This is the last part of my guest lectures on electron spin, in which I'll
focus on normally non-magnetic systems which are pushed out of spin degeneracy
equilibrium. By spin injection from an external
thermomagnetic source. I'm going to first tell you why we
want to study these systems, and why this is such a hard problem or at least why
the most straight forward approach to solving it is bound to fail.
Then I'll show you one particular way that I have personally used, to obtain
spin injection in silicon and germanium semiconductor devices.
And a little bit about what can be learned as a result.
To motivate the kinds of information about non-equilibrium spin transport
we're after, I want to appeal to the history of non-equilibrium charge
transport. Namely minority carriers and
semi-conductors. The seminal measurements were done by
Haynes and Shockley in the mid-20th century at Bell Labs.
In these timed domain experiments, a narrow pulse of minority electrons, were
injected into p-type semi-conductors filled with equilibrium holes.
An electric field carried these electrons to a charge detector, where the pulse
could be analyzed. By measuring the time of flight, they
could determine the minority carrier mobility or the proportionality constant
between applied electric field and electron velocity.
By measuring the spreading of the pulse in time, they determined the strength of
random thermal fluctuations from scattering, the minority carrier
diffusion coefficient. And by integrating over the pulse and
determining how many electrons made it without annihilating with a positively
charged hull, the minority carrier lifetime could be determined.
Without these values, none of the solid state devices we use today could be
designed and successfully made. So, if we speculate any use for
spin-polarized electrons out of equilibrium in semi-conductor devices.
Then we at least need to be able to measure the spin analogs of these
parameters. So, let's first look at why it's not so
easy to transfer the substantial spin and balance, spin polarization.
From a ferromagnet, into a non-magnetic electronic material, especially when
using a semiconductor in an effort to make, for example, new kinds of spin
transistors. The most straight forward way one might
naively try, is to make an ohmic or linearly resistive contact between the
ferromagnet and a semiconductor. Let's see how this works for plain old
charge injection. Ohm's law says that the charge current
density flowing, is proportional to the conductivity, and driven by a spacial
gradient of an electro-chemical potential.
This combines the effects of electric field, gradient of electro-static
potential, with flow from high concentration to low via random thermal
fluctuations comprising the random watts of diffusion.
If we incorporate the device geometry, we can use this expression to recover the
more familiar V equals IR form of Ohms Law.
In metal-semiconductor ohmic contacts, current is conserved across the interface
but the connectivity in the metal is large.
So, the potential energy provided by voltage q times v drops mostly across the
lower connectivity semiconductor. Now, if we want spin injection to
accompany this charge injection, then, the currents for spin up and down must be
different. Since the conductivities for up and down
are the same in a semi-conductor, the re, the respective electrochemical gradients
must be different. This, is what we need to happen on the
semi-conductor side. Spin up and down electrochemical
potentials have different gradients to drive asymmetric current densities
comprising a spin current. The unavoidable consequence of this
asymmetry, is an electrochemical splitting at the ohmic interface.
Using Ohm's law, we can obtain a relationship between current polarization
and this electrochemical potential splitting.
The first tern, in parenthesis, is due to the average potential drop over the
transport length scale, L. Note that, although, J up and J down are
not equal, their sum does equal the total charge current, as expected.
Now, we have to derive equivalent expressions, on the ferromagnet side, or
we will see the deleterious effect of the splitting on spin eject.
On the ferromagnet side, the electrochemical potential splitting
relaxes to zero in equilibrium, due to spin flips away from the interface.
The spin relaxation length scale is a so called spin diffusion length, lambda.
By once again applying Ohm's Law, we get the following expressions for the current
densities of spin up and, and down on the ferromagnet side.
Note that there are two important differences between these expressions for
the ferromagnet and the ones above describing transport in a semi-conductor.
First, the spin dependent connectivities are not equal to the ferromagnet, due to
their dependence on the asymmetric carrier densitites.
Second, and as a result, the deviation of the interface electrochemical potentials
from equilibrium are not symmetric. In other words, c-up is not equal to
c-down in this figure. However, because the ideal interface
preserves spin, the electrochemical potentials are continuous.
So, we do have the sum rule, giving us a total splitting, which we need to
determine the spin polarization, flowing across the interface in the
semiconductor. Using the definitions of both the
injected current polarization and the bulk ferromagnetic polarization, we can
derive a simple expression for the splitting on the ferromagnet side.
In intimate ohmic contact, the electrochemical potentials are
continuous, so this is the same as the splitting of the semiconductor side.
We can therefore substitute it into our previously-derived expression to obtain
this result. Note that this is very different from our
naive expectation, since it depends strongly on the magnitude of the
dimensionalist parameter, epsilon. The ratio of conductivities and transport
lengths across the interface. If epsilon is much less than 1, then,
only when the bulk magnetic polarization, beta, is approximately 1 a half-metallic
ferromagnet. Do we recover the desired case, where the
injected current polarization P, is approximately equal to the bulk
ferromagnetic polarization beta. Unfortunately, the bulk polarization of
typical ferromagnets is around 50%, so as this plot shows ohmic injection is doomed
unless epsilon is at least 0.01. However, the relevant materials
properties are not forgiving in this respect.
The ratio of conductivities between a semi-conductor and a ferromagnetic metal
is significantly below unity. Even for highly-doped semi-conductors,
and highly disordered, amorphous ferromagnetic metals.
Likewise, the ratio of length scales is small, due to the fast spin relaxation in
the ferromagnet, leading to a spin diffusion length Lambda of approximately
ten nanometers. Whereas in semiconductors with low spin
orbit interaction, such as silicon, transport lengths can be ten microns or
longer even at elevated ambient temperatures.
Therefore, even in the best scenario, epsilon is approximately ten to the minus
four. Leading to the negligible polarization
shown in the figure. Over the range of expected values for
epsilon, one needs bulk polarization of at least 95% for injected polarization of
greater than 10%, or so. So, elemental ferromagnets, Iron, Cobalt
and Nickel, are you useless for spin injection in the ohmic regime.
In fact, the problem is evident even graphically.
The splitting delta mu, which is necessary for a non-zero injected current
polarization, also tends to reverse the spin up electrochemical potential
gradient at the interface on the ferromagnet side.
Inhibiting injection of the very spin specise we want to inject into the
semi-conductor. Therefore, in order to maintain the
constraint of current conservation across the interface.
The steady state inter-facial splitting is small, and the injected current
polarization, p, is negligible. Modern techniques to overcome this
problem include quantum mechanical tunneling.
And, in my lab Ballistic hot electron injection, which circumvents the issues
relevant for ohmic injection, here. I'm not going to describe the details of
spin injection and detection, that's a whole other course in device physics and
magnetism. But rather, what we can learn from
measurements of spin transport and manipulation, by any means.
The key to extracting the most information from these measurements, is
exploiting a topic I mentioned several segments ago, spin procession.
Again, this is magnetic analog of a spinning top or gyroscope, with an off
axis gravitational force causing a mechanical torque.
In spin transport devices, we apply a magnetic field perpendicular to the
injective spin direction. But parallel to the transport direction
caused by electric fields and the spin with process in a plane.
The final spin procession angle, is determined by the product of spin
procession frequency determined by magnetic field strength.
About 28 gigahertz per Tesla in a material with weak spin orbit coupling
like silicon, and the transit time inversely proportional to electric field
strength. If we apply a perpendicular magnetic
field with the appropriate strength. Then you cause the spins to process an
average of 180 degrees, fully flipping with respect to their injected
polarization. Your experimental measurement of sigma z,
the spin along the initialization axis, will then vary.
Doubling the magnetic field doubles the procession frequency and therefore
results in an average procession angle of 360 degrees.
A coherent full rotation restoring the expectation value of sigma z.
As you can see from this actual experimental data, it doesn't matter
whether the field polarity is positive or negative.
In other words, it doesn't matter if the spin processes clockwise or counter
clockwise. Now if all electrons had the same transit
time from injector to detector, We would expect this cosine-like oscillation to
continue indefinitely for higher and higher orders of procession rotations.
But that's not what happens. In reality, not all electrons have the
same transit time due to random scattering processes.
Therefore, an uncertainty in transit time gives rise to an uncertainty in
procession angle. When the precession frequency grows in
higher an higher magnetic field, the affects of partial cancellation can be
seen an the oscillations diminish. We can model this measurement with a
transport simulation. Summing up the cosine like contributions
from electrons, or the distribution of arrival times, in order to fit the non
equilibrium spin mobility. And diffusion coefficients we're after.
However, there's a model independent method with far greater utility.
The key is to recognize that this integral summation, is really just a
fourier transform. Therefore, the oscillations we measure,
can be inverted to yield the empirical transport distribution without any model
dependence whatsoever. In this example we can see the effects of
increasing the electric field. Oscillation period increases, and the
number of oscillation themselves grows. But the transformed, clearly shows that
this is the result of smaller from mean and transit time and standard deviation.
This method of obtaining time of flight is called the Larmor clock.
We don't make a explicit measurement of transit time, we measure the angle of
rotation at a known angular velocity, the same way we measure time from an analog
clock. We know the rotation speed of the hand
for the clock, 360 degrees per hour for the minute hand, and infer time from the
instantatous oreintation. We're likewise measuring the spin
orientation and determining how long it processed in a known magnetic field.
For measurements of spin transport, we can correlate the transit time with final
spin polarization and extract the spin lifetime.
In silicon, we can see that, although the non-equilibrium lifetimes of hundreds of
nanoseconds are fairly long. In comparison to the momentum relaxation
time of picoseconds or less, they're strongly dependent on temperature,
increasing dramatically as the sample is cooled.
This demonstrates the importance of relaxation, via a nominally
spin-independent process, electrons scattering off of thermal phonons,
distortions in the crystal lattice. This electron-phonon spin relaxation
process, results from the fact that due to the weak but non-zero spin-orbit
coupling. The electron wave functions are not pure
spin eigenstates up and down. Rather, spin up has a small amount of
spin down and vice versa, but remain fully orthogonal.
We can calculate the transition rate between these states constituting a spin
flip, due to momentum scattering of these free electrons from wave vector k to k
prime. By using the so-called Fermi Golden rule.
This first order expression is proportional to the square of the matrix
element of a scattering potential, coupling the two initial and final
states. And the density of final states row.
Now, even if the scattering potential only couples states of different momenta
k. Due to the spin-orbit mixing of the wave
function, we see that there is a non-zero matrix element.
And this exactly equal to the quantity determining the spin preserving momentum
relaxation rate. The spin relaxation is therefore
proportional to the momentum relaxation, and also proportional to the square of
the typically small spin mixing amplitude.
This being the end of my contribution to this course, I'm obliged to acknowledge
support, not only from my experimental research on spin transport.
But also support for scientific outreach efforts for students and the public
outside my institution, the University of Maryland.
In particular, the National Science Foundation Career Award has made this
work possible. It's been my great pleasure to share this
quick story of electron spin with you. And I invite you to learn more about spin
through own study and perhaps even original research in Physics and
Engineering labs around the world.