!with: S. Syzranov, V. Gurarie

Transcription

1 Disorder-driven quantum transition in semiconductors and Dirac semimetals!with: S. Syzranov, V. Gurarie see: PRL, PRB 2015 $: NSF, Simons Foundation University of Utah, May 7, 2015

2 Outline Motivation Results Qualitative physics RG for dirty semiconductors and Dirac semimetals Critical density of states and transport Lifshitz tails Conclusions S. V. Syzranov, V. Gurarie, L. Radzihovsky, PRB 91, (2015) S. V. Syzranov, L. Radzihovsky, V. Gurarie, PRL 114, (2015)

3 Motivation Transport in lightly-doped (E F << W) conductors semiconductors, bilayer graphene: k k 2 graphene, Weyl semimetals, topological insulators: k k Cd 3 As 2

4 Motivation LETTERS PUBLISHED ONLINE: 29 JUNE 2014 DOI: /NMAT4023 Landau quantization and quasiparticle interference in the three-dimensional Dirac semimetal Cd3As2 Sangjun Jeon1, Brian B. Zhou1, Andras Gyenis1, Benjamin E. Feldman1, Itamar Kimchi2, C. Potter2,DOI: Quinn D. Gibson3, Robert J. Cava3, Ashvin Vishwanath2 and Ali Yazdani1* NATUREAndrew MATERIALS /NMAT4023 Condensed-matter systems provide a rich setting to realize d 1 2 Dirac and Majorana fermionic excitations as well as the As possibility to manipulatee them for potential applications3,4. It has recently been proposed that chiral, massless particles Cd known as Weyl fermions can emerge in certain bulk materials5,6 (above As plane) or in topological insulator multilayers7 and give rise to unusual Cd transport properties, such as charge pumping driven by (below As plane) a chiral anomaly8. A pair of Weyl fermions protected by + kd symmetry e ectively forming crystalline a massless Dirac kd c fermion has been predicted Γ to appear as low-energy excitations kz in a number of materials termed three-dimensional Dirac a = semimetals9 11. Here we report scanning tunnelling microscopy b = measurements at sub-kelvin temperatures and high magnetic II V semiconductor Cd3 As2. We probe this system c = fields on the k down to atomic length scales, and show that defects mostly influence the valence band, consistent with the observation of ultrahigh-mobility carriers in the conduction band. By combining Landau level spectroscopy and quasiparticle interference, we distinguish a large spin-splitting of the conduction band in a magnetic field and its extended Dirac-like dispersion above kz e the expected regime. A model band structure consistent with Several candidate materials, including Na3 Bi and Cd3 As2, were recently predicted10,11 to exhibit a bulk 3D Dirac semimetal phase with two Dirac points along the kz axis, stabilized by discrete rotational symmetry. Although photoemission measurements16 19 indeed observed conical dispersions away from certain points in the Brillouin zone of these materials, high energy resolution, atomically resolved spectroscopic measurements are needed to isolate the physics near the Dirac point and clarify the effect of material inhomogeneity on the low-energy Dirac behaviour. Lowtemperature scanning tunnelling microscopy (STM) experiments are therefore ideally suited to address these crucial details. Previously, Cd3 As2 has drawn attention for device applications owing to its extremely high room-temperature electron mobility20 (15,000 cm2 V 1 s 1 ), small optical bandgap20, and magnetoresistive properties21. The recent recognition that inverted band ordering driven by spin orbit coupling can foster non-trivial band topology renewed interest in Cd3 As2, which is the only II3 V2 semiconductor believed to have inverted bands. Updated ab initio calculations predict 3D Dirac points formed by band inversion between the conduction s-states, of mainly Cd-5s character, and the heavy-hole p-states, of mainly As-4p character11,22. However, the large unit cell

5 Results study: (ˆp) + V (r) = E at bottom of the band, E -> 0 S. V. Syzranov, V. Gurarie, L. Radzihovsky, PRB 91, (2015) S. V. Syzranov, L. Radzihovsky, V. Gurarie, PRL 114, (2015)

6 results: phase diagram non-interacting k k in random potential =2 d see also: SCBA: E. Fradkin 86 Ominato, Koshito, 14 graphene: Aleiner, Efetov, PRL 06 Ostrovsky, et al., PRB 06 WSM: Burkov, Balents PRL 11 Wan, et al, PRB 11 Goswami, et al., PRL 11 Nandkishore, et al., 14 Sbierski, et al., 14

7 results: critical density of states ρ(e) Weyl semimetal: z=d/2=3/2, ν=1/(d-2)=1 E d/z-1 ~ E E 2 γ-γ c -3/2 quantum critical t = γ-γ c γ-γ c zν E ignoring rare regions of strong disorder Lifshitz tails

8 results: critical conductivity Weyl semimetal: z=d/2=3/2, ν=1/(d-2)=1 D = 1 2 e2 v 2 = e2 v 2 2

9 results: nonanalytic mobility edge weakly-doped semiconductor 1 E E c z γ c 1 E E c A strength of disorder γ

10 Anderson localization noninteracting quantum motion in a random potential 2 2m 2 + V (r) = E V (r) r conventional wisdom (orthogonal) o d < 2: all states are localized o d > 2: localization-delocalization Anderson transition e r r 0 / ρ(e) smooth through E c 1 E E c A localized delocalized E c (Δ) mobility edge E

11 1, Lx 6ouss Wc. Z09 05 It o 8ox 116r,tO.~ [] Lorentz ~ 9 3.8~ 0.5 Mobility edge E c Z. Phys. B - Condensed Matter 66, (1987) Localization, Quantum Interference, and the Metal-Insulator Transition B. Bulka 1,, M. Schreiber 2, and B. Kramer 1 Physikalisch-Technische Bundesanstalt, Br 2 Insitut ffir Physik, Universit/it Dortmund, Received September 10, 1986 W!E] 05 wde.o) mobility edge E c (Δ) is a smooth function of disorder strength Δ ~ V EIV band edge at 6t Fig. l. Mobility edge trajectories Wc(E) for the box (O Wc(O)/V = 16.3 _+ 0.5), Gaussian (A Wc(O)/V _ 0.5), and the Lorentzian (I Wc(O)/V- 3.8 _+ 0.5) distribution

12 Model low-density electrons in a random potential, μ -> 0 (ˆp) + V (r) = E Z [d ]e is S = C (p) = dtd d r [ i t (ˆp) V (r)] p V (r)v (0) = (ˆp) = ˆp2 2m, semiconductor = v ˆp, Weyl semimetal treat via scaling, perturbation theory, and RG with ε=(2α-d)-expansion d (r)

13 Related problems Feynman s polaron coupled to a slow boson: o world-line self-avoiding polymer (degennes 72) H[r(t)] = 1 2 dt dr dt 2 + g dt dt d (r(t) r(t )) o singular corrections for d < 4 (=2α, α=2) o R G = (< r 2 >) 1/2 ~ L ν, ν = 1/2 + O(4-d) Interacting particles at low density (BCS-BEC): o exact T-matrix dg d =(2 d)g g2

14 The Physics via scaling (distinct from interference effects near Fermi surface: e.g. weak localization) Fermi liquids are protected by Fermi sea, with E F ~ W, qualitatively different for low doping, with E F << W Ψ V 0 ( /a) d ξ bound-state formation in random potential V 0 ( /a) d on scale ξ

15 ξ The Physics via scaling E V 0 ( /a) d/2 V(r) averaged over wavelength 1/k ~ ξ E K V eff d < 2α: disorder dominates -> bound state for arbitrarily weak V 0 loc = a!! V0 a 2 2 d 0 V 0 d > 2α: weak disorder irrelevant below critical V 0c = κ a -α = W 0 weak strong V 0 V 0c conventional mobility edge E a ξ loc ξ E loc weak-to-strong disorder quantum phase transition localized 1/ loc Lifshitz tails

16 scattering rate 1/τ(E): Perturbation theory S = C dtd d r [ i t (ˆp) V (r)] (E) d d q G(E) 1 (E q )q E d/ 1 1 (E) 1 E k E d/ 1 usually E-dependence ignored at E F Ioffe-Regel criterion γ < 1 vs γ > 1 : (E) (E) E E d/ 2 1 k (k) 1 kv(k) (k) d < 2α: (typical) disorder dominates at low E d > 2α: (typical) disorder is irrelevant at =(2 d) +...

17 Perturbation theory S = C dtd d r [ i t (ˆp) V (r)] Ioffe-Regel criterion, γ < 1 vs γ > 1 : (E) (E) E E d/ 2 1 k (k) 1 kv(k) (k) d < 2α: (typical) disorder dominates at low E d > 2α: (typical) disorder is irrelevant at low E (e.g., d > 4 =(2 d) γ What happens for stronger disorder for d > 2α? =(2 d) + c 2 0 γ weak γ c strong

18 RG conventional (Shankar) RG at Fermi surface E S = C dtd d r [ i t (ˆp) V (r)] E F W k f vacuum RG down to the bottom of the band E k W E F k f K 0 k

19 RG rescaling and tree-level RG flow: r = r b, t = t b z λ(b) = λ, κ(b) = κ b z-α, Δ(b) = Δ b 2z-d (b) = (b) (b) 2 = b2 d e momentum-shell RG flow: integrate out K 0 /b < k < K 0 S = C dtd d r [ i t (ˆp) V (r)] d d =, d d = +2 2

20 (b) = Mobility threshold & transition b 1 / c +( / c )b, (b) = (b) b 1/2, c = 2 renormalized Drude conductivity (γ < 1): D(b) = v(b)2 2 (b)b (d < 2α): >0 strong 0 γ(b) --> disorder dominates for k < K loc = K 0 1+ c 1/ --> mobility threshold of Anderson transition <0 (d > 2α) : weak γ c strong 0 γ(b) --> disorder-driven quantum transition T=0, μ->0

21 (d < 2α): >0 E c (γ) localized Mobility Edge strong 0 γ(b) conventional Anderson transition delocalized γ (d > 2α) : <0 E c (γ) weak γ c strong 0 γ(b) localized γ c delocalized nonanalytic mobility edge γ E c ( ) ( c) z, > c

22 ij( )= e2 2 Weyl semimetal: Conductivity de [n F (E) n F (E + )] d d r Trˆv ir G A (E +, r, r )ˆv jr G R (E,r, r) xx = e2 v 2 2 Tr ˆx = 1 2 e2 v 2 = e2 v 2 2 ˆx 1 = 2Im R = 2 = v 1 k + i 2

23 Scaling theory of critical transport (d > 2α): <0 c weak γ c strong 0 γ(b) 2 d (,T,µ) = (d 2) g(t z, µ/t ) (,0, 0) c (d 2) ( c,t,0) T (d 2)/z comparing with RG calculation, find (for 3d WSM):! z = 3/2, ν = -1/(2α-d) = 1 " ( E ~ k α /λ(k) ~ k d/2 ) see also: Goswami, Chakravarty, PRL 107, 2011

24 Density of states (E) = 1 Im dd r V G R(r, r,e) = (K E ) clean[ (K E )E] where: (b) = b 1 / c +( / c )b, (b) = (b) b 1/2, (K E )E K E

25 Scaling theory of density of states c, (E, ) k d k E k E 2 c E, for E c z z E d/z 1 f(e c 3/2, for E c z WSM: z=d/2=3/2 ν=1/(d-2)=1 z ) ρ(e) E Q critical t = γ-γ c E 2 γ-γ c zν ignoring rare regions of strong disorder Lifshitz tails (Nandkishore, et al.) E

26 Scaling theory of density of states c, E (E, ) k d k k z E d/z 1 f(e c semiconductor: z=α-ε/4, ν=-1/ε z ) quantum critical weak disorder γ c strong disorder t = γ-γ c

27 deeply lying states E < 0: Lifshitz tails (ˆp) + V (r) = E probability of V, size ξ: P (V ) e 1 R d d rv 2 e d ( / + E ) 2 V(r) Ψ ξ 1/ξ 2 E maximize over ξ: ρ(e) d < 2α: ξ ~ E-1/α --> (E) e E (2 d)/ Lifshitz tail E ρ(e) d > 2α: ξ ~ r 0 --> (E) e r d 2 0 our tail γ-γ c zν E

28 model: 1d chiral model M. Garttner, et al ĥ = ˆp sgn(ˆp)+v (x) E(k) -> long-range hopping J xx Ĥ = J xx â xâ x + V x â xâ x x,x x high-d transition in 1d: α <α c = d/2 = 1/2 o z = 1/2 k o ν = 1/(1-2α) o ρ(e) ~ E θ o θ = d/z 1 = 1, for γ = γ c o θ = d/α 1, for γ < γ c

29 Density of states 1d chiral model M. Garttner, et al ρ(e) ~ E θ θ = d/z 1 = 1, for γ = γ c θ = d/α 1, for γ < γ c W=0.9 DOS ρ(e) θ=0.93 θ=1.35 W=0.7 W= θ α =1/α-1=1.5 W=0.5 W= E

30 Density of states: 1d chiral model M. Garttner, et al ρ(e) ~ E θ θ

31 Summary and open questions studied transport in lightly-doped semiconductors, semimetals new weak-to-strong disorder-driven quantum phase transition RG and scaling theory --> conductivity, density of states 1 E E c z 1 E E c A numerics, experiments? role of rare regions near critical point? interplay with conventional Anderson localization? Coulomb impurities?

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33 Results: critical density of states c, (, )= d/z 1 f( 2 c z, for z z ) 3/2, for z WSM: z=d/2=3/2 ν=1/(d-2)=1 ρ(e) E E 2 δγ zν E ignoring rare regions of strong disorder Lifshitz tails (Nandkishore, et al.)

34 Results: critical conductivity of Weyl SM c, (, T) (d 2) g T z,µ z WSM: z=d/2=3/2 ν=1/(d-2)=1 c, at T =0 T 2/3, at = c z D = 1 2 e2 v 2 = e2 v 2 2

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