγ a ˜ units). The low-energy linear in the qy direction. c�ã σyy = units). The low-energy spectrum stays linearspectrum in the qystays direction.

h

1.2

γ

a ˜

∗ Without loss of generality, > 0.from When ∆ varies from negative positive values, a Without loss 1.5 of generality, we assume m∗ > we 0. assume When ∆mvaries negative to positive values, to a topological 1.0 transition from a semi-metallic phase with two Dirac insulator cones andwith a band insulator with aoccurs. gappedAt spectrum transition from a semi-metallic phase with two Dirac cones and a band a gapped spectrum transition, the spectrum is hybrid, a reminiscence point inphase, the semi-metallic phase, see the transition, the the spectrum is hybrid, a reminiscence of the saddle pointofinthe thesaddle semi-metallic see figure (1). 0.8 When ∆ < 0, the spectrum twoposition Dirac points thealong position which x axis is given b When ∆ < 0, the spectrum exhibits two Dirac exhibits points the of which the of x axis is along given the by ±q D with

1.0

qD

√ = −2m∗ ∆

qD

0.6 √ = −2m∗ ∆

(13)

0.4

0.5 uantity ∆ is varied inand sign atlinear the topological transition (arbitrary the spectrum Dirac points characterized by thethe velocity cx along andand the changes linear spectrum around these Diracaround points these is characterized byisthe velocity cx along x direction : the x directi n the qy direction. ! 0.2 ! −2∆ q D −2∆cx = qD = . . (14) cx = ∗ = ∗ ∗ m m ∗ ∗ ��� m m ��� m > 0. When ∆ varies from negative to positive values, a topological �1.0

�0.5

0.5

1.0

�1.0

�0.5

0.5

1.0

The two Dirac points are separatedoccurs. by a saddle two Dirac cones and a band insulator with a gapped spectrum At point at position qS = 0 whose energy is ∗±|∆|. The mass m

2 ∗ q , m∗ cx =∗ qD /m∗ ∆ 2 = −q ∗ D /2m qD , m∗ D cx = qD /m ∆ = −q /2m D q ,c m∗ = q /c ∆ = −c q /2 qD , cx D∗ m∗x = qD /cx D∗∆x= −cx qD /2 x∗ D2 m ,qcx = mq∗Dc = m ∆ cx= −m ∆∗= −m c /2 m∗ , cx c2 /2 p D x √ ∗ p √ ∗ ∗ m , ∆ c −2∆/m q ∗ x = D = ∗ −2m ∆ ∗ m , ∆ cx = −2∆/m qD = −2m ∆ ∗ qD = −2∆/c m = 2−2∆/c2x 4 cx , ∆ cxq,D∆= −2∆/c m∗x= −2∆/c x x ∗ 2 q m c 2 = −qD /2∆ D , ∗∆ x = −2∆/qD qD , ∆ m = −qD /2∆ cx = −2∆/qD

Conclusion • Significative difference with Dirac or non-relativistic excitations : anisotropic diffusion

∗ 2 ∗ qD , m∗ cx = qD /m ∆I:=In−q TABLE addition D /2m to the velocity cy , the universal Hamiltonian is described by two independent parameters (left column) TABLE I: In addition to the velocity cy , the universal Hamiltonian is described by two independent parameters (left column) from which other parameters be deduced qD , cx m∗ from = qDwhich /c ∆other =two −cparameters x two x qD /2 may bemay deduced m∗ , cx qD = m∗ cx ∆ = −m∗ c2 /2 p √ m∗ , ∆ cx = −2∆/m∗ qD = −2m∗ ∆ III. PROPERTIES OF THE UNIVERSAL HAMILTONIAN III. PROPERTIES OF THE UNIVERSAL HAMILTONIAN cx , ∆ qD = −2∆/cx m∗ = −2∆/c2x 2 qD , ∆ m∗ = −qD /2∆ cx = −2∆/qD

ty cy , the universal Hamiltonian is described by two independent parameters (left column) ay be deduced

•

Study of the topological phase transition

• •

Weak antilocalization (symplectic class)

ROPERTIES OF THE UNIVERSAL HAMILTONIAN

1: Evolution of the spectrum when the quantity ∆ is and varied and changes at the topological transition (arbitrary FIG. 1:FIG. Evolution of the spectrum when the quantity ∆ is varied changes in sign in at sign the topological transition (arbitrary The low-energy spectrum stays in linear the q direction. y units). units). The low-energy spectrum stays linear the qin direction. y

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∗ Without of generality, we assume > 0. When ∆ varies from negative to positive a topological Without loss ofloss generality, we assume m∗ > m 0. When ∆ varies from negative to positive values, values, a topological transition a semi-metallic phasetwo with two cones Dirac and cones and ainsulator band insulator a gapped spectrum At transition from afrom semi-metallic phase with Dirac a band with a with gapped spectrum occurs.occurs. At the transition, the spectrum is hybrid, a reminiscence of the saddle point in the semi-metallic phase, see figure (1). the transition, the spectrum is hybrid, a reminiscence of the saddle point in the semi-metallic phase, see figure (1). < 0, the spectrum exhibits two points Dirac points the position of along which the along the is x axis given ±qD with When When ∆ < 0,∆the spectrum exhibits two Dirac the position of which x axis givenisby ±qDbywith