Ultrahigh Mobility and Giant Magnetoresistance in a Dirac Semimetal

Tian Liang, Jun Xiong, Minhao Liu, N. P. O. Department of Physics, Princeton Univ.

Quinn Gibson, R. J. Cava Department of Chemistry, Princeton Univ.

1. Out of Z2-land. Physics of Weyl semimetal

2. Dirac semimetal candidates, ARPES results Cd3As2

3. High mobility, giant MR and pronounced MR anisotropy

4. Role of TRI breaking, Zeeman energy,

5. Chiral anomaly effects?

Stanford U, UC Berkeley, 2013

Supported by ARO-MURI, ARO and NSF NSF

Out of Z2-land

Z2 Land Bi2Se3

Bi2Te2Se

HgTe-CdTe

Weyl semimetal Dirac metal

Kagome Pyrochlore

Large U solids

f electron systems

Berry phase, Chern

flux

Cd3As2 Na3Bi

HgCr2Se4

Eu2Ir2O7

HerbertSmithite

β-BiO2 Cu(1,3 carboxylate)

Bi2Te3

Topological Crystalline Insulator

PbSnSe

Topological ideas are migrating from Z2-TRI matter to other classes of matter. The virus is the Berry curvature (Chern flux)

Weyl Semimetal

• Superficially, 3D analog of graphene, but w important differences:

• No mass term Mσz possible (compare Boron Nitride), so always massless

• Only way to remove nodes is by pairwise annihilation

• Requires broken TRI or broken Inversion • Must come in pairs with χ = ±1

• Regard as monopole source or sink of Berry curvature (Chern) flux

2-spinor 3D space

Chirality χ

F

Berry curvature

Wan, Turner, Vishwanath, PRB 2011

qy

qx

E

Chern flux

If TRI is restored Weyl points of opposite chirality mutually annihilate Left with a gapped spectrum

However, if crystalline symmetry prevents annihilation, we have a 4-comp Dirac metal

β-crystobalite BiO2 A3Bi (A = Na, K, Rb) Cd3As2

Dirac semimetal (3D analog of graphene)

E(k)

Gapped spectrum

Young, Kane et al., PRL (2012) Wang, Dai et al., PRB (2012) Bernevig et al., (2013)

kx

kz

kz

Each node is a 4-comp massless Dirac point Has 4-fold degeneracy (lifted by breaking of TRI or Inv.) 3D Analog of graphene

Dirac

0

(001)

(111)

Fluorite lattice CaF2

Approx. (112) in Cd3As2

Wikipedia

(100) (010)

IOP, Beijing

|S,½⟩, |P,3/2 ⟩, |S,-½ ⟩, |P,-3/2 ⟩

4x4 basis set

0 kx

kz

4-comp Dirac Little group C4 symmetry

Borisenko, Cava, arXiv 2013

Borisenko, Cava, arXiv 2013

vF = (6 – 10) x 105 cm/s SdH oscillations provide more accurate meas.

16 x 105 cm/s (Hasan)

Borisenko, Cava, arXiv 2013

Ultrahigh mobility in Dirac semimetal Cd3As2

Harvested crystals come in 2 sets, A and B Set A --- 50 µΩcm at 4 K Set B --- 30 nµΩcm at 4 K Mobility µ 15,000 to 26,000 cm2/Vs in A 106 cm2/Vs in B

In same league as Bi 2DEG in GaAs

300 0

Resis

tivity

ρ

T(K)

Crystals in Set A Resistivity ~ 50 µΩcm at 4 K RRR ~ 5 Mobility µ = 15,000 to 26,000 cm2/Vs

Hall density nH ~ 2x1018 2x1019 cm-3

Only one band (?) of carriers (n-type) Giant H-linear transverse MR Expt difficulty: Large Hall voltage skews the MR curves

Resis

tivity

ρxx

300 0 T(K)

9 Tesla

B=0

Transport Properties in Set A, Cd3As2

-10 -5 0 5 100.0

5.0x102

1.0x103

1.5x103

2.0x103

2.5x103

3.0x103

3.5x103

2.5K 5K 10K 20K 40K 60K 80K 100K 150K 200K 250K 300Kρ xx

( µΩ

cm

)

µ0H (T)

At all T 300 K, MR is much larger than T dependence

Giant MR extends above room temp

Sample A1

300 K

2.5 K

Resis

tivity

ρxx

300 0 T(K)

-10 -5 0 5 100.0

2.0x10-4

4.0x10-4

6.0x10-4

8.0x10-4

1.0x10-3

1.2x10-3

1.4x10-3

1.6x10-3

1.8x10-3

2.0x10-3

-95 -80 -65 -50 -35 -20 -5 -4 -3 -2 -1 0 1 2 3 4 5 10 25 40 55 70 85 90

MR

(Ω

cm

)

µ0H (T)

T=2.5KSample 7

Extreme MR anisotropy

E perp H

E||H

Giant H-linear MR is suppressed when H is parallel to E Rxx increases by 50-100 (H=0 to 10 T)

Sample A2

Rxx

(Ω c

m)

θ H

θ=90o

Sample A2

Fermi velocity derived from SdH amplitude vs temp

Dingle temp cyclotron mass mc. For Dirac dispersion mc = E/v2

Mechanisms for giant MR

1. Large H-linear positive MR H has component perp to c Scales like σ(H,θ) = σ(H/sinθ, 0) |θ| > 20o

However, Zeeman energy also contributes to MR In weak H, Zeeman energy term dominates (MR is isotropic)

2. Charge pumping between Weyl nodes predicted to give negative MR

Masked by much larger transverse positive MR. Need higher H?

Scaling fails for angle θ < 20o

Simple scaling σ(H,θ) = σ (H/sinθ, 0) Good for θ > 20o

(dotted curves)

(purely orbital MR) Fails for θ < 20o

Because Zeeman energy is important at low H These 2 effects mask charge pumping channel

Charge pumping and the chiral anomaly

Charge is pumped from left to right-handed Weyl node at the rate

Chiral (Adler-Bell-Jackiv) anomaly

Decay of neutral pi-meson π0 2 ϒ, non-conservation of axial current Why are all neutrinos left-handed? Goldstone mode of quark condensate Topological origin

E(k) E B

Landau levels

kz 0

Q

χ=1 χ=-1

Review: Hosur and Qi, arXiv:1309

E(k) E

B Landau levels

kz 0

Q

χ=1 χ=-1

Evidence for charge pumping and chiral anomaly?

E||B

When E||H, chiral anomaly term is a maximum. Appears to cancel nearly all of the giant H-linear transverse MR. May dominate at larger H.

Charge pumping?

3/2 -3/2

1/2 -1/2

∆kz

Linear MR from Zeeman splitting of FS? Dai et al, PRL (2012) |S,½⟩, |P,3/2 ⟩, |S,-½ ⟩, |P,-3/2 ⟩ 4x4 basis

B field also breaks TRI Weyl nodes move apart, Lifshitz transition

-3/2

3/2 1/2

-1/2

∆kz

EF

3/2

-3/2

1/2

-1/2

E

Tests need higher B and higher mobility

Is scattering from impurities suppressed at H =0?

ρxx

T

B=0

4 T

8 T 12 T

Giant B-linear MR observed with E perp. to B (100-fold in 15 T) Nearly T-independent except in weak B Impurity scattering dominates phonons at finite B Protection against impurity scattering at B = 0 Why is mobility so high in a unit cell with 80 atoms?

ρxx

B

300 K 4 K

Giant MR in sample B1

H

I Sample B1

Trans. MR very large 1000-fold incr at 15 T Rxx ~ H2.5 not H-linear

May have a second n-type band Mobility µ > 106 cm2/Vs

Quantum oscillations in high-mobility sample B1

SdH oscillations

H-3

H perp I

Sample B1

Quantum oscillations in high-mobility sample B1

See spin split (or valley split) at high fields N= 3 level is anomalously low in conductivity

Summary

Dirac semimetal Cd3As2 Ultrahigh mobility µ > 106 cm2/Vs (Set B) Moderately high µ ~ 20,000 cm2/Vs (Set B) Both sets show giant positive MR (100 to 1000 fold incr. in 15 T) Approximate scaling for |θ| > 20o observed --- orbital effect MR is isotropic for weak H Are carriers protected from scattering when C4 symmetry prevails? Charge pumping process needs higher H

END

Nature Commun. 3, 982 (2012)

Start with the 3D massive Dirac Hamiltonian at L

Surface states protectd by cryst symm.

Nature Materials 11, 1023 (2012)

SuYang Xu, M. Z. Hasan et al. Nat. Commun. 2012

Gap closing near 150 K in PbSnSe observed by ARPES Suyang Xu, Hasan, unpub.

Issues in Pb-based IV-VI rocksalts

1. Evidence that the bulk states near L point are massive Dirac states?

2. Transport properties in inverted phase relatively unexplored Can resolve smaller energy scales than ARPES Probe effect of intense magnetic field

3. Effect of gap inversion on transport? Hall, thermopower and Nernst?

Curves of thermopower Sxx and Nernst Sxy vs. H at selected T

Fits at low H

1) Quantum osc. observ. at 80 K as shoulder 2) Fit semiclassical expressns remarkably well at low H 3) Nernst signal changes sign at 180 K

Fits v. well to semiclassical expressions at low H

Sxx rises to plateau value ADH when μB>>1 |Sxy| attains max value ½A(DH-D) at μB = 1

𝐷𝐷 = 𝜕𝜕 𝑙𝑙𝑙𝑙𝑙𝑙/𝜕𝜕𝜕𝜕 𝐷𝐷𝐻𝐻 = 𝜕𝜕 𝑙𝑙𝑙𝑙𝑙𝑙𝑥𝑥𝑥𝑥/𝜕𝜕𝜕𝜕

~ 270 T μV

Semi classical Mott expression

Can “predict” Nernst curves from Sxx alone except, the sign is “wrong”!

Consistency checks between transport and ARPES

SdH osc., SF = 6 T ne = 8.2×1016 cm-3 per pocket

Hall coef., ne = 8.7×1016 cm-3 Thermopower slope dS/dT EF = 51 meV SdH v = 5.7×105 cm/s ARPES EF = 20—40 meV v = 5.6×105 cm/s (hole band)

Hasan

Nernst changes sign at gap inversion

Hall sign unaffected Thermopower sign unaffected Only Nernst signal changes sign

Nernst

Hall

Thermopower Nernst

Quantum oscillations in thermopower and Nernst signals

Large Hall angle μB ~ 220 at 20 T Strong quantum oscillations in Sxx and Sxy Sxx shows pronounced “jump” at 7 T (N=10 transition) Spin splitting relatively weak

Index plot of quantum oscillations

Slope yields SF = 5.95 Tesla ne = 8 x 1016 cm-3 per pocket close agreement with Hall nH

Fermi Surface section

Period is independent of tilt angle 4 isotropic FS pockets

Non-degeneracy of the n=0 Landau level for massive Dirac fermions

Simplest case: 2D on honeycomb (boron nitride)

Missing levels

Missing N=0 Landau level at K for E<0

Degeneracy of n=0 Landau level

Since ne and B1 are known, we can determine gs

gs = spin degeneracy of each Landau level E00

Experiment yields gs = 1 for N=0 Landau level Non degenerate

Thermoelectric response in quantum limit

Weyl semi-metal H = px σx + py σy + pz σz

Basis is a 2-spinor in 3D k space Massless (topological, not accidental) Come in pairs with opp. chirality Chemical potential locked at Dirac points System must have broken TRI and Inversion sym.

Weyl points are monopoles of Berry curvature BB = x A

Weyl Semimetal – monopoles in k-space

BB

Proposed candidates Pyrochlore iridates (A2Ir2O7 (Vishwanath et al. 2010) HgCr2Se4 (Xi Dai, Z. Fang PRL 2011)

kz

ky kx Ferromagnetic state (broken TRI)

Vishwanath, Nat Phys 2010

Weyl points

ky kx

E

-10 -5 0 5 100.0

5.0x102

1.0x103

1.5x103

2.0x103

2.5x103

3.0x103

3.5x103

2.5K

ρ xx (µ

Ω c

m)

-3/2

3/2 1/2

-1/2

∆kz

EF

3/2

-3/2

1/2

-1/2

E

Summary In PbSnSe (x = 0.23), have a 4 small FS pockets with very high mobility. Nernst signal changes sign at gap inversion temp. The N = 0 Landau level is non-degenerate, consistent with massive Dirac Linear-H thermopower in quantum limit In Cd3As2, observe unusual giant, linear H magnetoresistance Consistent (?) with breaking TRI in Dirac topological semimetal