Introduction to the Spintronics 2014 at Korea University Report
Deadline
31st
3, Oct., 2014
Oct., 2014, Submission: e-mail to suzuki-y@mp.es.osaka-u.ac.jp
1. Choose two problems from following 4 problems and answer to them.
2. Write your impression/opinion of/to the lecture.
I. Magnetic Crystal
For a linear chain of atoms with a single spins on, show that a spin wave state is an
eigen state of the following Heisenberg Hamiltonian,
 
 Sˆ  Sˆ ,i Sˆ ,i 1  Sˆ ,i 1 Sˆ ,i  Sˆ ,i Sˆ ,i 1  Sˆ ,i 1

H   J  S i  S i 1   J    , i

 Sˆ z ,i Sˆ z ,i 1 


2
2
2i
2i
i
i 

2
 Sˆ z S , S z  S z S , S z , Sˆ S , S z  S S  1 S , S z

 Sˆ  Sˆ  iSˆ , Sˆ , Sˆ   Sˆ , Sˆ S , S  S  S S  S  1 S , S  1
x
y

z


z
z
z
z
 


.
And obtain dispersion relation of the spinwave (magnon).
Hint: Spinwave state
 n  Sˆ, n  S , S , j , single spin(chang e of spin direction) is excited at position n.

j
 
 
 ik  x
 k   e n  n , Spin wave state
n


II. Please answer to following questions about spin-dependent diffusive transport.
Take a z-axis as a quantization axis and define ↑ and ↓ as upwards and downwards
spins along the z-axis. We also define electro-chemical potentials in a ferromagnetic
material with z-upwards magnetization as
 s   s  e . Here, s is ↑ or ↓.  s is the
electro-chemical potential, e elemental charge,  electric scalar potential. For steady
states, the electro-chemical potential satisfies following relations. Here, the system is
assumed to be uniform along y- and z-directions,
 d2
        / l 2 ....(1) ; Diffusion and relaxation of spins

 dx 2 
,
 2
d

        0
....(2) ; Conservati on of charges (charge neutrality )

 dx 2  
Where l is spin-diffusion length,
ｓ is a electric conductivity for an electron system
with spin s. The current density for the electrons with spin s is given by j s 
1)
 s d s
e
dx
.
Imagine an Fe wire with uniform magnetization along +z-direction, which
cross-section is 1.0 [m2]. Conductivities of the wire for two spin sub-channels are
   7.0 [ 1 m 1 ] and    3.0 [ 1 m 1 ] . Calculate voltage appeared in each
1.0 [mm] along the wire when we pass 1.0 [mA] through the wire.
2)
Next, we think about a half-metal wire with the same cross-sectional area.
Half-metal is a material in which ↑ spin band is metallic but ↓ spin band has Fermi
energy
in
a
band
gap.
Here,
let’s
assume
that
our
material
has
   1.0 [ 1m 1 ],    0.0 [ 1m 1 ] ，and a gap in ↓ spin band is 100 [meV]. In
addition, the Fermi level is at center of the gap under zero bias voltage.
Now, we magnetize a half of the wire (x<0) towards +z direction, and the rest (x>0)
towards +z direction. Please discuss about I-V characteristics of the system using a
view graph.
3)
Then we change a left part of the wire (x>0) by a non-magnetic Cu wire with
   Cu ,   Cu ,   60.0 [ 1m 1 ] . Assuming l  0.1 [ m] of a spin-diffusion
length in Cu, explain a distribution of the electro-chemical potential of the system
under a constant current of 1 [mA] using a graph.
4)
We continue to discuss about a hetero-connection described in the question 3).
From charge conservation (neutrality condition), we may easily show following
relations;
J

     e A x ; x  0
….(3)

 1   1   e J x  b ; x  0
 2  2 
A
Also, from spin-relaxation equation, we may get;


   ae  x / l
; x  0 …(4)
where a and b are unknown parameters. J  J   J  is a total current. A is a
coross-sectional area of the wire.
In x>0, express  and
5)
 using a, b, J , A, x and e,  , l .
At x=0, ↓ spin current is zero because of the gap. Using this condition obtain
spin-splitting in the electrochemical potential at the interface, a.
6)
The spin-accumulation caused by the spin-splitting results in an appearance of
additional interface resistance. Please estimate its value.
III. From ballistic transport to diffusive transport
We think about an electric conduction in an
(a)
e-
one-dimensional wire with one quantum channel
(Fig. 1 (a)). Wire with a scatterer (Fig.1 (b)) may
have
following
conductance
given
by
V
A
(b)
r
Landauer-Buettiker formula:
2e 2
T.
h
G
t
1
(c)
(1)
2
t
t
1
Here, e is the elementary charge, h Planck
constant, T transmittance of the scatterer. T is
2
r
r
Fig 1.Ballistic transport in one
dimensional wire
expressed by using amplitude transmittance t as
follows; T  t .
2
If there are two scatterers (Fig. 1 (c)), by considering multiple scattering between them,
we get following transmittance.
G
2e 2
T2
,
h 1  r 2 e 2 i 2
(2)
where r is an amplitude reflectance and  is a phase change because of an electron
travel from one scatterer to another.
1) Show derivation of equation (2). Show a graph of conductance as a function of a
distance between two scatterers. Obtain minimum and maximum conductance.
2) For a case of random distribution of scatteres, we may take average with respect to
the phase change and obtain following conductance.
G
1
2

2
0
d
2e 2
T2
h 1  r 2 e 2i
2

2e 2 T 2
2e 2 T

.
h 1 R2
h 1 R
(3)
Show that the equation (3)
is equivalent to a diffusive
(a)
R
sum of the multiple scattering
problem that is illustrated in
T
R
1
T
1 R
R
R
R
T
T2
T
TR 2T
T
TR 4T
T
TTn
T2
1 R2
Fig. 2 (b).
3) Assuming that a system with
(b)
R
Rn
n scatterers has diffusive
Rn
transmittance of Tn , obtain
diffusive resistivity of the
system with n+1 scatterers
Tn
R
R
T
TRRnTn
T T RRn 2 Tn
TTn
1  RRn
Fig. 2 Diffusive multiple scattering. (a) Two
equivalent scatterers. (b) Addition of a scatterer.
(Fig. 2 (b)).
By considering T1  T , derive following expression of the resistivity in the system
with N scatterers.
Resistivit y 
h h 1 T

N.
e2 e2 T
(4)
This equation provides Ohm’s law with an interface resistance.
IV. Spin-dependent tunneling
Tunneling barrier
with spin momentum,
We have a spin-polarized tunneling barrier. The

barrier has its own spin angular momentum, S .
The
barrier
transmits
up-spin
Fig. A
electron
↑
,
,
completely without change in phase neither in
amplitude.
While
for
down-spin
amplitude
transmittance
and
,
electrons,
amplitude
↓
reflectance are t and r, respectively.
(1) Calculate the transmittance, t  ,   , for an
2
electron with arbitrary spin.
Hint: spin function for an electron with a
spin that points
 , 
t,
r,
Fig. B
↑
?
direction in the polar
?
coordinate is expressed as:
 ,  cos

2
up  ei sin

2
Plane L
down .
Plane R
(2) By comparing the transmittance for up-spin and down-spin electron, show following
relation between the magneto-resistive effect defined as follows and the r and t.
t   0,    t    ,  
2
MR 
t    ,  
2
2

r
t
2
2
.
Hint: you may use unitarity condition of r and t.
(3) For incident electrons with pure spin-polarization along
 , 
direction, estimate

spin currents at L and R planes and obtain the spin-transfer torque exerted on S ,
assuming conservation of the total spin angular momentum of the system. Here, we
have one electron par second as for the incidence.
(4) Estimate the magneto-resistive effect for a current incidence with incomplete
spin-polarization that is expressed by following density matrix,
u
  
m
m
.
d 

S