Supplementary materials
Spin current formation at the Graphene/Pt interface for magnetization
manipulation in magnetic nanodots
A.M. Shikin1, A.A. Rybkina1, A.G. Rybkin1, I.I. Klimovskikh1, P.N. Skirdkov2,3,4, K.A. Zvezdin2,3,4,
A.K. Zvezdin2,3,4
1
Saint Petersburg State University, Saint Petersburg, Peterhof, Ulyanovskaya str. 1, 198504
Russia
2
A.M. Prokhorov General Physics Institute, Russian Academy of Sciences, Moscow, Vavilova st.
38, 119991 Russia
3
Moscow Institute of Physics and Technology, Dolgoprudny, Institutskiy per. 9, 141700 Russia
4
Russian Quantum Center, 143025 Skolkovo, Moscow Region, Russia
Spin current developed at the Graphene/Pt interface due to application of an electrical or
thermal gradient.
The process of the spin generation in the graphene-derived system with strong spin-orbit
(Rashba) interaction can be described by the following Hamiltonian:

H SO   ( p ) Iˆ   px y  p y x  (1),
where  ( p)   k is the dispersion law for the electrons of graphene, Iˆ is the Identity matrix,
 is the Rashba constant (  k F =80 meV),  x , y are the Pauli matrixes. In the case of the
presence of an electrical or temperature gradient we can describe the spin accumulation in the
system with strong spin-orbit interaction, using the Boltzmann equation in time-relaxation
approximation with additional force Fx , with FxU  eEx in case of applied electric field Ex , and
FxT   T x     T  if it caused by the temperature gradient, where  is a chemical
potential. The solution of this equation can be represented in the form:
f
f  f 0   x Fx 0
(2),

where  is the relaxation time, f is the distribution function of electrons, f 0 is the unperturbed
distribution function of electrons. Application of the thermogradient T x or electric field Ex
along x direction causes a spin current alone the x axis. In our case spin current can be defined as
follows:


e
  f 0 
x
x
x
2   f 0
jS  j  j  B 2 Fx  d k 

 (3),
 k x k x k x k x 
where B   n 2     F  , n is the electron density in graphene, f 0 is the unperturbed
distribution function of majority/minority electrons. The following relations can be obtained
from Hamiltonian (1):
  
 kx 
 kx 
 k    k     k 
 
 
 x
(4).
 

 f 0  f 0      k x
 k x

k
2
1
Using the relations (4), the spin current in case of the electrical field can be rewritten as:
2 x 2
jSx 
j 
 Ex (5),
 e

where  is the graphene conductivity, jex is the electric current along the x-axis. In case of the
thermogradient the spin current can be rewritten as:
2 x
2 T
jSx 
je   S
(6),

 x
en 2 
where S 
is the Seebeck constant.
 F T
In addition to the spin current, under application of the electric field Ex or the temperature
gradient T x along x direction an uncompensated density spin appears in the y direction
(because the spin of electron in the Rashba model is locked perpendicular to the momentum).
The spin accumulation   B  d 2 k  f   f   ey , where e y is the unit vector along the y-axis,
after substitution of (2) can be transformed to the following form:
n
F
(7).
 x   z  0 and  y  
F x
Substituting the expression of Fx for the cases of thermogradient and applied electric field, we
can obtain spin accumulation in following form:
ne

T
U
T
S
 y  
Ex and  y  
(8).
2
e 
x
F
Application of the Graphene/Pt interface for the magnetization dynamics of the (Ni-Fe)nanodots array due to spin-orbit torque effect.
Spin current in Graphene/Pt plane enters to the NiFe contact and affects the effective magnetic
field in NiFe. Effective Hamiltonian in this case can be described as:
H   ( p)  H R  JM 0 
(9),
where M 0  M M S is the unit vector along the magnetization direction. The last term is related
to the induced magnetic field,  is the uncompensated spin, M S is the saturation
magnetization of magnetic material, J is the exchange integral. The second (Rashba) term
H R    k  z0   leads to the appearance of the effective magnetic field BSO :
 kF
J
BSO  
P  je  z 0  j 0  , where j 0 =1, P 
(10).
e M S
F
This additional effective field leads to appearance of the spin torque, acting on magnetization in
NiFe layer. Corresponding spin torque per unit volume is given by:
 kF
k
TSO   M  BSO  
Pje  M  j 0   F Pje  M 0  j 0  (11).
e M S
e
2