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Xiaofeng Jin Physics Department, Fudan University, Shanghai, China

The Hall Effects Edwin Hall Never imagined

1. Introduction

2. Our approach

3. Conclusions

Santander, June 22, 2017

1. Introduction

1879 � 1880 � 2004 �

Lorentz Force �

1879 � 1880 � 2004 �

Spin-Orbit Coupling �

1879 � 1880 � 2004 �

1879 � 1880 � 2004 �

Lorentz Force �

1879 � 1880 � 2004 �

Spin-Orbit Coupling �

1879 � 1880 � 2004 �

1879 � 1880 � 2004 �

Lorentz Force �

1879 � 1880 � 2004 �

Spin-Orbit Coupling �

1879 � 1880 � 2004 �

J. Inoue and H. Ohno, Science, 309, 2004 (2005)

Spin-orbit coupling VH

Vxx

I

B

23

2' '

( ( )) ( )

2 Im | | ' ' | |( )

( )

zxy n n

n

x yzn

n n n n

e d f

n v n n v n

σ ε

ω ω≠

= − Ω

Ω = −−

∑∫

∑k k

k k k

k k k kk

hBerry curvature

int constantσ =

Jungwirth, Niu, MacDonald (2002), Onoda & Nagaosa (2002)

( )( )anomal u

no s

kd tdt k

vr ε→→

→

∂=

∂+

dttrdrB

r

rVdttkd )()()()(

→→→

→

→→

×−∂

∂=

(1) Karplus-Luttinger Intrinsic (1954) Anomalous velocity

2int int xxρ σ ρ=

r- space curvature

G. Sundaram and Q. Niu, Phys. Rev. B 59 (1999) 14915.

k- space curvature

( )( )nd k tkdt

→→ →

−Ω ×

(2) Skew-scattering (Smit, 1955)

(3) Side-jump (Berger, 1970)

( ) ( )dV rdr

⋅s L

( ) ( )dV rdr

⋅s L

ah xxρ αρ=

2ah xxρ βρ=

21 2ah xx xxc cρ ρ ρ= + Intrinsic or Extrinsic ?!

21 2ah xx xxc cρ ρ ρ= +

(2010) 2010

21 2ah xx xxc cρ ρ ρ= +

21 2ah xx xxc cρ ρ ρ= +

21 2ah xx xxc cρ ρ ρ= +

Euro. Phys. Lett. 103 (2013) 47003

(1) Karplus-Luttinger Intrinsic (1954)

(2) Skew-scattering (Smit, 1955)

(3) Side-jump (Berger, 1970)

2int rinsic xxbρ ρ=

2side jump xxρ βρ− =

skew xxρ αρ=

Theoretically:

All based on single type of scatters !

(1) Karplus-Luttinger Intrinsic (1954)

(2) Skew-scattering (Smit, 1955)

(3) Side-jump (Berger, 1970)

0xx xx xxTρ ρ ρ= +

(Matthiessen's rule)

In real materials:

2int rinsic xxbρ ρ=

2side jump xxρ βρ− =

skew xxρ αρ=

Theoretically:

Key Issue: should

scale with or or ? xxρ 0xxρ xxTρ

ahρ

2. Our approach

Substrate

FM film Capping

6mm

1.5mm

0( , , )ah xx xxT xxfρ ρ ρ ρ=

Tuning with impurity and temperature Tuning with thickness and temperature

0 20 40 60 80 100

5

10

15

20

25

Thickness (nm)

ρxx @ 5K

ρ xx (µΩ

cm

)Fe

e-

R=1 at interface

D.Z. Hou et al., J. Phys. Cond. Matt., 24 (2012) 482001

A. Cottey, Thin solid films 1, (1967) 297-397

e-

R<1 at interface

D.Z. Hou et al., J. Phys. Cond. Matt., 24 (2012) 482001

A. Cottey, Thin solid films 1, (1967) 297-397

P.N. Dheer Phys Rev (1967)

zahxy MRBR += 0ρj

B

j

B

ahρ

j

)( xxah f ρρ =

M⊥

)( xxah f ρρ =

0xxρ

xxTρ

0xx xx xxTρ ρ ρ= +

Matthiessen's rule :

impurity --> skew ! Phonon --> no skew !

A. Crepieux et al., PRB, 64 (2001) 014416

21 0 1 2'ah xx xxT xxc c cρ ρ ρ ρ= + +

Experimental verification !

21 2ah xx xxc cρ ρ ρ= +

Contradictory to the traditionally used scaling:

1c

1 'c

2cFe

21 0 1 2'ah xx xxT xxc c cρ ρ ρ ρ= + +

21 0 2ah xx xxc cρ ρ ρ= +

1 0xxc α βρ= +

21 0 2ah xx xxc cρ ρ ρ= +

2 20 0 2ah xx xx xxcρ αρ βρ ρ= + +

20

0

0.7 3.7xx

xx

βραρ

= →

3 1 12 int 1.1 10rinsicc b cmσ − −= = ≈ × Ω

Dheer, PR 156, 637 (1967)

1 2 20 0 2( )ah xx xx xx cσ ασ βσ σ− −= − + −2 2

0 0 2ah xx xx xxcρ αρ βρ ρ= + +

Ni,Co

Y.  Shiomi,  Y.  Onose  and  Y.  Tokura,  PRB  79,  100404(R)  2009

Intercept:      0.3%:  0.99×103  Ω-­‐1cm-­‐1,      1%:  1.07×103  Ω-­‐1cm-­‐1  

Fe

Euro. Phys. Lett. 103 (2013) 47003

A System without Intrinsic AHE

Cu ~42%Ni Ni

Ferromagne:c  –  paramagne:c  cri:cal  point  

~42%Ni

S.A.Aherm et al., Proc. Royal Soc. London, 248 A, (1958) 145

MgO  anealed  @500°C  

NiCu  as  grown

34%Ni 2 20 0ah xx xx xxbρ αρ βρ ρ= + +

00

0xx

xx

AH βραρρ

+=

200 xxxxAH βραρρ +=

At  5  K

2 20 0ah xx xx xxbρ αρ βρ ρ= + +

41 42 43 44 453.0x10-4

4.0x10-4

5.0x10-4

6.0x10-4

7.0x10-4

8.0x10-4

9.0x10-4

ρ AH

5K / ρ xx

5K

ρxx5K(µΩ cm)

4.9 nm 6.0 nm 7.1 nm 8.2 nm 9.3 nm 10.4 nm 11.5 nm 12.6 nm 13.7 nm 14.8 nm

α β  (Ω-­‐1cm-­‐1) -­‐(5.4±0.2)x10-­‐3 140±5

00

0xx

xx

AH βραρρ

+=

T=5K

| 𝛽  𝜌↓𝑥𝑥0 2  |/| 𝛼  𝜌↓𝑥𝑥0 | =𝟏.𝟏

Fe/MgO(001) films

0 2 4 6 8 10 12

1.2

1.4

1.6

   8  nm  13  nm  18  nm  23  nm  28  nm  33  nm

σxx2 (1010 Ω-2cm-2)

-σA

H(1

03 Ω-1cm

-1)

0 .2 0 .4 0 .6 0 .8 1.0

1.2

1.4

1.6

1.8

 8  nm  13  nm    18  nm  23  nm    28  nm  33  nm  fit  

−σAH0-­‐  α

Feσ x

x0

-­‐1σ x

x2  (10

3  Ω

-­‐1cm

-­‐1)

σxxσxx0

-­‐1  

AH conductivity in Kubo-Streda formula

𝜎↓𝐴𝐻↑ = 𝜎↓𝐴𝐻↑𝐼 +𝜎↓𝐴𝐻↑𝐼𝐼 

𝜎↓𝐴𝐻↑𝐼 = 𝑒↑2 /2𝜋𝐴 ⟨𝑣↓𝑥 𝐺↓↑𝑅 ( 𝜀↓𝐹 )𝑣↓𝑦 𝐺↓↑𝐴 ( 𝜀↓𝐹 )⟩

velocity vertex: intraband vertex needs to be dressed

scattering effects in the Fermi surface term

note that 1/ 𝜌↓𝑥𝑥 ≅ 𝜎↓𝑥𝑥 ∝𝜏↑𝑡𝑟 

∼ 𝜌↓𝑥𝑥↑−1 

∼ 𝜌↓𝑥𝑥↑0 

intra

inter

Theory with multiple competing scatters !

+ 𝜎↓𝐴𝐻↑𝐼𝐼  𝑐 intrinsic Berry curvature

∑↓𝑖 𝑐↓𝑖 𝜌↓𝑖 / 𝜌↓𝑥𝑥 

( ∑↓𝑖𝑗 𝑐↓𝑖𝑗 𝜌↓𝑖 𝜌↓𝑗 + ∑↓𝑖∈𝑆 𝛼↓𝑖 𝜌↓𝑖 )/ 𝜌↓𝑥𝑥↑2 

Three groups of diagrams

− 𝜎↓𝐴𝐻 =𝑐+ ∑↓𝑖 𝑐↓𝑖 𝜌↓𝑖 / 𝜌↓𝑥𝑥 +( ∑↓𝑖𝑗 𝑐↓𝑖𝑗 𝜌↓𝑖 𝜌↓𝑗 + ∑↓𝑖∈𝑆 𝛼↓𝑖 𝜌↓𝑖 )/ 𝜌↓𝑥𝑥↑2  𝜌↓𝐴𝐻 ≅− 𝜎↓𝐴𝐻 𝜌↓𝑥𝑥↑2 =𝑐𝜌↓𝑥𝑥 + ∑↓𝑖 𝑐↓𝑖 𝜌↓𝑖 𝜌↓𝑥𝑥 + ∑↓𝑖𝑗 𝑐↓𝑖𝑗 𝜌↓𝑖 𝜌↓𝑗 + ∑↓𝑖∈𝑆 𝛼↓𝑖 𝜌↓𝑖 

•  partial resistivities 𝜌↓𝑖  as scaling variables •  coefficients 𝑐↓𝑖 , 𝑐↓𝑖𝑗 , and 𝛼↓𝑖  not depend on disorder concentration

for two scattering sources: one static (impurity) and one dynamic (phonon)

𝜌↓𝐴𝐻 =𝛼𝜌↓𝑥𝑥0 + 𝛽↓0 𝜌↓𝑥𝑥0↑2 +𝛾𝜌↓𝑥𝑥0 𝜌↓𝑥𝑥𝑇 + 𝛽↓1 𝜌↓𝑥𝑥𝑇↑2 

a quadratic surface passing through origin in (𝜌↓𝑥𝑥0 , 𝜌↓𝑥𝑥𝑇 , 𝜌↓𝐴𝐻 ) space

𝛽↓0 =𝑐+ 𝑐↓0 + 𝑐↓00  𝛾=2𝑐+ 𝑐↓0 +𝑐↓1 +𝑐↓01  𝛽↓1 =𝑐+ 𝑐↓1 + 𝑐↓11 

2 20 0 0 1 0ah xx xx xx xx xxd dρ αρ βρ ρ ρ ρ= + + +

1 2 2 10 0 0 1 0( )ah xx xx xx xx xxd dσ ασ βσ σ σ σ− − −= − + − −

1 2 2 10 0 0 1 0( )ah xx xx xx xx xxd dσ ασ βσ σ σ σ− − −= − + − −

Au

A. Hoffmann, IEEE transactions on magnetics 49, 5172 (2013)

2 20 0 2ah xx xx xxcρ αρ βρ ρ= + +

Spin Hall Effect

L.  Q.  Liu,  et  al,  Science,  336,  555  (2012)  

Spin injection with FM/Cu/NM bridge Method 1: non-local spin valve

FM/NM bilayer driven by Brf

Method 2: spin pumping

E. Saitoh, et al . Appl. Phys. Lett 88, 182509 (2006).

NM/FM bilayer structure Method 3: spin transfer torque

L. Liu, et al., Science 336, 555 (2012)

Method 4: spin Seebeck effect

S. M. Rezende., et al Phys. Rev. B 89 014416. (2014) S. Y. Huang, et al . Phys. Rev. Lett. 107, 216604 (2011)

S. O. Valenzuela, et al . Nature 442, 176-179 (2006) T. Kimura, et al, Phys. Rev. Lett. 98, 156601 (2007) Y. Niimi, et al, Phys. Rev. Lett. 106, 126601 (2011)

Au

Generation

Detection

Shunting

Interface

A. Hoffmann, IEEE transactions on magnetics 49, 5172 (2013)

60 nm thick Au film

w

L

0 100 200 3000.3

0.4

0.5

0.6

0.7

0 100 200 300

0.00

0.01

0.02R nl(Ω)

Rsq(Ω)

Temperature(K) Temperature(K)

ll sqVR RI d

ρ= ⇒ = nl

nlVRI

=

Device E-beam Lithography Lift-off SiO2/Si

L = 150nm W= 120nm

Dr. Y. Tian Dr. L. Ye

Dr. G. Su Dr. J.L. Xu

Dr. Y.F. Li

Dr. L. Wu

Dr. D.Z. Hou

Dai Tian Caigan Chen

3. Conclusions

Extrinsic “Intrinsic” (1) Y. Tian et al., Phys. Rev. Lett., 103, 087206 (2009)

(2) L. Ye et al., Phys. Rev. B, 85, 220403(R) (2012)

(3) D.Z. Hou et al., J. Phys. Cond. Matt,24 (2012) 482001

(4) L. Wu et al., Phys. Rev. B, 87 (2013) 155307

(5) J.L. Xu et al., Appl. Phys. Lett., 102 (2013) 162401

(6) G. Su et al., Phys. Rev. B, 90 (2014) 214410

(8) D.Z. Hou et al., Phys. Rev. Lett., 114, 217203 (2015)

(7) Y.F. Li et al., Euro. Phys. Lett. 110 (2015) 27002

(9) L. Wu et al., Phys. Rev. B, 93 (2016) 214418

(10) D. Yue et al., J. Phys. Soc. Jpn. 86 (2017) 011006