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Berry phase mediated Anomalous Thermoelectric and magnetic response in 2D Topological Insulators

Panagiotis Kotetes

Institut f¨ ur Theoretische Festk¨

• rperphysik, Karlsruhe Institute of Technology
• G. Sch¨
• n

(Karlsruhe Institute of Technology)

• A. Shnirman

(Karlsruhe Institute of Technology)

• G. Varelogiannis

(National Technical University of Athens)

NanoCTM meeting: Balaton Hungary, June 16, 2011

Karlsruhe Institute of Technology Panagiotis Kotetes Berry phase effects in 2D Topological Insulators

Making a long story, short...

Brief introduction to 2D Topological Insulators Sleuthing for unique fingerprints of Topology The emergence of an Anomalous Nernst effect and Orbital Magnetization Original Motivation: numerous experimental observations of a Giant Nernst signal in strongly correlated electronic systems The conditions for a Giant Nernst Signal in Chiral states of matter Anomalous thermoelectricity and magnetic response in planar Topological Semiconductors

Karlsruhe Institute of Technology Panagiotis Kotetes Berry phase effects in 2D Topological Insulators

2D Topological Insulators and bulk-boundary correspondence

Definition of a Topological Insulator A state characterized by Topologically protected edge modes 2 fundamental systems: Anomalous Quantum Spin Hall Insulator → Time-Reversal Anomalous Quantum Hall Insulator → No Time-Reversal

X.-L. Qi and S.-C. Zhang, Physics Today Karlsruhe Institute of Technology Panagiotis Kotetes Berry phase effects in 2D Topological Insulators

2D T-invariant Topological HgTe-QW Semiconductors

The Hamiltonian must include the following features Spin-Orbit coupling Band Gap near the Γ point Time-reversal symmetry

Karlsruhe Institute of Technology Panagiotis Kotetes Berry phase effects in 2D Topological Insulators

BHZ Quantum Spin Hall Insulator lattice model

BHZ lattice model Hamiltonian: H(k) =

• H(k)
• H∗(−k)
• ,

    |k, ml = 0, ms = + 1

2 >

|k, ml = +1, ms = + 1

2 >

|k, ml = 0, ms = − 1

2 >

|k, ml = −1, ms = − 1

2 >

    with

• H(k) = ε(k) + g(k) · τ

where ε(k) = C − 2D(2 − cos kx − cos ky) g(k) =

• A sin kx, A sin ky, −2B
• − M

2B + 2 − cos kx − cos ky

• Band Gap at the Γ point 2E(0) ≡ 2|g(0)| = 2|M|
• B. A. Bernevig, T. L. Hughes, S.-C. Zhang, Science 314, 1757 (2006)

Karlsruhe Institute of Technology Panagiotis Kotetes Berry phase effects in 2D Topological Insulators

Berry Curvature in k-space and Topological charge

For each block, the avoided band touching at the Γ point

• X

M

• Energy Bands

generates a finite Berry Curvature for each band in a block Ωz

ν(k) = −ν

2 ˆ g(k) · ∂ˆ g(k) ∂kx × ∂ˆ g(k) ∂ky

• ν = ±

The Topological (Monopole) Charge that sources the Berry Curvature is equal to

• N = − 1

2π

• d2k Ωz

−(k) = 1

and provides the # of protected edge modes per block!

Karlsruhe Institute of Technology Panagiotis Kotetes Berry phase effects in 2D Topological Insulators

Berry phase effects on thermoelectric transport and magnetic response

The finite Berry curvature acts as a k−dependent magnetic field leading to Anomalous Charge Hall effect with Hall conductivity σxy = −e2

• 1

N

• k,ν

Ωz

ν(k)nF[Eν(k)]

For T = 0 and µ = 0 σxy = −n e2

h , n = 1

Anomalous thermoelectric effect with Hall conductivity αxy = e TN

• k,ν

Ωz

ν(k)

• Eν(k)nF[Eν(k)] + kBT ln
• 1 + e−βEν(k)

Finite Orbital Magnetization Morb = e N

• k,ν

Ωz

ν(k)

• E(k)nF[Eν(k)] + kBT ln
• 1 + e−βEν(k)

Karlsruhe Institute of Technology Panagiotis Kotetes Berry phase effects in 2D Topological Insulators

Berry-Curvature-originating fingerprints, for detecting Topological Order in 2D

Anomalous Quantum Hall state: Single block Hamiltonian! Anomalous Hall effect Anomalous charge Thermoelectric effect Finite Orbital Magnetization Anomalous Quantum Spin Hall state: Two block Hamiltonian with opposite Berry curvature per block! Anomalous Spin Hall effect Anomalous Spin Thermoelectric effect Finite additional Zeeman Magnetization due to Orbital effects!

Karlsruhe Institute of Technology Panagiotis Kotetes Berry phase effects in 2D Topological Insulators

Thermoelectric Transport and Nernst Signal

Constitutive relations for thermoelectric charge transport Jx = σxxEx + σxyEy + αxx (−∂xT) Jy = σyxEx + σyyEy + αyx (−∂xT) with J ≡ charge current, E ≡ electric field, T ≡ temperature Thermopower S ⇒ longitudinal voltage appearing for J = 0 Nernst signal N ⇒ transverse voltage appearing for J = 0 S ≡ Ex ∂xT and N ≡ Ey −∂xT Anomalous N → Bz = 0

Karlsruhe Institute of Technology Panagiotis Kotetes Berry phase effects in 2D Topological Insulators

Quasiparticle and Vortex sources of a Nernst signal

Quasiparticles

1

Transverse velocity due to the Lorentz force ⇒ N ∼ Bz

2

Nernst signal takes both signs depending on Doping

3

Nernst signal strongly linear in Temperature

4

Single band metals show a tiny Nernst signal ∼ nV/K due to Sonheimer cancellation

• E. H. Sondheimer, Proc. R. Soc. London, Ser. A 193, 484 (1948)

Superconducting Vortices

1

Normal Core Entropy + Vortex attached Flux ⇒ αxy = 0 ⇒ N ∼ Bz, B. D. Josephson, Physics Letters 16, 242 (1965)

2

Only Positive Nernst signal !!!!

3

Nernst signal non-linear in Temperature

Karlsruhe Institute of Technology Panagiotis Kotetes Berry phase effects in 2D Topological Insulators

Chirality driven Nernst signal

Chirality ≡ Finite Angular Momentum

1

Violation of Time-Reversal ⇒ σxy(Bz = 0) = 0 and αxy(Bz = 0) = 0 ⇒ Anomalous Hall + Nernst Effects!

2

“Magnetic-field” in k-space: the Berry curvature Ωz(k).

3

The Nernst signal takes both signs !!!!

4

Large Fermi-Surface ⇒ N linear in Temperature Examples

CuCr2Se4−xBrx: Spinel Ferromagnet + Spin-Orbit coupling

Wei-Li Lee, S. Watauchi, V. L. Miller, R. J. Cava and N. P. Ong, Science 303, 1647 (2004)

Heavily-Doped Chiral dxy + idx2−y2 Density Wave

• C. Zhang, S. Tewari, V. M. Yakovenko and S. Das Sarma, Phys. Rev. B 78, 174508 (2008)

But what happens in the Strongly Insulating limit????? Chirality Induced Tilted-Hill Giant Nernst Signal: PK and G. Varelogiannis, PRL 104, 106404 (2010)

Karlsruhe Institute of Technology Panagiotis Kotetes Berry phase effects in 2D Topological Insulators

Giant Tilted-Hill Nernst signal in High-Tc cuprates

Giant N in Pseudogap + Superconducting regimes Tilted-Hill (peaked) temperature profile Positive Nernst signal Enhanced Diamagnetism in the pseudogap phase Diamagnetism scales with the Nernst signal

Yayu Wang, Lu Li, and N. P. Ong, PRB 73, 024510 (2006) Karlsruhe Institute of Technology Panagiotis Kotetes Berry phase effects in 2D Topological Insulators

Giant Tilted-Hill Nernst signal in the heavy fermion compound URu2Si2

The non-SC order in the phase diagram ≡ “Hidden Order” (HO) Giant N in the Hidden Order Tilted-Hill temperature profile No Diamagnetism! For low T, the HO condenses in a SC state, possibly Topological

• Y. S. Oh et al, PRL 98, 016401 (2007)
• R. Bel, H. Jin, K. Behnia, J. Flouquet

and P. Lejay, PRB 70, 220501 (2004) Karlsruhe Institute of Technology Panagiotis Kotetes Berry phase effects in 2D Topological Insulators

Chiral dxy + idx2−y 2 Density Wave

Chiral D-Density waves have been recently proposed for understanding the Pseudogap regime in the cuprates (PK and G. Varelogiannis 2008 & S. Tewari et al. 2008) and the Hidden Order (PK, A. Aperis and G. Varelogiannis 2010) The very-same interactions promoting unconventional superconductivity, also favour Chiral Density Wave formation Half-filled single band square lattice model: H0 = −2t

• k

(cos kx + cos ky)c†

kck

Enhanced tendency towards an Insulating Chiral D-Density Wave due to perfect nesting Formation of a Topological Insulating Condensate

Karlsruhe Institute of Technology Panagiotis Kotetes Berry phase effects in 2D Topological Insulators

Pairing interactions and Mean-field decoupling

Intersite extended Hubbard interactions up to n.n.n. Hint =

• <<i, j>>
• Vij ni nj + Jij

Si · Sj

• Driving effective interaction

∼

• k,k′ Vk,k′c†

kck+Qc† k′+Qck′

Chiral d-density wave “Anomalous” Terms ∆(k)c†

kck+Q + h.c.

Chiral D-Density Wave Order Parameter: ∆(k) ∼

k′ Vk,k′ < c† k′+Qck′ >

⇒ ∆(k) = ∆1 sin kx sin ky − i∆2(cos kx − cos ky)

Karlsruhe Institute of Technology Panagiotis Kotetes Berry phase effects in 2D Topological Insulators

Mean-field Hamiltonian of a chiral d-density wave

Nambu isospinor Ψ†

k = (c† k c† k+Q), k ∈ reduced B.Z.

We obtain a pseudospin-1

2 system for each k-point

H(k) =

• ε(k) + g3(k)

g1(k) − ig2(k) g1(k) + ig2(k) ε(k) − g3(k)

• = ε(k)Iτ + g(k) · τ

g1(k) = ∆1 sin kx sin ky, g2(k) = ∆2(cos kx − cos ky), g3(k) = −2t(cos kx + cos ky) and ε(k) = −µ. 2-Band Energy Spectrum: ν = ± → Eν(k) = ε(k) + ν|g(k)|

Karlsruhe Institute of Technology Panagiotis Kotetes Berry phase effects in 2D Topological Insulators

Steps for calculating the Tilted-Hill Giant Nernst signal

1 We obtain self-consistently the Chiral Order Parameter PK and G. Varelogiannis, PRB 80, 212401 (2009) 2 Anomalous thermoelectric Hall conductivity

αxy = e TN

• k,ν

Ωz

ν(k)

• Eν(k)nF[Eν(k)] + kBT ln
• 1 + e−βEν(k)

3 σxx and αxx in the Boltzmann approximation

σxx = −e2

• 1

N

• k,ν

n′

F [Eν(k)] τν(k)

• ∂Eν(k)

∂kx 2 αxx = +ekB

• 1

N

• k,ν

n′

F [Eν(k)] τν(k)

• ∂Eν(k)

∂kx 2 Eν(k) kBT

4 We ignore quasiparticle Hall conductivities 5 For finite Bz = 0, Eν(k), τν(k), DOS get modified

• D. Xiao, M.-C. Chang and Q. Niu, Rev. Mod. Phys. 82, 1959 (2010)

Karlsruhe Institute of Technology Panagiotis Kotetes Berry phase effects in 2D Topological Insulators

Mechanism of the Chirality Induced Tilted-Hill Giant Nernst signal

For low temperatures, the strongly insulating character leads to the condition σxx << σxy providing S ≃ αxy/σxy and N ≃ −αxx/σxy. A Thermoelectric crossing point emerges at σxx = σxy, where S = N. After the the crossing, σxx >> σxy, provides S ≃ αxx/σxx, N ≃ αxy/σxx. Crucial: N = S. The usually high values of S, unavoidably lead to an enhancement of the Nernst voltage. S = αxxσyy + αxyσxy σxxσyy + σ2

xy

N = σxxαxy − αxxσxy σxxσyy + σ2

xy

(t = 250meV, µ = 0, a = 5˚ A, dx2−y2 = 53meV, τ = 10−13s, Bz = 5T and dxy ≃ 22meV) 120 x Hall Π2 N ΜVK S ΜVK

I II

15 20 25 30 35 40 45 20 40 60 80 100 120

TK

a.

PK and G. Varelogiannis, PRL 104, 106404 (2010) Karlsruhe Institute of Technology Panagiotis Kotetes Berry phase effects in 2D Topological Insulators

Properties of the Novel Anomalous Nernst Effect

Emergence of a Thermoelectric Point Around the vicinity of this point, the Nernst signal exhibits a peak due to the crossover behaviour, leading to the Tilted-Hill profile The Nernst signal may be inverted by tuning the chemical potential or the doping of the sample

Μ4.8meV Μ2.9meV Μ0.9meV Μ1meV Μ3meV

20 40 60 80 100 100 50 50 100

TK Nernst signal ΜVK

b.

Karlsruhe Institute of Technology Panagiotis Kotetes Berry phase effects in 2D Topological Insulators

Prediction of a giant Nernst Signal in the Quantum Anomalous Hall state

To engineer the long-sought Quantum Anomalous Hall state we can start from either a planar Anomalous Quantum Spin Hall state or by the surface of a T -invariant 3D Topological Insulator which is described by a helical-liquid H(k) = −µ + kyσx − kxσy . In both case T -invariance must become violated via the following routes

1 magnetic impurities 2 perpendicular to the surface Zeeman field 3 Ferromagnetic coating

T -violation leads to an effective BHZ single block model

Karlsruhe Institute of Technology Panagiotis Kotetes Berry phase effects in 2D Topological Insulators

Results on the Anomalous Nernst signal in the Anomalous Quantum Hall state

We observe that the Anomalous Nernst signal becomes giant

• f the order mV /K

There exist a thermoelectric crossing point where N = |S|. The Nernst response may be directly tuned by the doping of the system

M1.2meV CΜ1meV N S

5 10 15 20 25 30 50 100 150 200

TK N, S ΜVK CΜ1meV CΜ1meV CΜ0meV

5 10 15 20 25 30 100 50 50 100

TK Nernst Signal ΜVK M = −1.2meV , τ = 10−13s, a = 0.65nm Karlsruhe Institute of Technology Panagiotis Kotetes Berry phase effects in 2D Topological Insulators

Berry Curvature driven Zeeman magnetization in the Anomalous Quantum Spin Hall state

The additional contribution to the Zeeman magnetization can be in principle detected due to the temperature dependence that it demonstrates (although weak) its doping dependence that controls its sign!

CΜ52.6meV CΜ26.3meV CΜ0meV

50 100 150 200 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

TK Orbital Magnetization ΜB

Karlsruhe Institute of Technology Panagiotis Kotetes Berry phase effects in 2D Topological Insulators

Conclusions

We demonstrated a novel source of Giant Thermoelectricity that is dictated by a Tilted-Hill temperature profile and

• riginates form a well insulated Chiral state in the case of

strongly correlated systems The Anomalous Quantum Hall state could be detected due to a giant Nernst signal and an accompanying Thermoelectric crossing point The Berry Curvature induced Zeeman magnetization constitutes a magnetofingerprint for the detection of an Anomalous Quantum Spin Hall state through its temperature and doping dependence

Karlsruhe Institute of Technology Panagiotis Kotetes Berry phase effects in 2D Topological Insulators

Thanks for your attention!

Karlsruhe Institute of Technology Panagiotis Kotetes Berry phase effects in 2D Topological Insulators