Quantum Transport :Atom to Transistor,Supplementary Topic: Spin
Transcript of Quantum Transport :Atom to Transistor,Supplementary Topic: Spin
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Quantum Transport:Quantum Transport:Atom to Transistor
Prof. Supriyo DattaECE 659Purdue University
Network for Computational Nanotechnology
04.30.2003
Lecture 42: Supplementary Topic: SpinRef. Chapter 5.4 & 5.5
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In this lecture, well discuss
electron spin
Recall, when deriving current
through a small structure, like
that show on the right, we first
obtain a Hamiltonian [H].
Usually, however, theeigenenergies of [H] represent
two degenerate spin levels
The proper Hamiltonian for degenerate
spin levels is twice as big
with no coupling between the spin up andspin down portions
Small Device
1 2
[H]
HO
OH
Spin00:00
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We want to consider the conditions
for which the spin levels are not
degenerate. There are two cases
by which this arises
Case 1: Strong Electron-Electron
Interactions (i.e. Coulomb Blockade)
Because electrons are forced to
interact strongly instead of 2
degenerate levels
we have two split levels
the first level fills
and forces the
second up
Important Example: Magnetism.
All electrons want to have the same
spin due to electron-electroninteractions.
Strong ElectronInteractions
02:20
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Case 2: Spin Orbit Interaction
This has nothing to do with
electron-electron interactions and isthe focus of the remainder of this
lecture.
Continuing with spin-orbit
interaction
Ordinarily the Hamiltonian is
When a magnetic field is applied,
say in the direction, up and down
spin levels do not remain degenerate
so we add
where B is the Bohr magneton
constant . A well knownexample of this is the Zeeman effect
+
+
UmpO
OUmp
2
22
2
Z
Z
B
B
B
0
0
z
mq 2/h
Spin Orbit Interaction06:08
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If we apply a magnetic field in the x-
direction as well the additional term
becomes
To summarize, the magnetic field
gives
and the magnetic field gives
These matrices have the same
eigenenergies but different
eigenvectors. The eigenenergies
are
+BBz, -BBz,
+BBx, -BBx
x
01
10xBB
+
ZX
XZ
BBB
BB
z
1001
ZBB
Eigenenergies08:55
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The eigenvectors are
direction
direction
One might imagine that we could
have written the additional term as
this model does explain theZeeman effect but fails under what
is known as the Stern-Gerlach
experiment
+
+22
22
0
0
xz
xz
B
BB
BB
x
z +BBz
-BBz
0
1
1
0
+BBx
-BBx
1
1
1
1
Note the different eigenvectors
Eigenvectors11:15
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Example Continued:
And if we add another
inhomogeneous, say in the x-
direction, the beam splits again
In the Stern-Gerlach experiment a
beam of electrons is injected into an
inhomogeneous magnetic field, due to
spin interaction the beam splits after
entering the magnetic field.
Example: A beam of electrons split by
an inhomogeneous magnetic fieldz
0
1
1
0
Bz
high
low
1
1Bz
high
low
1
1
1
1
1
1
Bx
high
low
Stern-Gerlach Experiment12:39
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Overall the correct term,
valid for any direction, added
to the Hamiltonian is
Note, regardless of magnetic
field direction the splitting
effect is the same though theeigenvalues may vary
Often this matrix is written as
where
and
+
ZyX
yXZ
BBiBB
iBBB
Bvv
zyx
zyx
BzByBxBzyx
++=++=v
v
+=
=
=
=
zyx
yxz
yxz
BiBBiBBBB
i
i
vv
0
0
01
10
10
01
Eigenenergies16:59
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Moving on we can define
spin-orbit interaction
Recall, an electric field may
be expressed as
and a magnetic field as
A scalar potential, U, is
simply added into the
Schrdinger equation
But a vector potential, , is incorporated
as a dot product with the momentum vector
, i.e.
t
AE
=
vv
AB
vvv
=
Av
pv
( )
( )
++
++
Um
Aqp
U
m
Aqp
20
0
2 2
2
vv
vv
Vector Potential19:40
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Interestingly, even when there is
nomagnetic field a strong electric
field can induce an effectivemagnetic field on an electron. This
phenomena occurs due to
relativistic interactions between an
electron and the applied electric
field. Interaction between the
electric field of the nucleus and
electrons of an atom is a very
good example of this. We call this
effect Spin-Orbit Interaction
Spin-Orbit Interaction is incorporated
into the system Hamiltonian with the term
Note: Even in a weak electric field spin-
orbit interaction is present but generally
is too small to deserve any consideration
22mc
pEB
vvv
Spin-Orbit Interaction23:42
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Spin-Orbit coupling plays a
fundamental role in our understanding
of semiconductor physics
Given a semiconductor with an E-k
diagram like the following
If we were to examine the area
right around the dot we would find
that conduction band was formed of
|s> orbitals and the valence band of
|p> orbitals (|px>,|py>,|pz> or in
another notation |x>, |y>, |z>)E
k
Semiconductors25:34
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Ignoring spin orbit coupling one might
assume semiconductor valence bands
of the form |x>, |y>, and |z> separately.If this were the case then a transition
from the conduction band, |s>, to say
|x> would be polarized in the x
direction. Furthermore, since there is
no preference in transition to the
valence band y-polarized and z-
polarized light should be equally likely
resulting in overall isotropic emission
However, this is not the case,
emission is not isotropic but in fact
circularly polarized. We cannotignore spin-orbit
coupling/interactions. When
included, spin-orbit coupling
provides an entirely different (but
correct) set of valence bands that
accurately predict the circularly
polarized light emitted from in
semiconductors
Isotropic Emission27:24
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The first valence band
derived with spin-orbit
coupling is
and
we call this the heavy hole
band. We will ignore the
other valence bands (light
hole) for now since most
optical transitions occur to
and from the heavy hole band
Heavy Hole Band
E
k
Note: The |s>
conduction orbital is
not usually affected by
spin-orbit coupling
+
0
yix
yix
0 Ex:
will produce circularlypolarized light in the x-
y plane. So we see
what a dramatic
physical effect spin-
orbit coupling can have
+
00
yixsto
Correct Valence Bands29:03
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Important Point: Even when
electrons travel at a fraction of
the speed of light relativistic
phenomena can produceobservable effects on spin
Note: In somesemiconductors, such as InAs,
spin-orbit interaction can alter
even the conduction band
So how does one make sense of the spin-
orbit interaction term
To begin, look at the vector term in theHamiltonian (ignoring the spin-orbit term
and any scalar potential U):
where I is a 2x2 matrix
22mc
pEB
vv
v
BIm
AqpH B
vvvv
+
+=
2
2
Spin-Orbit Interaction34:00
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Note that
is equivalent to
where is composed of the Pauli spinmatrices
Furthermore, without any vector potential
,
( )BI
m
AqpB
vvvv
++
2
2
( )[ ]m
Aqp
2
2vvv+
v
Av
( )
=
=
mp
mp
m
pH
20
02
2 2
22vv
But if we include and apply a
few dot and cross product
identities to the familiar
term in is produced
Point: is related to thegeneral vector Hamiltonian and
does not simply appear from
nowhere
1
2
Av
BB
vv
BBvv
Vector Hamiltonian37:00
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Now, the spin-orbit interaction term
is relativistic and can only be derived from the
full relativistic Hamiltonian
The full relativistic Hamiltonian, otherwiseknown as the Dirac equation, is
Note: I is the 2x2 identitiy matrix, c is the
velocity of light, and is replaced by
when a magnetic field is applied
=
Imcpc
pcImcH
2
2
vv
vv
So how do we get the
familiar, low velocity,
Schrdinger equation
from the Dirac equation?
22mc
p
B
vvv
pv
Aqpvv
+
Im
pH
2
2
=
Deriving the Spin-OrbitTerm
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Recall, the E-k
relationship at low
electron energies, forwhich
is valid looks like
Whereas the high energy Dirac equation produces
an E-k relationship of the form
Im
pH
2
2
=
E
k
E
k
- - -
-
+--
-when an electron
moves up it leaves
a position
+mc2
-mc2
bottom band
filled with
electrons
E-k Relationships42:00
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E
k
- - - -+--
-when an electronmoves up it leaves
a position
+mc2
-mc2
bottom band
filled withelectrons
Note however, that moving to the top band
requires an energy equal to 2mc2 (on the
order of 1MeV)
For the most part, outside
high energy experiments, only
a few electrons reside in the
top-band. Importantly, we cangreatly simplify the Dirac
equation by concentrating on
the few electrons in the top
band and ignore the bottomband completely. Based on
this assumption we can derive
the familiar low energy
Hamiltonian.
Only a Few at the Top43:44
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Conceptually, in the low energy regime we
can concentrate on term in the Dirac
Equation
and treat terms , , and as a self-
energy
Recall, for any system, H, coupled to alarger system, HR, we may define a self-
energy
Coupled Systems
System Large System
H HR
Imcpc
pcImc2
2
vwvw
1 2
3 4
+= 1)( RHE
For the Dirac equation
and
2mcH
pc
R =
== +vv
( ) ( )pcmcEpcvvvv
+= 2
1
Dirac Equation Self-Energy
45:40
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Furthermore, since the electrons we
are concentrating on, which lie in the top
band, have an energy around mc2 we
may make the approximation E=mc2.Thus,
So, overall the non-relativistic first-
order Hamiltonian is
( ) ( )
( )m
p
pcmc
pc
2
2
1
2
2
vv
vvvv
=
=
( )m
p
ImcH 2
2
2
vv
+=
But mc2I, as constant, may be
ignored which gives the familiar
To include spin-orbit interactions
we need to improve our
approximation for E, that is E
mc2, to something more accurate
( )m
pH
2
2vv =
First Order Self-Energy47:00
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Including a more accurate representation of E complicates the algebra
considerably, nonetheless by including a next order approximation of E we
are able to derive the familiar spin-orbit interaction term
from the Dirac equation
Main Point: Spin-orbit interactions, magnetic interaction, vector potentials,
etc. all follow from the Dirac equation and are all fundamentally related
22mc
pE
B
vvv
Closing Comments48:10