QUANTUM THEORY OF SPIN AND ANOMALOUS HALL EFFECTS IN GRAPHENE

Mirco Milletarì

Recent Progress in Spintronics of 2D Materials Hsinchu, Taiwan, 14/11/2016

Phys. Rev. B 94, 134202 (2016)

Phys. Rev. B 94, 201402(R) (2016)

COLLABORATOR

Aires Ferreira University of York, UK

OUTLINE

•Spintronics and Hall effects

•Why Graphene?

•Full Quantum approach (theory)

HALL EFFECTS (CLASSICAL): A BRIEF SURVEYHall

Hall Voltage No Spin Currents

H

E

•Lorentz Force

•Metals…

•No Time Reversal

Anomalous Hall

Hall Voltage Spin Polarized Current

M

E

•Spin-orbit+imbalance

•Ferromagnets…

•No Time Reversal

Spin Hall

No Hall Voltage Pure Spin Current

E

•Spin-orbit

•SOC materials

•Time Reversal

WHY GRAPHENE ? Open surface + carbon chemistry: engineering of SOC

(comparable to diamond)�SO ' 10meV

Local SOC: Lattice distortion

Avsar et al (Nat. Comm. 5, 2014)

Global SOC: Proximity effect to 2D TMDs

WHY GRAPHENE ? - RESONANT SKEW SCATTERING -Spin dependent scattering: semiclassical contribution �SH / ⌧SS

Skew scattering in a semimetal is strongly energy dependent

vanishing DOS

⌫0(✏)

✏

Wehling et al (PRL 105, 2010) Stauber et al (PRB 76, 2007) Ferreira et al (PRB 83, 2011)

Electron scattering easily resonate with the impurity

T (✏) =1

1V0

�GR0 (✏)

GR0 (✏) = F0(✏)� i⇡ ⌫0(✏)

First Born approximation is never truly valid in graphene!

RESONANT SKEW SCATTERING: AN EXAMPLE

Strong left/right asymmetry near resonances! 0.72 0.78 0.84V0 (eV)

-0.05

0

0.05

0.1

γ

SOC ~ 10 meV

Proximity effect to nanoparticles/clustersFerreira et al (PRL 112, 2014)

Skewness (spin Hall angle at T=0)

0.72 0.78 0.840

5

10

(u

nits

of R

)

V0 (eV)

σ

EF = 0.1 eV

�So

= 25meV

QUANTUM SIDE JUMP: SEMICLASSICAL PICTURE

The side-shift is due to the “anomalous” velocity (Karlplus and Luttinger 1954)

�v / � ^rVimp

…that is related to the Berry connection

see e.g. Sinitsyn et al. PRB 73, 075318 (2006) & 75 045315 (2010)

�rk0,k = �r+1 � �r�1 / � ^ �k

SS ⇠ 1/nimp QSJ ⇠ n0imp

At the semiclassical level, the two processes are treated as independent…

…however, this is never true quantum mechanically!

QUANTUM SIDEJUMP V.S. SKEW SCATTERING

Q: Is there a crossover between a SS and a QSJ dominated regime at fixed impurity density?

Q: What is the validity of the commonly employed Gaussian approximation?

intrinsic-like, SO componentscalar component

MODEL SYSTEM: ADATOM DECORATED GRAPHENE

M = ↵0 �0 + ↵3 �3

moments [20] which potentially spoil the time-reversalsymmetry protecting the QSH effect. Moreover, adatomsgenerically mediate both intrinsic and Rashba spin-orbitcoupling. The latter is believed to be detrimental to theQSH phase [6], and previous work has indeed establishedthat certain kinds of adatoms do generate substantialRashba coupling in graphene that typically overwhelmsthe intrinsic contribution [21–23]. (As Ref. [23] showed,however, magnetic adatoms inducing strong Rashba cou-pling can induce an interesting ‘‘quantum anomalousHall’’ state in graphene.) The adatoms may also favorcompeting, ordinary insulating states depending on theirprecise locations in the lattice. And finally, since spin-orbitcoupling is generated nonuniformly in graphene, the stabi-lization of a QSH phase even in an otherwise ideal situationis unclear a priori.

After an extensive search using tight-binding and first-principles analyses, we have found that two elements,indium and thallium, are capable of stabilizing a robustQSH state in graphene. Neither element forms a magneticmoment, and although they do generate significant Rashbacoupling, for symmetry reasons this remarkably does notsuppress the QSH state. We find that gaps many orders ofmagnitude larger than that predicted in pure graphene canform even with coverages of only a few percent; forexample, at 6% coverage, indium yields a gap on the orderof 100 K, while for thallium the gap approaches roomtemperature. These predictions revive graphene as a viableQSH candidate, and they can be verified by probing the gapand associated edge states using spectroscopic and con-ductance measurements.

II. PHYSICS OFA SINGLE HEAVYADATOM

To set the stage, let us first briefly review the Kane-Melemodel [6] describing pure, undoped graphene with spin-orbit coupling. The Hamiltonian can be expressed asHKM ¼ Ht þHso, where Ht describes the usual nearest-neighbor hopping and Hso encodes intrinsic spin-orbitcoupling. In terms of operators cyr! that add electronswith spin ! to site r of the honeycomb lattice and Paulimatrices sx;y;z that act on the spin indices, Ht and Hso

explicitly read

Ht ¼ #tX

hrr0iðcyr cr0 þ H:c:Þ; (1)

Hso ¼ "so

X

hhrr0iiði#rr0c

yr szcr0 þ H:c:Þ: (2)

Here and below, spin indices are implicitly summed when-ever suppressed. In Eq. (2) the sum runs over second-nearest-neighbor lattice sites, and #rr0 are signs that equalþ1, if an electron hops in the direction of the arrows inFig. 1(c), and #1 otherwise. Thus Hso describes ‘‘chiral’’spin-dependent second-neighbor electron hopping. When"so ¼ 0, the band structure exhibits the familiar gapless

Dirac cones centered on momenta &Q, resulting in semi-metallic behavior. Turning on "so ! 0 generates an energygap [6] ! ¼ 6

ffiffiffi3

pj"soj at the Dirac points, transforming the

system into a (very fragile [7–11]) QSH insulator.Importantly, if mirror symmetry with respect to thegraphene plane is broken, then Rashba coupling—whichinvolves spin flips and thus breaks the U(1) spin symmetryenjoyed by H—will also be present [6]. Rashba couplingcompetes with the intrinsic spin-orbit term in pure gra-phene, and, beyond a critical value, it closes the gap anddestroys the QSH state.If heavy adatoms are to stabilize a more robust QSH

phase in graphene, then, at a minimum, they should benonmagnetic (to preserve T ) and modify the physics nearthe Dirac points primarily by inducing large intrinsic spin-orbit coupling. The latter criterion leads us to focus onelements favoring the ‘‘hollow’’ (H) position in the gra-phene sheet indicated in Fig. 1(a). Compared to the‘‘bridge’’ (B) and ‘‘top’’ (T) positions, adatoms in theH position can most effectively mediate the spin-dependentsecond-neighbor hoppings present in the Kane-Melemodel, while simultaneously avoiding larger competingeffects such as local sublattice symmetry-breaking gener-ated in the T case.

b) c)

a)

T

H

4

2

B

16

5

3

FIG. 1. Adatoms in graphene. (a) Depending on the element,adatoms favor either the high-symmetry ‘‘bridge’’ (B), ‘‘hollow’’(H), or ‘‘top’’ (T) position in the graphene sheet. (b) Detailedview of an H-position adatom, which is best suited for inducingthe intrinsic spin-orbit coupling necessary for stabilizing thetopological phase. The desired spin-orbit terms mediated bythe adatom are illustrated in (c). Red and yellow bonds representthe induced second-neighbor imaginary hopping, whose sign isindicated by the arrows for spin-up electrons. For spin-downelectrons, the arrows are reversed.

WEEKS et al. PHYS. REV. X 1, 021001 (2011)

021001-2

moments [20] which potentially spoil the time-reversalsymmetry protecting the QSH effect. Moreover, adatomsgenerically mediate both intrinsic and Rashba spin-orbitcoupling. The latter is believed to be detrimental to theQSH phase [6], and previous work has indeed establishedthat certain kinds of adatoms do generate substantialRashba coupling in graphene that typically overwhelmsthe intrinsic contribution [21–23]. (As Ref. [23] showed,however, magnetic adatoms inducing strong Rashba cou-pling can induce an interesting ‘‘quantum anomalousHall’’ state in graphene.) The adatoms may also favorcompeting, ordinary insulating states depending on theirprecise locations in the lattice. And finally, since spin-orbitcoupling is generated nonuniformly in graphene, the stabi-lization of a QSH phase even in an otherwise ideal situationis unclear a priori.

After an extensive search using tight-binding and first-principles analyses, we have found that two elements,indium and thallium, are capable of stabilizing a robustQSH state in graphene. Neither element forms a magneticmoment, and although they do generate significant Rashbacoupling, for symmetry reasons this remarkably does notsuppress the QSH state. We find that gaps many orders ofmagnitude larger than that predicted in pure graphene canform even with coverages of only a few percent; forexample, at 6% coverage, indium yields a gap on the orderof 100 K, while for thallium the gap approaches roomtemperature. These predictions revive graphene as a viableQSH candidate, and they can be verified by probing the gapand associated edge states using spectroscopic and con-ductance measurements.

II. PHYSICS OFA SINGLE HEAVYADATOM

To set the stage, let us first briefly review the Kane-Melemodel [6] describing pure, undoped graphene with spin-orbit coupling. The Hamiltonian can be expressed asHKM ¼ Ht þHso, where Ht describes the usual nearest-neighbor hopping and Hso encodes intrinsic spin-orbitcoupling. In terms of operators cyr! that add electronswith spin ! to site r of the honeycomb lattice and Paulimatrices sx;y;z that act on the spin indices, Ht and Hso

explicitly read

Ht ¼ #tX

hrr0iðcyr cr0 þ H:c:Þ; (1)

Hso ¼ "so

X

hhrr0iiði#rr0c

yr szcr0 þ H:c:Þ: (2)

Here and below, spin indices are implicitly summed when-ever suppressed. In Eq. (2) the sum runs over second-nearest-neighbor lattice sites, and #rr0 are signs that equalþ1, if an electron hops in the direction of the arrows inFig. 1(c), and #1 otherwise. Thus Hso describes ‘‘chiral’’spin-dependent second-neighbor electron hopping. When"so ¼ 0, the band structure exhibits the familiar gapless

Dirac cones centered on momenta &Q, resulting in semi-metallic behavior. Turning on "so ! 0 generates an energygap [6] ! ¼ 6

ffiffiffi3

pj"soj at the Dirac points, transforming the

system into a (very fragile [7–11]) QSH insulator.Importantly, if mirror symmetry with respect to thegraphene plane is broken, then Rashba coupling—whichinvolves spin flips and thus breaks the U(1) spin symmetryenjoyed by H—will also be present [6]. Rashba couplingcompetes with the intrinsic spin-orbit term in pure gra-phene, and, beyond a critical value, it closes the gap anddestroys the QSH state.If heavy adatoms are to stabilize a more robust QSH

phase in graphene, then, at a minimum, they should benonmagnetic (to preserve T ) and modify the physics nearthe Dirac points primarily by inducing large intrinsic spin-orbit coupling. The latter criterion leads us to focus onelements favoring the ‘‘hollow’’ (H) position in the gra-phene sheet indicated in Fig. 1(a). Compared to the‘‘bridge’’ (B) and ‘‘top’’ (T) positions, adatoms in theH position can most effectively mediate the spin-dependentsecond-neighbor hoppings present in the Kane-Melemodel, while simultaneously avoiding larger competingeffects such as local sublattice symmetry-breaking gener-ated in the T case.

b) c)

a)

T

H

4

2

B

16

5

3

FIG. 1. Adatoms in graphene. (a) Depending on the element,adatoms favor either the high-symmetry ‘‘bridge’’ (B), ‘‘hollow’’(H), or ‘‘top’’ (T) position in the graphene sheet. (b) Detailedview of an H-position adatom, which is best suited for inducingthe intrinsic spin-orbit coupling necessary for stabilizing thetopological phase. The desired spin-orbit terms mediated bythe adatom are illustrated in (c). Red and yellow bonds representthe induced second-neighbor imaginary hopping, whose sign isindicated by the arrows for spin-up electrons. For spin-downelectrons, the arrows are reversed.

WEEKS et al. PHYS. REV. X 1, 021001 (2011)

021001-2

Weeks et al. PRX 1, 021001 (2011), Pachoud et al. PRB 90, 035444 (2014)

For p-orbitals adatoms, only intrinsic like SO coupling.

Hamiltonian density

H = †(x)n

�i v �a@a � �0 ✏+ V (x)o

(x)

Random Potential

V (x) = R2 MNiX

i=1

�(x� yi)

The effective low energy theory is

DISORDER AVERAGE: WHAT DO WE REALLY WANT TO DESCRIBE?

M = ↵0 �0 + ↵3 �3

V.S.

P is a Matrix valued distribution. In this case, Born approximation always fails S. Hikami and A. Zee, Nuclear Physics B 446, 337 (1995)

Mred = ↵0 �0 Mblue = ↵3 �3

Correlated random variablesP (M) 6= P (Mred)P (Mblue)

Independent random variablesP (M) = P (Mred)P (Mblue)

Model BModel A

THE DIAGRAMMATIC APPROACH (LRT)

jzy vx

vxj zy

+= +

+ + + +...

(a)

(b)

j zy vx(a)

= +δvx

(b)

= +δvx vx δvx

δvx vx

SH-conductivity

Correlated processes (model A) are described by the complete series of non-crossing diagrams. See also Chunli’s talk afterwards, PRB 94, 085414 (2016)

T

T ⇤

+=

+ ...+ +

+

+

�SH =✏ �v202n v ⌘

+n✏ �v22 + 2 (v + �v10) ⌘

2 v ⌘� �v20

✓

1

⇡v+

⌘m

2 v ⌘2

◆

o

⌘ S(✏)/n+Q(✏)

skew scattering

Anomalous

A

B

Diffusion process: incoherent scattering

kF l � 1

All order SH-conductivity

Milletari & Ferreira PRB 94, 134202 & PRB 94 201402(R) (2016)

CROSSOVER

0.25% (atomic ratio)nimp = 1013cm�2

R = 4nm

↵3 = 0.01 eV

Crossover is only revealed within the self consistent approach

Milletari & Ferreira PRB 94, 134202 & PRB 94 201402(R) (2016)

QUANTUM COHERENT PROCESSES

JzyJz

y vxv

x

Jzy v

x

vx vxb

+j zy j zy

avxj zy

vxj zy vxj zy+

+

In the AHE at Gaussian level: Ado et al. EPL 111, 37004 (2015)

Gaussian

(eV )

Coherent SS dominates the anomalous contribution. Absent in Gaussian!

(eV )

1

2�XSH

jzy

vx

kF |x� x

0| . 1

O(n0imp)

Naively O(n2imp)

MM & Ferreira PRB 94, 134202 & PRB 94 201402(R)(2016)

AHE IN MAGNETISED GRAPHENEYIG-graphene heterostructure

Z. Wang et al, PRL 114, 016603 (2015)

optical image of a graphene device on YIG/GGG beforethe top gate is fabricated. Room-temperature Raman spec-troscopy is performed at different stages of the devicefabrication. Representative spectra are shown in Fig. 1(c)for the same graphene device on SiO2 (before transfer)and YIG (after transfer), and for YIG/GGG only. Graphene/YIG shows both the characteristic E2g (∼1580 cm−1) and2D peaks (∼2700 cm−1) of single-layer graphene as well asYIG’s own peaks, suggesting successful transfer. We alsonote that the transfer process does not produce any meas-urable D peak (∼1350 cm−1) associated with defects [17].Figure 1(d) is a schematic drawing of a top-gated transferreddevice on YIG/GGG.Low-temperature transport measurements are performed

in Quantum Design’s physical property measurementsystem. Figure 2(a) is a plot of the gate voltage dependenceof the four-terminal electrical conductivity scaled by theeffective capacitance per unit area Cs. Since different gatedielectrics are used in the back- and top-gated graphenedevices, Cs is calculated based on the quantum Hall datawhich agrees with the calculated value using the nominaldielectric constant and the measured dielectric film thick-ness. Before transfer, the Dirac point is at ∼ − 9 V and thefield-effect mobility is ∼6000 cm2=V · s. After transfer, theDirac point is shifted to ∼ − 18 V. The slope of the σxx=Csvs Vg curve increases somewhat, indicating slightly highermobility, which suggests that the transfer process, the YIGsubstrate, and the top-gate dielectric do not cause anyadverse effect on graphene mobility. At 2 K, the mobilityimproves further, exceeding 10 000 cm2=V:s on the elec-tron side. Well-defined longitudinal resistance peaks andquantum Hall plateaus are both present at 8 T as shownin Fig. 2(b), another indication of uncompromised device

quality after transfer. In approximately 8 devices studied,we find that the mobility of graphene/YIG is eithercomparable with or better than that of graphene=SiO2.To study the proximity-induced magnetism in graphene,

we perform the Hall effect measurements in the field rangewhere the magnetization of YIG rotates out of plane overa wide range of temperatures. Nearly all graphene/YIGdevices exhibit similar nonlinear behavior at low temper-atures as shown in Fig. 2(c). Figure 2(d) only shows theHall data after the linear ordinary Hall background [thestraight red line in Fig. 2(c)] is subtracted. In ferromagnets,the Hall resistivity generally consists of two parts [18]:from the ordinary Hall effect and the anomalous Hall effect(AHE), i.e., Rxy ¼ RHðBÞ þ RAHEðMÞ ¼ αBþ βM, hereB being the external magnetic field, M being the magneti-zation component in the perpendicular direction, and α andβ are two B-and M-independent parameters, respectively.The B-linear term results from the Lorentz force on onetype of carriers. Higher-order terms can appear if there aretwo or more types of carriers present. The M-linear termis due to the spin-orbit coupling in ferromagnets [18]. Theobserved nonlinearity in Rxy suggests the following threepossible scenarios: the ordinary Hall effect arising frommore than one type of carriers in response to the externalmagnetic field, the same Lorentz force related ordinary Halleffect but due to the stray magnetic field from the under-lying YIG film, and AHE from spin-polarized carriers. Thenonlinear Hall curves saturate at Bs ∼ 2300 G, which isapproximately correlated with the saturation of the YIGmagnetization in Fig. 1(a). This behavior is characteristic

FIG. 2 (color). (a) The gate voltage dependence of the deviceconductivity scaled by the capacitance per unit area for thepretransfer (293 K, black) and transferred devices (300 K, red;2 K, green) with the same graphene sheet. (b) Quantum Halleffect of transferred graphene/YIG device in an 8 T perpendicularmagnetic field at 2 K. (c) The total Hall resistivity data at 2 K(anti-symmstrized) with a straight red line indicating the ordinaryHall background. (d) The nonlinear Hall resistivity after the linearbackground is removed from the data in (c).

FIG. 1 (color). (a) Magnetic hysteresis loops in perpendicularand in-plane magnetic fields. Inset is the AFM topographic imageof YIG thin film surface. (b) Optical image (without top gate) and(d) schematic drawing (with top gate) of the devices aftertransferred to YIG/GGG substrate (false color). (c) Room temper-ature Raman spectra of graphene/YIG (purple), graphene=SiO2

(red), and YIG/GGG substrate only (blue).

PRL 114, 016603 (2015) P HY S I CA L R EV I EW LE T T ER Sweek ending

9 JANUARY 2015

016603-2

anomalous component

optical image of a graphene device on YIG/GGG beforethe top gate is fabricated. Room-temperature Raman spec-troscopy is performed at different stages of the devicefabrication. Representative spectra are shown in Fig. 1(c)for the same graphene device on SiO2 (before transfer)and YIG (after transfer), and for YIG/GGG only. Graphene/YIG shows both the characteristic E2g (∼1580 cm−1) and2D peaks (∼2700 cm−1) of single-layer graphene as well asYIG’s own peaks, suggesting successful transfer. We alsonote that the transfer process does not produce any meas-urable D peak (∼1350 cm−1) associated with defects [17].Figure 1(d) is a schematic drawing of a top-gated transferreddevice on YIG/GGG.Low-temperature transport measurements are performed

in Quantum Design’s physical property measurementsystem. Figure 2(a) is a plot of the gate voltage dependenceof the four-terminal electrical conductivity scaled by theeffective capacitance per unit area Cs. Since different gatedielectrics are used in the back- and top-gated graphenedevices, Cs is calculated based on the quantum Hall datawhich agrees with the calculated value using the nominaldielectric constant and the measured dielectric film thick-ness. Before transfer, the Dirac point is at ∼ − 9 V and thefield-effect mobility is ∼6000 cm2=V · s. After transfer, theDirac point is shifted to ∼ − 18 V. The slope of the σxx=Csvs Vg curve increases somewhat, indicating slightly highermobility, which suggests that the transfer process, the YIGsubstrate, and the top-gate dielectric do not cause anyadverse effect on graphene mobility. At 2 K, the mobilityimproves further, exceeding 10 000 cm2=V:s on the elec-tron side. Well-defined longitudinal resistance peaks andquantum Hall plateaus are both present at 8 T as shownin Fig. 2(b), another indication of uncompromised device

quality after transfer. In approximately 8 devices studied,we find that the mobility of graphene/YIG is eithercomparable with or better than that of graphene=SiO2.To study the proximity-induced magnetism in graphene,

we perform the Hall effect measurements in the field rangewhere the magnetization of YIG rotates out of plane overa wide range of temperatures. Nearly all graphene/YIGdevices exhibit similar nonlinear behavior at low temper-atures as shown in Fig. 2(c). Figure 2(d) only shows theHall data after the linear ordinary Hall background [thestraight red line in Fig. 2(c)] is subtracted. In ferromagnets,the Hall resistivity generally consists of two parts [18]:from the ordinary Hall effect and the anomalous Hall effect(AHE), i.e., Rxy ¼ RHðBÞ þ RAHEðMÞ ¼ αBþ βM, hereB being the external magnetic field, M being the magneti-zation component in the perpendicular direction, and α andβ are two B-and M-independent parameters, respectively.The B-linear term results from the Lorentz force on onetype of carriers. Higher-order terms can appear if there aretwo or more types of carriers present. The M-linear termis due to the spin-orbit coupling in ferromagnets [18]. Theobserved nonlinearity in Rxy suggests the following threepossible scenarios: the ordinary Hall effect arising frommore than one type of carriers in response to the externalmagnetic field, the same Lorentz force related ordinary Halleffect but due to the stray magnetic field from the under-lying YIG film, and AHE from spin-polarized carriers. Thenonlinear Hall curves saturate at Bs ∼ 2300 G, which isapproximately correlated with the saturation of the YIGmagnetization in Fig. 1(a). This behavior is characteristic

FIG. 2 (color). (a) The gate voltage dependence of the deviceconductivity scaled by the capacitance per unit area for thepretransfer (293 K, black) and transferred devices (300 K, red;2 K, green) with the same graphene sheet. (b) Quantum Halleffect of transferred graphene/YIG device in an 8 T perpendicularmagnetic field at 2 K. (c) The total Hall resistivity data at 2 K(anti-symmstrized) with a straight red line indicating the ordinaryHall background. (d) The nonlinear Hall resistivity after the linearbackground is removed from the data in (c).

FIG. 1 (color). (a) Magnetic hysteresis loops in perpendicularand in-plane magnetic fields. Inset is the AFM topographic imageof YIG thin film surface. (b) Optical image (without top gate) and(d) schematic drawing (with top gate) of the devices aftertransferred to YIG/GGG substrate (false color). (c) Room temper-ature Raman spectra of graphene/YIG (purple), graphene=SiO2

(red), and YIG/GGG substrate only (blue).

PRL 114, 016603 (2015) P HY S I CA L R EV I EW LE T T ER Sweek ending

9 JANUARY 2015

016603-2

B

THEORY (PRELIMINARY)TR-invariance is broken by the exchange term

H� = †� �

n

� i v �i@is0 � ✏ �0s0 + � (� ⇥ s)z +��0 szo

�

Highly non-trivial vertex renormalization

vx

= �v01�0s1 + �v02�0s2 + �v10�1s0 + �v20�2s0 + �v13�1s3 + �v23�2s3 + �v31�3s1 + �v32�3s2

-6000 -4000 -2000 0 2000 4000 6000-200

-100

0

100

200

B (G)

RAHE(Ohm

)

Boltzmann calculation (16 relaxation times!)

(resonant impurities)

� = 5meV

� = 3meV

THANK YOUPhys. Rev. B 94, 134202 (2016)

Phys. Rev. B 94, 201402(R) (2016)

SUMMARY MESSAGES

Different levels of failure of Born approximation

QSJ and SS are never truly distinguishable

Coherent processes may dominate the physics