Non-equilibrium thermoelectric approach:

Part II: Spintronics

Jean-eric.wegrowe@polytechnique.edu

Transport Equations including magnetic degrees of freedom

Two different ensembles of degrees of freedom: ferromagnetic and spin attached to charge carriers

•1) Hall effect (Ohm’s law ): the role of electric screening

•2) Ohm’ law in Ferromagnetc materials: the role of symmetries:

Anisotropic magnetoresisance, anomalous Hall effect, and Planar Hall effect.

•3) Internal degrees of freedom (spin-flip relaxation) : Role of the relaxation

3a) The two channel model for the Giant Magnetoresistance (GMR)
3b) The two channel model for the Spin-Hall effect (SHE) 31 Slides

•4) Cross-effect between ferromagnetic degrees of freedom and spins of the carriers.
at the end of the next Lecture about ferromagnetism (Thusday February 1)

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End of
Part I
Hall effect:
the cross-coefficient couples x and y space directions
Fully antisymetric tensor (coss product):
Ohm’s law:
Onsager reciprocity relation II

Time-inversion symmetry 
at micro scale with  rotation Lorentz force:

Isotropic otherwhise B
Vectorial form:

Projection over x:

Problem of inrterpretation: the Hall field y cannot produce the current

Jx
(cause ≠ consequence)
galvanostatic mode Entropy production implicit here.
Matrix inversion:
Inversion of the 2 by 2 matrix : Defines an « arrow of time »
(causality)
even at stationary state!

Conclusion:
The courant Jx generates the Hall field y
specific statut of
Hall field for Jy=0 (condition for stationary states: see next slides) the transport equations
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End of
Last Lecture Hall effect including diffusion
Accumulation of electric charges at stationary regime:
Chemical potential n(y) : density of consequence of the application of the magnetic field
electric charges

V(x,y) Electric potential like a

n0 : electro-neutrality
Capacitor Jx0
In degenerate metal (Fermi temperature)

in non-degenerate semiconductor Mobility

n > 0

Ohm’s law: Notation:

Conductivity:
usually  n(y)= n(y) – n0 => electric field along y

Poisson law Maxwell-Gauss:

Onsager
reciprocity
relations

Imposed
by the generator
Induced by
Charge
Drift-diffusion equation for the Hall effect: accumulation
Drift Diffusion Drift Diffusion
Two diffusion coefficients :
(Einstein’s relation)
Well-known equation (validation of the thermokinetic approach)

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 Stationary states : Jy = 0
and Hall-angle

--------------
Transport equation:
The Hall effect is defined by two transport coefficients:  and 

Stationarity condition:
This leads to two possible stationary states: Field lines

(1) and +++++++++++++ ++

Exercise
(2) --------------
Solution (1) confirmed by the minimization of the Joule power
under current injection (beyond this lecture).
Warning: in the Wikipedida article « Hall effect », the animation is strongly missleading!
Current lines
Definition of the Hall angle H:
+++++++++++++ ++
The charge accumulation (consequence)
counterbalances the effect
Stationarity condition of the Lorentz force (cause)
Lechatellier-Lenz principle
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 Out-of-equilibrium charge-accumulation at the edges
The electric charges are renewed permanetly: surface current confined within the Deby-Fermi lenght

Jx0

n > 0

Variational methode: minimization of the power dissipated under constraints.

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 Surface current in Hall-effect :
measured recently (breaking news)

Caution: should still be confirmed…

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 How much power can be injected from the latteral
edges to a load circuit?

« Maximum-power-transfer
theorem »: max at Rl = RH

contact of the lateral edges

Work in progress (see SHE below) : proposition for internship

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 Drude’s microscopic model * triple vector product:

including magnetic field:

Newton’s law
with damping Stationary regime: (1)
(Langevin equation)
velocity v, mass M, mag. field B, elec. field , relaxation time  implicit equation
Eq. (1) :
where and Puting in factor!

Development of the double cross-product (removing the cross-product):

Eq. (1)
cross vector b:
*
(1) with
re-insertion into the expression (1): factorisation:

explicit equation

Eq.(1) reads actualy:

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 Ohm’s law including magnetic field:
the Lorentz Magnetoresistance

(1)

The transport equation takes the form:

(2) Drude conductivity

Magnetoresistance Hall effect

(measured along x)

Lorentz Magnetoresitance
Eq. (2) shows that the magnetoesistance was missing
in the slides 2 and 3 about the Hall effect,
because we assumed

Observed in
non-magnetic
Next slides: generalization of the Hall effect single crystal
with in a ferromagnet.

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 Non-equilibrium thermoelectric approach:
Part II: Spintronics

Transport Equations including magnetic degrees of freedom

Two different ensembles of degrees of freedom: ferromagnetic and spin attached to charge carriers

•1) Hall effect (Ohm’s law ): the role of electric screening

•2) Ohm’ law in Ferromagnetc materials: the role of symmetries:

Anisotropic magnetoresisance, anomalous Hall effect, and Planar Hall effect.

•3) Internal degrees of freedom (spin-flip relaxation) : Role of the relaxation

3a) The two channel model for the Giant Magnetoresistance (GMR)
3b) The two channel model for the Spin-Hall effect (SHE) 31 Slides

•4) Cross-effect between ferromagnetic degrees of freedom and spins of the carriers.
at the end of the next Lecture about ferromagnetism (Thusday February 1)

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 Role of Anisotropic magnetoresistance (AMR)
symmetry and anomalous Hall effect (AHE)
Generalization of the Ohm’s law in a conducting ferromagnet (whatever the mechanisms).
The Lorentz magnetoresistence (previous slide) was a restrictive case (quasi-ballistic case).

Electric field (+ diffusion) Jx

Ohm’s law
a) Basis
Onsager reciprocity
If relation
Isotropic at B=0
Anisotropy m = B
Tensorial form (whatever the basis):
(see previous slide)
b)
Totally antisymmetric tensor
Vectorial form: Projection operator

Note that it is equivalent to apply the rotation:

Conclusion: same form as the equation obtained with the Drude approach (slides 8 and 9).
All microscopic mechanisms that are compatible with the symmetries
are included in the phenomenological transport equation.
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 Anisotropic electric transport (read heads, HDD)
Only the symmetry matters (whatever the mechanisms involved )

Galvanostatic (Hall bar) matrix inversion : Inversion of

the 2 by 2 matrix

Anisotropic Magnetoresistance
Anomalous Hall effect Spherical coordinates (r=1):
and « Planar Hall effect »

Longitudinal Experiments: Transverse

Jx

Anisotropic Magnetoresistance
AMR
V « Planar Hall Effect » « Anomalous Hall
Effect »

Vectorial sensor!
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 APPLICATION: AMR for integrated
magnetic sensors (HDD before 1998)
Read head

Magnetic tape

AMR read head: Magnetic recording

about 2% resistance variation Hard Disk Drive (HDD)
vs. Flash technology (SSD)

GMR
(second part
 of this lecture)

AMR

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 Measurements of AHE + PHE as a function
of the magnetic field in NiFe ferromagnet
Out of plane field: H ≠ 

Planar Hall > Anomalous Hall Measured of Vy

Transverse
The magnetization curves
voltage Vy
M(H,)
will be calculated in Lecture IX!

Calculated from
Applied magnetic field

Consequences
of the symmetries
PhD dissertation (including microscopic invariance
Benjamin Madon (X10) under time-inversion)!

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 Righi-Leduc effects and Nernst – Ettingshausen effects
Hall configuration for thermal and thermoelectric transport

3D configuration space: N Arrhenius

E

Peltier coefficient Peltier and Seebeck

same effect

Nernst effect
In the relevant basis :

Graz 1887
Ludwig Boltzmann and coworkers
Augusto Righi
1850–1920
In the orthogonal basis
Righi-Leduc effect Univ. Palermo, Padoua,
Bologna

In ferromagnetic material (anisotropy):

Anisotropic, Anomalous and Planar-Hall for Hall, Righi-Leduc and Nernst effects:
54 « effects »? But the nomenclarure is clear from the transport equations (« only » 9 transport coef). 15
 « Explosion » of « effects » in spintronics:
lack of unifying theory or absence of nomenclature?
Different « new effects » associated with the term
``Spin-Hall'' includes (this is not an exhaustive liste):
About thermoelectric effects:
- Inverse Spin-Hall (ISH), - Anisotropic Seebeck
- Anomalous Spin-Hall (ASH), - Anosotropic Peltier
- Abnormal Spin-Hall - Planar Nernst
- Spin-Hall magnetoresistance (SHR) - Anomalous Nernst
- Anomalous Spin-Hall magnetoresistance (ASHR) - Spin-Seebeck
- Magnetic Spin-Hall (MSH) - Spin-Peltier
- Inverse magnetic Spin-Hall (IMSH) - Spin-Nernst
- Quantum Spin-Hall (QSH) - Nernst magnetoresistance
- Quantum anomalous Spin-Hall (QASH) - Spin pumping
- Topological Spin-Hall (TSH) -…
- Phonon Spin-Hall (PSH)
- Photon Spin-Hall (spin-Hall effect of light)
- Magnon Spin-Hall (MSH)
- Magnon Spin-Hall magnetoresistance (MSHR)
-…
(1) Change of boundary conditions => new name (≠ new effect)

(2) Different microscopic mechanisms (simple exemple next slide)

Different mechansisms can be presented as different « effects ».
But the transport equations are already riche anough (e.g. 54 possible names!)
16
 Non-equilibrium thermoelectric approach:
Part II: Spintronics

Transport Equations including magnetic degrees of freedom

Two different ensembles of degrees of freedom: ferromagnetic and spin attached to charge carriers

•1) Hall effect (Ohm’s law ): the role of electric screening

•2) Ohm’ law in Ferromagnetc materials: the role of symmetries:

Anisotropic magnetoresisance, anomalous Hall effect, and Planar Hall effect.

•3) Internal degrees of freedom (spin-flip relaxation) : Role of the relaxation

3a) The two channel model for the Giant Magnetoresistance (GMR)
3b) The two channel model for the Spin-Hall effect (SHE)

•4) Cross-effect between ferromagnetic degrees of freedom and spins of the carriers.
at the end of the next Lecture about ferromagnetism (Thusday February 1)

14/02/23 PHY581b Wegrowe 17

 The two spin channel model:
introduction of the spins attached to the charge carriers

Spin polarization =
internal degrees of freedom At each points
{x,y,z}
of the charge carriers.
This was also the case for
electrons and holes in Lecture VII

Two conducting
Spin-channel model:

First descriptions of the GMR:

- Johnson and Silsbee 1985 x
(Thermodynamic approach with affective field)

- Van Son, van Kempen, Wyder 1987

(introduction of « spin-pumping » )

- Valet and Fert 1993 (Boltzman equations)

G
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 Ferromagnetic Junction at nanoscale:
Intuitive picture of a supplementary dissipation
GMR JUNCTION
Conductivity asymmetry:
switch with a magnetic field

Ferromagnet up Ferromagnet down

R + R J
GMR
Switch with the = R/R (%)
External magnetic field Ferromagnet up Ferromagnet up
NO JUNCTION
= Reference R
(necessary)
R : supplementary dissipation due to spin-flip scattering
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 Role of internal degrees Spin-dependent electric transport
of freedom « SPINTRONICS »
The two channel model: the current is spin-polarized Nobel 2007
A.Fert / P. Grünberg
=> two species of charge carriers for the discovery or the giant magnetoresistance

No charge accumulation
(≠ semiconductor)!
1988
Two species of charge carriers: up and down
and spin-polarized current

Kinetic coupling:
through internal degrees of freedom

Reaction:
reaction rate

reactant product
Extent of the reaction

Chemical affinity of the spin-flip scattering:

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 The two spin channel model
Chemical potential Chemical affinity of the
spin-flip reaction:
No charge
accumulation (n=0 )

reactant product

Extent of the reaction:

Ohm’s law:
G
Reaction rate:
Transport equation in the spin space

Thermodynamic conjugate variables:

: Onsager coefficent: Supplementary power dissipated :

Conservation equations with spin-flip

Charge (charge is « consevred »)

Spin (spin are «not conserved »

Spin-current in the restrictive sense)
Note that the number of spins is « conserved » in the sense that the chemical potential is not zero

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 Spin accumulation and GMR
First law of thermo + continuity equation: Entropy production  ≠ 0 :
Out of equilibrium resistance
1 interface

+ Second law => Onsager matrix:

Ohm’s law spin up

Ohm’s law spin down
Spin-relaxation

Stationary regime: => Spin accumulation 2 interfaces …

EXERCICE

lsf is typically of the order of tens of nanometers in metals

at room temperature, to hundreds of nm at law T.
Interface Magnetoresistance
Voltage drop
generated by
the interfaces

The spin-polarized electric current forces the spin to flip at the interface.
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The « non-local » geometry:
« diffusive pure spin-current »
It is difficult and rather subtle.
But we have now the tools to understand this effect.
Conventional
measurements

Non-local
measurements

Conventional Non-local GMR

measurements No current of electric charges but spin
diffusion current = pure spin-current
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 Role of Non-local configuration:
diffusion the lateral spin-accumulation
Complement (1)
(not for exam) Ohm’s law: Maxwell-Gauss:

Inside the non-Ferromagnetic branche:

Current injection
Transport (2)
I Equations in
Ferromagnet 1 The N-F bar

M Conservation law at stationary state:

Non-Ferromagnetc
(3)
bar Screening equation (inserting (2) and (1) into (3))
Spin diffusion
along y

-M M where is the Debye-Fermi length

Ferromagnet 2

Bulk approximation
V
(negligible charge accumulation)
Probe: mesure of the GMR
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 The 4 channels model:
Two relaxation processes (Mott’s mechanism): Channels = bands
Complement
(not for exam)

s = Chemical affinities of the relaxation (1)

 = Chemical affinities of the relaxation (2)
Two relaxation channels = two coupled diffusion equations
= two spin-accumulation processes @ the interface

Out of equilibrium AMR: GMR

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 Non-equilibrium thermoelectric approach:
Part II: Spintronics

Transport Equations including magnetic degrees of freedom

Two different ensembles of degrees of freedom: ferromagnetic and spin attached to charge carriers

•1) Hall effect (Ohm’s law ): the role of electric screening

•2) Ohm’ law in Ferromagnetc materials: the role of symmetries:

Anisotropic magnetoresisance, anomalous Hall effect, and Planar Hall effect.

•3) Internal degrees of freedom (spin-flip relaxation) : Role of the relaxation

3a) The two channel model for the Giant Magnetoresistance (GMR)
3b) The two channel model for the Spin-Hall effect (SHE)

•4) Cross-effect between ferromagnetic degrees of freedom and spins of the carriers.
at the end of the next Lecture about ferromagnetism (Thusday February 1)

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 Spin-Hall effect (SHE) in 3 actes.
Current induced « transverse » spin accumulation

Acte 1
1971
Introduction of
The transport equations
of both charge
and spin currents.

See 3 slides below

Google 01/2020:
1337 citations

The SHE : mixt of the classical Hall effect and the Giant Magnetoresistance

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 Independent rediscovery …
Acte 2 Inspired by the Anomalous MR and GMR
(no reference to D-P papers)
Inspired by Hirsh 1999 and the GMR 2000

Incidentaly:
Author of
the h index!

Spin-accumulation
GMR-like
(No charge accumulation)

844 citations
2800 citations (Google Scholar 01/2021)
Stationary state (2)
Stationary state (1)
Spin-dependent Electric field Pure spin-current
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 Measuring the SHE: spin to charge conversion device
Magneto-optics measurements
Acte 3 J. Wunderlich et al. Y.K. Kato, …, D. D. Awschalom Science (2004)
« Observation of spin-Hall effect in semiconductore »

1400 citations 2200 citations

Valenzuela and Tinkhamn, Nature (2006)

« Direct electronic measurement of the spin-Hall effect »

Full electric
(non-local inverse SHE)
Google 1200 citations

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 r t 1 Hall effect including diffusion
Pa
Chemical potential
n(y) : density of
electric charges Accumulation of electric charges at stationary regime

V(x,y) Electric potential

n0 : electro-neutrality
In degenerate metal (Fermi temperature) Jx0
in non-degenerate semiconductor Mobility
n > 0
Ohm’s law: Notation:
Conductivity: n(y)= n(y) – n0 => electric field along y
usually 
Poisson law
Maxwell-Gauss:
Onsager
reciprocity
relations

Imposed
by the generator
Induced by
Charge
Drift-diffusion equation for the Hall effect: accumulation

Two diffusion coefficients

Drift Diffusion Drift Diffusion (Einstein’s relation)

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 Definition of the SHE : spin-dependent transport equations
(in non-ferromagnetic heavy metals)
Microscopic mechanism: spin-orbit coupling

Spin-dependent charge density:

Electric neutrality
(if no other source of
spin-current)
Slide 21: Hall effect:

for each spin-channel (add ), with a change of sign

Dyakonov-Perel equations (1971) Two spin-channel model.

Demonstration
Ohm’s law: next slide

Spin-dependent chemical potential.

indexj =z fixed (same « electric field » as in GMR)
SHE
 ≈ so D = kT Equivalent to Dyakonov- Perel equations if

31
 Spin-Hall effect (SHE) : two times the Hall effect
Two Hall subsystems
with opposite (spin-orbite) magnetic fields

Ohm’s law (mobility):

H

-H

Maxwell-Gauss:

Chemical potential:

Drift-diffusion equations for the SHE:

Same electric field as in GMR

Diffusion constants:

Equivalent to Dyakonov- Perel equations if 32

 Two possible stationary states!
Wegrowe « Twofold stationary states in the classical spin-Hall effect » J. Phys. Condens. Matter 29 (2017)

( in ferromagnets)
If the edges are symmetric:

Superimposing (a) and (b):

≠0 Stationary regime reached
before summing (intrachannel)

This is mesured
(1)
Two times the Hall effect. Two stationary states for the subsystems

What is the correct stationary state? if

-Electronic relaxation time << spin relaxation => (1) Stationary regime reached
- Continuity equation with spin-flip => (2) after summing (interchannel)

However, the continuity equation is « pure spin-

coupled to the electromagnetic field (2) current »
(Poisson equation). Role of surface current?
Like for Corbino disk 33
(no charge accumulaion)
 Variational methode for the SHE: the role of surface current
(see slide 5)

The stationary state (minimization of the power functional) :

The spin accumulation is linear
Spin-flip control parameter:

- Non-zero pure-spin transverse current if large enough spin-flip (small-enough ratio l/lsf )
- Non-zero pure-spin longitudinal current whatever the spin-flip

Next: what is the efficiency of a spin-Hall converter?

Work in progress. Call for internship!
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 Linear spin-accumulation observed in dopped Ge
with magneto-optic microscope

linear spin-accumulation

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 Summary Part II
- Lorentz force and Newton’ law: Hall effect deduced from Drude’s model

- General (universal) approach: many « effects » deduced from the symmetries

- Spintronics: spin-flip scattering described as transport equation in the spin space

- Immediate consequence : geant magnetoresistance (GMR) at an interface

- Spin-Hall effect :

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