MSE 7025 Magnetic Materials

(and Spintronics)

Chi-Feng Pai cfpai@ntu.edu.tw

Lecture 12: Transport measurements Episode I: Family of the Hall effect

Course Outline • Time Table

Week Date Lecture

1 Feb 24 Introduction

2 March 2 Magnetic units and basic E&M

3 March 9 Magnetization: From classical to quantum

4 March 16 No class (APS March Meeting, Baltimore)

5 March 23 Category of magnetism

6 March 30 From atom to atoms: Interactions I (oxides)

7 April 6 From atom to atoms: Interactions II (metals)

8 April 13 Magnetic anisotropy

9 April 20 Mid-term exam

10 April 27 Domain and domain walls

Course Outline • Time Table

Week Date Lecture

11 May 4 Magnetization process (SW or Kondorsky)

12 May 11 Characterization: VSM, MOKE

13 May 18 Characterization: FMR

14 May 25 Transport measurements in materials I: Hall effect

15 June 1 Transport measurements in materials II: MR

16 June 8 MRAM: TMR and spin transfer torque

17 June 15 Guest lecture (TBA)

18 June 22 Final exam

Transport measurement

• In solid state physics and condensed matter physics, one wants to study the “transport properties.”

I

V

Transport measurement

• In solid state physics and condensed matter physics, one wants to study the “transport properties.”

Transport measurement

• In solid state physics and condensed matter physics, one wants to study the “transport properties.”

I

VH

, , , ...T T

Transport measurement

• Out-of-plane field

– Hall voltage VH

– Hall effect

– Anomalous Hall effect

• In-plane field

– Resistance ~ V/I

– Magnetoresistance

– Anisotropic MR

I

V

H I

V

H

x xx xJ E y yx xJ E

Transport measurement

• Out-of-plane field

• In-plane field

I

V

H I

V

H

The Hall effect

• Family of the Hall effect

– Ordinary Hall effect (metals, 1879)

– Anomalous Hall effect (magnetic metals, 1880)

– (Extraordinary Hall effect)

• Modern family of the Hall effect

– Quantum Hall effect (2DEG, 1985 Nobel prize in physics)

– Fractional quantum Hall effect (2DEG, 1998 Nobel prize in physics)

– Quantum anomalous Hall effect (topological insulators, 2010s)

– Spin Hall effect (metals, 2000s)

– Quantum spin Hall effect (topological insulators, 2010s)

The Hall effect

• The Hall effects

Weng et. al., From the Anomalous Hall Effect to the Quantum Anomalous Hall Effect, arXiv (2015)

The Hall effect

• The Hall effects

Cui-Zu Chang and Mingda Li, Journal of Physics: Condensed Matter, 28, 12 (2016)

The Hall effect

• Edwin Hall, Johns Hopkins University, 1879 – PhD student under Henry Rowland

– Read Maxwell’s Electricity and Magnetism and had doubts

– Carried out experiments

– Published “On a new action of the magnet on electric currents” at 24

• Experimental

1y HH

x z

E V tR

j B IB ne

The Hall effect

• Edwin Hall, Johns Hopkins University, 1879 – PhD student under Henry Rowland

– Read Maxwell’s Electricity and Magnetism and had doubts

– Carried out experiments

– Published “On a new action of the magnet on electric currents” at 24

• Experimental

1y HH

x z

E V tR

j B IB ne

0.0 0.5 1.0 1.5 2.0

0

20

40

60

RH=0.510

-11m

3C

-1

VH (

V)

V

H (V

)

W(4nm)

Bext

(T)

RH=10.10

-11m

3C

-1

0

2

4

6

8

10

W(10nm)

The Hall effect

• Edwin Hall, Johns Hopkins University, 1879 – PhD student under Henry Rowland

– Read Maxwell’s Electricity and Magnetism and had doubts

– Carried out experiments

– Published “On a new action of the magnet on electric currents” at 24

• Experimental

0.0 0.5 1.0 1.5 2.0

0

20

40

60

RH=0.510

-11m

3C

-1

VH (

V)

V

H (V

)

W(4nm)

Bext

(T)

RH=10.10

-11m

3C

-1

0

2

4

6

8

10

W(10nm)

The Hall effect

• The Hall conductivity

Schultz et. al., PRB 45, 10886 (1992)

2 32

0 33 2

kdfe d kk v k

d

2 32 3

3

1

6 2

k

H

dfe d kk v k

k d

2

0

1/ 0H HR

ne

2

0 *

Fne k

m

Free-electron approximation F

The Hall effect

• The Hall conductivity – Cannot be directly calculated from free electron model

– Related to the details of Fermi surface

http://www.phys.ufl.edu/fermisurface/

Anomalous Hall effect

• Anomalous Hall effect – Also observed by Edwin Hall

– The Hall effect in (ferro) magnetic materials

– Or the Hall effect in materials with intrinsic magnetic moments

0.0 0.5 1.0 1.5 2.0

0

20

40

60

RH=0.510

-11m

3C

-1

VH (

V)

V

H (V

)

W(4nm)

Bext

(T)

RH=10.10

-11m

3C

-1

0

2

4

6

8

10

W(10nm)

0.0 0.5 1.0 1.5 2.0 2.5

0

200

400

600

VH (

V)

Bext

(T)

VH (V

)

CFB(1.7nm)/W(4nm)

0

20

40

60

80

100

CFB(10nm)/W(10nm)

W thin films W/CoFeB bilayer films

Anomalous Hall effect

• Anomalous Hall effect – Also observed by Edwin Hall

– The Hall effect in (ferro) magnetic materials

– Or the Hall effect in materials with intrinsic magnetic moments

xy O z A zR H R M

Co thin films CoFeB PMA thin films

A OR R

Anomalous Hall effect

• Anomalous Hall effect – Also observed by Edwin Hall

– The Hall effect in (ferro) magnetic materials

– Or the Hall effect in materials with intrinsic magnetic moments

xy O z A zR H R M

2 2/xy xy xx xy xx xy xy Weng et. al., From the Anomalous Hall Effect to the Quantum Anomalous Hall Effect

A OR R

AR M

0xy O z A zR H R M

Anomalous Hall effect

• From spin-orbit interaction

• Possible mechanisms – Intrinsic

– Side-jump (Extrinsic)

– Skew-scattering (Extrinsic)

N. Nagaosa et al., ROMP 82 1539 (2010)

S. Onoda et al., PRL 97 126602 (2006)

(Berry curvature)

Anomalous Hall effect

• From spin-orbit interaction

• Possible mechanisms – Intrinsic

– Karplus-Luttinger (1954)

– “Anomalous velocity” or “Berry phase curvature”

N. Nagaosa et al., ROMP 82 1539 (2010)

(Berry curvature)

Anomalous Hall effect

• From spin-orbit interaction

• Possible mechanisms – Intrinsic

– Spin-dependent Hall “angle”

(Berry curvature)

Ferromagnetic material

Paramagnetic material

Non-zero

Anomalous Hall effect

• Calculations

N. Nagaosa et al., ROMP 82 1539 (2010)

Tons of Green’s function calculations…

But wait, what is a spin current?

• (charge) Current

• Spin-polarized current

• Spin current

e

e

e e

e

e

e

e

e

e

e

e

ˆ ˆsJ s v

But wait, what is a spin current?

• Anomalous Hall effect

• Spin Hall effect

e

e

e

e

e

e

e e

e

xJ

e

xJ

0

0

e

y

s

y

J

J

0

0

e

y

s

y

J

J

How to measure, anyway?

• Which of the following is the recommended way to measure Hall effect?

How to measure, anyway?

• Van der Pauw method

– Proposed by L. J. van der Pauw in 1958 (Philips)

– If you have an extended film or a uniform sample with no holes on it

– If you want to measure both resistivity and Hall coefficient

The ideal bridge-type sample Arbitrary-shaped sample

How to measure, anyway?

• Van der Pauw method

• Resistivity Relative errors if P is not ideal

How to measure, anyway?

• Van der Pauw method

• Hall effect (coefficient) Relative errors if P is not ideal

How to measure, anyway?

• Van der Pauw method

, ,exp exp 1MN OP NO PM

d dR R

,

1H MO NP

dR R

ne B

ne

1

Get (Resistivity)

Get (major carrier density)n

Get (mobility)

How to measure, anyway?

• But of course, Hall-bars are always better (if you have time and money!)

How to measure, anyway?

• In mesoscopic regime, one has to be careful…

Quasi-ballistic transport

G. Mihajlovic et. al., PRL 103, 166601 (2009)

(But experimentally not the case)

What can we extract (from AHE signals)?

• Many useful information

cH

anH

Out-of-plane field AHE voltage

In-plane field AHE voltage

In-plane field 2nd harmonic AHE voltage

SOT

DLHSOT

FLH

The spin Hall effect

e-

e-

e-

e-

Js Je

J. E. Hirsch, Phys. Rev. Lett. 83 1834 (1999)

ˆS SH eJ J

M. I. Dyakonov and V. I. Perel, JETP 13 467 (1971)

Spin-orbit interaction

The spin Hall angle

/SH s eJ J

Possible mechanisms

• Intrinsic

• Side jump

• Skew scattering

( )1( )n k e

r E kk

extrinsic

Berry curvature

N. Nagaosa et al., Rev. Mod. Phys. 82 1539 (2010) T. Fujita et al., J. Appl. Phys. 110 121301 (2011)

Demonstration of the SHE

Spin Hall Effect in semi conductor (GaAs) - Optical detection Y.K. Kato et al., Science 306 1910 (2004)

Spin Hall Effect in metallic system (Al) -Transport measurement -Inverse SHE voltage S.O. Valenzuela et al., Nature 442 176 (2006)

4~10Al

SH

The SHE in Pt

• Pt is a heavy element large spin-orbit interaction possible large intrinsic spin Hall effect1

• Experiments were carried out

by several groups with different

techniques2

1Guo et al., Phys. Rev. Lett. 100 096401 (2008) 2Kimura, T., et al, Phys. Rev. Lett. 98, 156601 (2007) Ando, K. et al. Phys. Rev. Lett. 101, 036601 (2008) Mosendz, O. et al., Phys. Rev. Lett. 104, 046601 (2010)

θSH = 0.003 ~ 0.08 (?)

The spin Hall conductivity (intrinsic)

• If due to intrinsic mechanism

• Spin current operator

The calculated SHC of transition metals

• The spin Hall conductivity calculated for 4d 5d elements

• ab initio calculation: θSH(Ta)<0 and θSH(Pt)>0

for highly resistive case, θSH(Ta) can be large

Tanaka, T. et al, Phys. Rev. B 77, 165117 (2008)

Demonstration of the GSHE in beta-Ta

DC current induced SHE-ST switching

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

60

80

100 B

ext = -3.5 mT

dV

/dI

(k

)

IDC

(mA)

Schematics of a SHE-3T device

-Ta 0.12 0.04SH

Pt 0.06SH

e-

Ta

e-

e- Je

Js

CoFeB

CoFeB

MgO

V

QHE and QSHE

• Next semester: Contemporary Solid State Materials in a Nutshell

(Regular insulators)

(High field effect, Landau level)

(Topological insulators)

0sJ

0sJ

QHE and QAHE

• Next semester: Contemporary Solid State Materials in a Nutshell

QHE and QAHE

• Next semester: Contemporary Solid State Materials in a Nutshell

http://www.sp.phy.cam.ac.uk/

QHE and QAHE

• Next semester: Contemporary Solid State Materials in a Nutshell

Y. Tokura group (2014)

topological insulator Crx(Bi1 − ySby)2 − xTe3

Qi-Kun Xue group, Science (2013)

QHE and QAHE

• Next semester: Contemporary Solid State Materials in a Nutshell

Topological (skyrmion) Hall effect

• In skyrmion lattice

• In a magnetic heterostructure