Jordan University of Science and Technology
Name: - Ala'a Ali Abd-alrahman Qudah.
ID: - 20053092012.
Advisor :-Dr Hasan Al-Khateeb.
Presented to Dr Abdullah Obiedat.
Seminar (phy 791 ).
Research of Spin Polarized Electron.
Ala'a Qudah
1
CONTENTS
Introduction ……………………………………………………………2
Background………………………….………………………………..3
Optical Spin Orientation ………………………..…..………………4
Transport and Escape………………………………………………6
Important of Polarized electrons………………….………………7
Stokes Parameters…………………………………………………..9
Appendix A………………………………………………..…………12
References……………………………………………….…………..15
2
Spin Polarized Electron
Introduction
Although spin polarized electron (SPE) beams now have
wide application in condensed matter, atomic and molecular,
and high energy physics, historically, the development of a
suitable SPE source is certainly not a trivial matter. There is no
simple polarized filter for electrons equivalent to a calcite prism
for light or a Stern-Gerlach magnet for atoms. A number of spin
dependent processes have been tried in attempts to produce
beams of spin polarized electrons [1]. Among these are
scattering from unpolarized target, photoemission from
polarized atoms, Fano effect, and photoemission from gallium
arsenide GaAs [2]. For most applications, photoemission from
GaAs and related materials provides the most suitable source of
polarized electrons.
A source of spin polarized electrons can be characterized
by a number of parameters which allow one to determine how
well it will meet the requirements of a particular application. The
most of these parameters is the polarization itself which define
as (see the appendix)
P
N  N 
,
N  N 
…………… ……. .
(1)
where N  ( N  ) are the number of electrons with spins parallel
(anti-parallel) to a quantization direction. From this definition
we can say that an electron beam is spin polarized if there
exists a direction for which the two possible spin states (spin
up, spin down) are not equally populated.
The ideal polarized electron source would produce a beam with
the maximum polarization, P  1. As an example, suppose we
have an ensemble of 100 electrons, 80 electrons with spin in the
3
z-direction (spin up) and 20 electrons with spin in –z-direction
(spin down). The polarization of electrons along z-axis is P =0.6.
Spin polarization of the electron beam is a vector quantity and
given by

P  Pxiˆ  Py ˆj  Pz kˆ
………..………..………..……………..
(2)
and the total degree of polarization can be written as
P  Px  Py  Pz
2
2
2
……………………………………. ….
(3)
Regarding the direction of the electron spin with respect to its
momentum, we can consider a longitudinally or transversely
polarized beam of electron. For longitudinal, the spins are
parallel or anti-parallel to the momentum of electrons, but for
transverse the spins are perpendicular to the momentum of
electrons.
Background
In the mid-1970s Pierce, Meier, and Zurcher developed an
intense source of polarized electrons based on photoemission
from p-type GaAs [3]. In the last three decades, the use of
this source has become widespread in many areas of
research mainly because of the following advantages: narrow
energy width, quantum efficiency, good degree of polarization
compared with other type of polarized electron sources, high
brightness (brightness B =J/πα2, where J is electron current
density and α is half-angle of the emission cone is formed by
the electron beam emerging from the GaAs), favorable figure
of merit (which is equal to P2I , where I is the current of the
polarized electron), and the possibility of the reversing
electron polarization optically without changing the beam
characteristics. This source have disadvantages such as it
requires an ultra-high vacuum (a pressure of order of 10-10
4
Torr) as well as cleaning (heat cleaning the crystal by
passing current through it) and activating (adding layers of
Cs and oxygen) the crystal.
Photoemission from GaAs can be described in a particular
straight forward way by Spicer’s three step model [4,5]:
Photoexcitation, transport, and escape. The polarized electrons
are generated in the photexcitation process. The transport and
escape strongly effect the quantum efficiency (the number of
electrons emitted per incident photon).
Optical Spin Orientation
GaAs is a direct gap semiconductor with a minimum band
separation Eg at Γ as in the E(k) plot of the energy bands
versus crystal momentum k
show in figure 1.
The
conduction band is a two-fold degenerate s1/2 level. The
spin-orbit interaction splits the six-fold degenerate for p-state
of the valence band into a four-fold degenerate p3/2 level and
two-fold degenerate p1/2 level. The difference of energy
between the two levels p3/2 and p1/2 is ∆ = 0.34 eV. The
energy band gap between valance and conduction bands is
Eg = 1.42 eV. The relative allowed transition probability
between the valence and conducting bands are given by the
numbers in the circles.
The only way to make polarized electrons is to produce
different populations of the degenerate mj states in the s1/2
conduction band. This requires the use of circularly polarized
light to derive the electrons from the valence band to
conduction band.
For example, right-handed circularly
polarized light with an energy of about 1.4ev can derive two
possible transitions because of the selection rules ∆mj=+1.
The first transition, │3/2 -3/2 > → │1/2 -1/2 >, has an
intensity three times greater than the second transition │3/2
-1/2 > → │1/2 1/2 >. This leads to the net electron spin
5
polarization P = (3-1)/(3+1) =50%. Similarly, by using lefthanded circularly polarized light (the selection rule ∆mj=-1),
the net polarization becomes P = -50%.
With more energetic light, the transition of the electrons
from the p1/2 valence band can occur. This will create an
equal population of electrons in the two states of the
conducting band, and leads to zero net electron polarization.
Therefore, it is necessary to choose the light energy of a laser
hν in the range E< hν < Eg + ∆ in order to get the maximum
electron polarization. It is possible for the electron to flips it is
spin through collisions in the conducting band and /or at the
crystal surface, the measured electron polarization of the
GaAs crystal is always less than the maximum theoretical
Value (50%) [6].
Figure 1: Energy bands in GaAs and the technique for
producing polarized electron source.
6
Transport and Escape
The electron affinity χ of p-type GaAs is large (about 4eV).
It represents the difference in energy between the vacuum
level (where electrons are free) and the conducting band
prevents electrons from leaving the crystal surface. As
shown in figure 2, adding layers of cesium to the crystal will
reduce the electron affinity to values close to zero.
Furthermore, adding layers of cesium and oxygen will reduce
the electron affinity to negative values (negative electron
affinity (NEA)) [7]. NEA GaAs is able to extract photoelectrons
with high quantum efficiency.
Adding several layers of cesium and oxygen (is called
activating the crystal) is able to increase the photocurrent of
electrons to many orders of magnitude (see figure 3)
Figure 2: the band structure of p-type GaAs near the surface
after adding layers of Cs as well as Cs2O.
7
Figure 3: Schematic of the activation of the GaAs by adding
alternating cycles of Cs and O2.
Important of polarized electrons
1) it is important to know the spin state of incident electron
just as it is important to electrons with well-defined energy
and momentum .
2) studying
spin-dependent
electron-atom
scattering
processes such as spin-orbit interaction between the
incident electron and the atom as well as the exchanges
between them.
3) knowing the spin of electron will save us from losing
information abut the system due to averaging over any
parameter depends on the electron spin
8
4) the polarized-electron atom collisions can provide valuable
information about collision mechanisms and electron
correlation.
Many studied have been conducted in the field of polarized
electron-noble gas atom. They focus on measuring the
polarization of the emitted light from excited states as shown in
this reaction.
electron ()  Atom  ( Atom )  electron  photons( )
The polarization of the light emitted from excited state
provides us with a lot of information about the processes during
the collision between the electron and atom.
To describe the polarization of the light completely, one must
measure the stockes parameters P1, P2, and P3. The P1 and P2
represent the components of linear polarization, where P3 gives
the circular polarization. For example, in 1969 Fargo and wykes
[8] pointed out that if atoms are oriented by spin exchange
processes, the oriented atoms will emit circularly polarized light
in the direction of the electron spin polarization , and P3 will be
proportional to the polarization of incident electron.
In 1980 Eminyaus and Lamped measured the effect of
longitudinally polarized electron exciting a zinc target. The first
experiment that measured extensively the integrated Stokes
parameters for mercury using polarized electrons was done by
Wolcke in 1983 [9,10].
Gay et al measured the polarization of the fluorescence
emitted by the noble gases He, Ne , Ar ,Kr and Xe . Using
Stokes parameters as tools to study the spin-dependent
phenomena of polarized electrum-atom collision has been
discussed in detail by Bartschat et al [11,12]. For electronphoton non-coincident experiment where the scattered electron
are not detected, they study the effect of the incident electron to
on the angular distribution and the polarization of the emitted
photons. Also , they introduced the integrated state multipoles
9
Tkq (J ) as a tool to describe the physical characteristics of the
excited states and their relationship with stokes parameters.
For transversely polarized electron- atom experiment the
three measured integrated stokes parameters of fluorescence
from excited state of an atom can be described by four
independent state multipoles T00 ( J ) , T11 ( J ) , T20 ( J ) , and T21 ( J ) .
The T00 ( J ) is called a monopole moment and it is proportional
to the total cross section σ. The T11 ( J ) is called magnetic dipole
T20 ( J ) and T21 ( J ) are called electric quadrapole
moment,
moments. The state multipole can reflect the spin dependence is
excitation collisions, they provide information about anisotropy
and geometric detail of excited states and they yield the relative
differential cross section of magnetic substates.
Stokes Parameters
To determine the polarization properties of the light
completely, a set of three independent quantities must be
measured. These quantities are called Stokes parameters and
they are related to the absolute intensity of linearly and
circularly polarized light.
To determine these parameters, we have to measure the
intensity of the light after passing through rotatable retarder
and linear polarizer (this combination of optical elements is
called optical polarimeter) as shown in figure 4.
10
Figure 4: Analyzing system for the polarization of the light.
The light propagates along y-axis. The retarder has a fast axis
that makes an angle  with respect to the z-axis, and the linear
polarizer has a transmission axis that makes an angle  with
respect to z-axis.
The intensity of the transmitted light I ( ,  ,  ) of the
incident light beam which has Stokes parameters P1, P2 and P3
can be written [13]:
I P
I ( ,  ,  )  1  1 cos2(   ) cos2   sin 2(   ) sin  cos 
2 I
P
+ 2 cos 2(   ) sin 2   sin 2(   ) cos 2  cos 
I
P 
+ sin 2(   ) sin   3  ,
I

11
………...
(4)
where I is the total intensity of the light, and  is the retardance
of the retarder.
To measure the linear polarization of light P1 and P2, we
have to remove the retarder or set the values of  and  to
zero.
P1 and P2 are the linear polarization fractions
corresponding to 0º and 90º as well as 45º and 135º,
respectively.
I P (0  )  I P (90  )
P1 
I P (0  )  I P (90  )
(5)
……..
I P (45  )  I P (135  )
P2 
……...
I P (45  )  I P (135  )
(6)
And the circular polarization friction of the light can be written
as
P3 
I (0  ,45  ,90  )  I (0  ,135  ,90  )
I (0  ,45  ,90  )  I (0  ,135  ,90  )
.. ……. .
(7)
Furthermore, it possible to express the intergrated Stokes
parameters in terms of state multipoles [14]:
1
P1  3 / 2 
J
1
P2  3 / 2 
J
1
P3   3 / 2 
J
1
J
2 
 T ( J ) /  , … …..
J f  20
1
J
2 
 Re T21 ( J ) /  ,
Jf 
1
J
12
2 
 Im T21 ( J ) /  ,
Jf 
.
……….
(8)
(9)
(10)
where
J J
2(1)
1 1
I
T00 ( J ) 

3 2J  1
6 J
(11)
J
K
J
TKQ ( J )   (1) J  M (2 K  1)
M
M  M  Q
f
1
J
2 
 T (J ) ,
J f  20

 JM  JM ,

(12)
:::are 3-j coefficients, and K and Q take the values
 J  K  J and  K  Q  K , J and Jf are the total angular
momentum for the initial and final states of the system
respectively.
Appendix A: Quantum Mechanics Background
In quantum
mechanics, the spin can be represented by the

operator S which can be written in terms of pauli operator:
  
S   . The matrix representations of the Cartesian component
2
of the pauli operator can be given by:
0  i
 0 1
1 0 
,  y  
,  z  
.
 x  
1 0 
i 0 
 0  1
(13)
these operators have meaning from their applications to the two
component wave functions
 a1 
  which
 a2 
represents the two
possible orientation of the electron spin. For example, the
eigenvalue equation
1  1 0  1 
1 
 0  1 0   0 
1 
    1 ,  z    
    1 ,
 z    
 0   0  1  0   0 
1   0  1 1 
 0
(14)
1 
this means that   is an eigenfunction of  z with eigenvalue +1
0
1 
(or  / 2 ) and   belongs to eigenvalue -1.
0
 
13
We can use these two states as a basis for representing
the
a 
general
state    1  as
 a2 
a
linear
superposition
 a1 
1 
 0
   a1    a2   .
,
For normalized
this leads to
 0
1 
 a2 
a 
2
2
2
   a1* a2*   1   a1  a 2  1. The a 1 represents the probability of
 a2 
1 
finding the electron in the state   ”spin up”, and a 2 represents
0
2
1 
the probability of finding the electron in the state   ”spin
0
down”.
The spin state can be pure spin state of mixed spin state.
The system of electron is said to be in a pure spin state if all the
electrons are in the same spin state. The polarization of the
electron can be defined as an expectation value of pauli spin
operator



 a 
P        a1* a2*   1  .
 a2 


(15)
For mixed spin state, the beam of electrons is partially
polarized. This means that the electrons are statistical mixtures
of spin states. In this case, the polarization of the total system
is the average of the polarization vectors P (n ) of individual
systems which are in pure spin states  (n) :

(n)

P   ( n ) P   ( n )  ( n )   ( n ) ,
n
n
(16)
where the
 (n) 
 (n ) is
the weighting factors, which given by
(n)
N
, where N (n ) is the number of electron in the state  (n) .
(n)
N
n
To describe the polarization of electron we have to use the
density matrix  which defined as
14
 a( n) 2
 1
   
n
 a1( n )*a2( n )

a1( n ) a2( n )* 
.
2

a2( n )

(n)
(17)
The relationship between
polarization is given by
the
density
matrix
and
the


P  tr . Also, it is possible to write the elements of the density
matrix in terms of the components of the polarization
0
1 1  P
.
1  P 
2 0
  
(18)
This form of density matrix illustrates the meaning of
2
polarization. a1( n ) is the probability that the eigenvalue  / 2 will
be obtained from a spin measurement in z-direction for nth
2
system. This probability is equal to  ( n ) a1( n ) , which can be
n
expressed as N  /( N   N ) . Similarly

n
( n)
2
a2( n) 
N
is the
N  N 
probability that the value of   / 2 will be obtained. Therefore,
N
1
 (1  P).
N  N  2
This is lead to well-known definition of
polarization of an ensemble of electrons in a mixed spin state
P
N  N 
.
N  N 
15
References
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[2] A .Green ,Gallium Arsenide as a Source of Spin Polarized
Electrons (1996).
[3] D.T.Pierce,F. Meier, and Zurcher, phys. Lett.A51, 465
(1975).
[4]
H.W.B.Skinner
and
E.T.S.Appleyard,Proc.Roy.
Soc.A.117,227(1927).
[5] J.R.Oppenheinner,Z.Phys.43,27(1927).
[6] K. Jost, J. Phys. E 12, 1006 (1979).
[7] J.R.Oppenheinner,Proc.Nat.Acd.Sci.13,800 (1927).
[8] P.S. Farago and J. S. Wykes, J. Phys. B2, 747(1969).
[9]
J.E.
Furst.T.J.Gay,W.M.K.P.Wijiayaranta,
J
Phys.B25,1089(1992)
[10]T.J.Gay,J.H.Furst,K.W.Tranthan
and
W.M.K.P
Wijayaratna, Phy,Rev A 53,163(1996).
[11]K.Bartschat,K.Blum,G.F.Hanne,and
Kessler,J.Phys.B14,3761(1981)
[12]K.Barstchat,K.Blum.Z.Phys .A304,85(1982)
[13] H.M. Al-Khateeb , Angular Momentum Partitioning and
hexacontatetrapoles in impulsively – excited argon ions .
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Lett.59,1413 (1987).
16