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Summary
Core Theme
This course, "Quantum Transport," introduces the fundamental concepts and experimental phenomena of how charged particles behave as quantum mechanical objects when flowing through conductors, bridging theoretical physics with modern technological applications.
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yeah okay welcome everyone to uh physics
3102 Quantum
transport uh this is uh my very first
class uh as a professor and it's a first
lecture that I give so it is uh hi come
on in so I'm very excited uh and uh I'm
uh doubly excited because uh I have a
chance to give the this class on a topic
that is dear to my heart and is closely
related to my own research and uh I know
that it's closely related to the
research of many of you um and for for
those of you who come from other
disciplines I hope uh I will be able to
make you excited about uh this
wonderful um field of physics it's I
think as of a few years ago it is no
longer a subfield but is a fully-fledged
uh independent field which is uh very
important in modern science and
potentially in technology today's
lecture is uh an introduction and uh it
will be um about uh the scope of the
course and uh of this field and uh to to
get all of you also excited for the
upcoming semester uh and I will explain
how the course will proceed and give you
some examples of of what I mean by Quantum
Quantum
transport okay our uh we meet twice a
week uh there was a typo on the web uh
the course actually goes till 2:45 not
2:15 so it's a little bit longer uh than
some of you might have thought um and uh
yes my name is Sergey uh please just
call me Serge we are all upper level uh
graduate students so no need for uh
ceremony you can uh feel free to stop by
by my office or by my lab or send me an
email anytime you want about the course
any questions you might have I'll be
them okay first of
all um there is by now quite a bit of different
different
literature on Quantum transport and the
synonyms for this field such as um
nanoscale transport mesoscopic physics
uh those are all mean roughly the same
thing so there is a number of very good
textbooks um and uh here I list
um all the literature that I I think uh
I you could find useful though I think
course will be largely self-contained
but um I think to to get more background
and to have something to read I strongly
encourage you to uh look at these books
and uh um some of them uh especially
these upper two are available at the
library and also in my lab so you can
come and borrow or send me an email uh
if you want to use these books so the
the I listed them in a chronological
order the upper two are the most recent
ones and they also make a very nice set
because uh this book um Quantum
transport it is a written by theorists
uh and uh so it has a lot of formalism
and uh equations and uh uh you know
consistent approach so for those of you
who are theorists you might enjoy this
book uh more and this book is written by
a career experimentalist a very
excellent physicist but also a great
teacher and uh so he focuses on uh
experiments that are key to this field
uh and at the same time I want to say
that none of these books is complete
otherwise I would have given you one for
example uh in this course I'm hoping to
touch on uh some of the effects related
to Super
conductivity but this book uh strictly
talks about semiconductors yeah so the
some of the effects are not in this
book um okay now here are three other
sources uh these are a little bit uh
more dated so so they don't have all the
most recent
experiments uh but they are very good um
this is
um um these are lecture notes from Del
University which uh I might be able to
offer you if you ask me for them uh so
they um course will follow these uh in
many instances and these are written
very well for graduate students who are
starting in the field so this will be a
good option uh this is
a very famous textbook in our field uh
which explains uh some of the similar
stuff to this book and this book it is
written by a theorist who is in
electrical engineering so it combines
kind of the approaches of these two
books uh uh this one is a open uh Source
you can find it on the
archive uh and this is uh already 20
years old but it summarizes the
foundational experiments that were done
in the ' 80s and early '90s in the field
of quantum
transport um I have a pleasure of
knowing uh most of these people uh
actually personally so the I don't know
suprio dat though I've attended his
seminars uh but um I've worked with uh
juli nazarov and blanter and Carlo Bayer in
in
Del um I know thomasin and I know and
case Harmons is also from the so
actually a large fraction of this
material is from the Netherlands where I
uh used to work before coming to pit uh
and uh that is a perhaps a coincidence
but uh it also allows me to understand
very well what these people are talking
about and hopefully you will enjoy these
books as well and some of the lecture no
uh lecture slides and notes that I will
be showing are also from courses
similar course courses developed in Del
and in einhoven in the Netherlands uh
but uh of course they are reshaped for
okay uh the course uh uh mostly relies
on the knowledge of uh the basic Quant
mechanics yeah so not even Advanced
quantum mechanics but uh the simplest
Concepts such as uh representation of
particles as
waves um confinement Quantum confinement
so particle in a box problem and a box
can be onedimensional two-dimensional or
three-dimensional those are very important
important
examples um yeah waves that are free
propagating and uh uh things like uh two
level systems for example spin one half
physics Dynamics in two level systems
this will become important later in the
course um a little bit of knowledge of
solid state physics uh will be very
helpful uh though I will probably try to
cover most of it but if you have uh some
idea about band structure in metals and
semiconductors about the behavior of fir
electron gas density of States if those
concepts are familiar to you um it will
be easier for you to navigate the course
actually this is uh uh very remarkable
that so little background is needed to
understand such an advanced field which
is at the Forefront of modern science it
is um enabled by modern
technology so all these very simple
Concepts uh you can actually literally
Implement them in real nanoscale devices
so you can actually observe the most
simple examples from quantum mechanical
textbook uh where which maybe at the
time where you taken a Quantum mechanic
mechanics course you would have thought
that this is a completely gunkan
experiment such as a Quantum levels of a
particle in a box etc etc but here with
a with a new technology which I will
highlight today a little bit you can
implement the simplest models such as a
electron propagating in a straight line
uh in like in a wave guide and and so
this is a
this is why we don't need to go into too
much details of these uh solid state
course okay um there will be homework
this is an advanced course uh but uh I I
think this homework should be uh um fun
for you and the homework as well as the
final exam will be uh like playing a
professor I say like kids play a doctor
or a or or they play in a store or they
sell and give Goods we will play a
professor so we will review real scientific
scientific
papers and we will also write proposals
for experiments right so they will do
some of the components of a principal
investigator yeah so the homework is uh
that you should pick up one
paper it could be recent could be old
but it has to be on this website yeah
pick one paper read it try to understand
it you don't have to understand it fully
but make some effort and then write down
your thoughts um they could be uh just
something like I didn't understand
anything but I have these questions or
it can be a full evaluation of the paper
the paper is flawed in such and such way
it is not a good paper or this is a
wonderful paper a template for everyone
to to follow so and any anything
anything like that counts um and uh so
this is like a review that someone would
submit to uh a journal like a referee
report uh but then uh um you will not
just send this report to me uh you will
post it on a new experimental platform
what we call archive plus I will show it
in the next slide uh but basically it is
a this website with the ability to
comment on papers yeah so then uh all of
you will see each other's reviews and
you can also engage in discussions like
for example oh I disagree with this
review I think this is a wonderful paper
or you can answer each other's questions
if someone didn't understand uh the
paper maybe another person understands
so this kind of interaction you can also
just evaluate papers with one click on
Facebook you can add a like to the
paper um and uh this uh archive plus is
something that uh uh we started with a
couple of PhD students in Del and now we
are in a testing phase yeah so you will
be the focus group so at the end of the
course I would appreciate if you have
any thoughts feedback uh on this idea on
the implementation as well um I would
appreciate that so so this is how it
looks uh this is this is going to be
your homework um how many of you have uh
looked at archive.org at least once in your
your
life 100% excellent so basically all of
you recognize this template this is the
the red bar and the then the abstract of
a page of an archive and here you can
download the paper and read it now what
the archive plus does it it gives you
the same web page but it adds certain
elements so in this design they're
marked with little
pluses yeah so at the front you have uh
at the top you have a certain menu but uh
uh
here we added a little square where you
can uh like and add the paper to your
library if you want to read it later um
and here at the bottom uh you can leave
your comment so here uh this this paper
I found it interesting and uh I left a
comment here yeah so by the way I will
also participate in this and I'll try to
get your uh supervisors to join this
website so we have some critical mass
and we have fun with it yeah so here I I
left a comment I will not show you the
uh whether I like the paper didn't like
the paper um but in principle this will
be a for now a closed project so you can
express your opinions freely the authors
of the paper will not will not see them
unless you comment on each other's
papers of course
um and the way it works is it it is a
Shell through which you browse
archive so if this is a archive.org and
it has a a storage for papers with uh um
abstracts attached to them then uh when
you log in through archive plus it
retrieves papers from archive.org yeah
and then it adds in your screen so it's
like a proxy server right it adds in
your screen the ability to do all these
things yeah so it is a like a small
social network for scientists uh where
we can quickly exchange opinions about
the papers yeah ideally you put
something on the archive and then boom
you get 10 comments the next day if it's
a if it's an interesting
paper so this will be the
homework I propose that we do one review
per week yeah per person so this we will
have some fun with
it and the final project is a uh
proposal for a new experiment yeah so um
by the time that we finish with this
course we will know the scope of this
field and we should be able to come up
with a with a new idea has not been done
before that is interesting to try to at
least to think about maybe it's not
completely realistic but a little bit
feasible yeah and it should be
exciting um I will help you throughout
the semester so you can talk to me about
your project one important condition is
that it cannot be uh strongly connected
to your current PhD research yeah so
that that makes it too easy maybe you
have been in your group for a couple of
years you thought about uh your project
and you already have an idea that is
ready to
go uh we will not allow that
uh so it has to be from another field
but from the area of quantum transport
which is pretty big and uh yeah we have
plenty of time to to come up with it and
um importantly it is has to be an
experiment yeah so not a theoretical
idea or I will propose a certain
calculation but it has to be a
measurement and this is related to the
uh main focus of this course which will
be an experimentalist perspective on the
field yeah so the the the bulk of the
course will be about me uh telling you
about various exciting experiments from
this field that shape the field and the
concepts that we need to understand this
experiments yeah so this will be a
experimental course uh focused on also a
little bit on methods which are used to
do experiments and also on the on
Staring at the experimental data uh
trying to read into it uh trying to read
enough into it but not too much and
things like that
okay so this slide uh is um designed to
uh illustrate the difference uh between
classical and Quantum transport yeah the
quantum transport is two words Quantum
and transport and transport refers to
the type of measurement that we will be
studying it is always basically the same
measurement it is a study of how charged
particles flow through a conductor yeah
so if the most basic example of
Transport is when you uh hook up a
battery to a resistor and you measure
the current that flows through this
resistor this is a transport measurement
of course uh unless it's a very special
resistor it is a classical transport
measurement and that is because
electrons when they flow through a
resistor maybe at room temperature for
sure uh they are can be regarded as
little balls particles and not as
waves yeah so to go to Quantum
transport electrons or holes charged
particles flowing through our conductor
have to behave like quantum mechanical objects
objects
yeah so for example here are um two
samples which um the the shape of them
looks identical it is some sort of a
two-dimensional cavity it is a rather
perfect cavity because uh here in the
classical regime when an electron enters
into this cavity it just bounces off the
walls there is absolutely no scattering
or interaction between different
electrons uh so this is a
ballistic cavity but it is classical
because all we need to know about
transport of these electrons to model
them is that they are particles with a
certain velocity and they bounce in this
cavity yeah so then if we know that and
the shape of the cavity we can calculate
what will be the current flowing through
this cavity for a given voltage bias
that is applied to the
cavity now this is an example where we
have done something uh to this
system um and uh the same sample um we
cannot anymore treat this electron as a
classical Ball but we have to uh
describe this
entire cavity with a wave
function yeah so this is an example of a
a wave function formed by um
interferences of
uh uh electron uh living inside this
cavity so then the proper description
for this sample would be that um a
particle enters this wave function and
then it has a certain lifetime inside
this wave function and it tunnels out
and in principle if we had a probe maybe
some needle uh that comes in from here
we could map out this wave function
where it has Peaks and where it has dips
and so the the field is dedicated to uh
finding that um see when you have to use
a wave function description of uh
electrons then many non-trivial and
often counterintuitive effects can be
observed in real samples yeah so what uh
can someone name me one reason why this
electron would stop being a a little
ball uh classical particle and uh go and
become a wave function what what can we
to to make it uh smaller than the
wavelength of an electron for example
ideas lower the temperature lower the
temperature exactly temperature is a is
an energy scale um and uh it uh smears
the quantum effects yeah so the the high
temperature makes Quantum objects look
classical so you would have to shrink
the temperature such that the energy
scale related to this wave function is
larger than the temperature yeah okay so
the so the we already uh just with these
two examples we have determined that uh
we have to be probably at some low
temperature to study Quantum transport and
and
or we have to study small objects Maybe
okay um just at the beginning of the
course um I want to uh show you a couple
of slides that um you know provide a
real world motivation for these studies
Beyond fascinating fundamental physics
that can be discovered and it's uh most
of first and foremost uh the uh progress
in uh electronics
electronics
now so um You probably all seen some
version of this graph this is the mors
law as postulated empirically by Moore
that every two years the number of
transistors on a chip uh doubles and we
are still living in the Moors
law um for already about 50 years and it
is expected to continue like that for at
least a few more
years and um so U
basically um it is important uh for the
semiconductor industry for information
technology to continue this law because
um if you make more transistors and if
you make them smaller the computers
become more powerful yeah so the smaller
transistors means that electrons can
cross them faster and that means we can
increase the
frequency of the transistors yeah and uh
more transistors means we can do more
calculations uh maybe simultaneously
yeah so it's a it's very important uh
for U uh various uh applications in uh
in this industry uh but uh since we were
following this law already for 50 years
uh we have got to a stage where the
circuits uh that uh actually in any of
our computers or even by now in our cell
phones are extremely complicated and
they're also extremely miniaturized yeah
so this is a uh one uh Intel chip uh
which has I don't know billions of
transistors on it and U so it is a
highly complex uh circuit but if you
look at just one
transistor yeah a transistor always has
a source and a gate and a
drain yeah so then uh just one
transistor uh in this circuit has
already become so
tiny yeah that um know here are some
numbers the channel length so the gate
length is 32 nanometers and so we are in
a nanometer scale and we're not too far
from the wave length of an electron in a
semiconductor yeah actually in fact we
about the right
size for
that and so we are already dealing with
a nanotechnology here and we must
already include Quantum effects when we
function and so it's important to
understand uh what happens if electrons
are considered as quantum mechanical
objects to improve and uh improve the
technology so uh here is um um a
specific example which um um also shows
uh some of the challenges for going
further further with camos and it is a
uh this is a computer simulation of uh a
a single transistor a single mosfet
transistor metal oxide
semiconductor Field Effect
transistor yeah so it has a important
elements source and drain so you can
send electrons in and then they go along
this channel here from source to drain
and then um there is a gate which can
change the whether electrons go through
or not yeah so it's an uh electrostatic
uh gate like in a
capacitor um it can change the basically
the occupation of uh electrons
underneath the gate and so it can
completely block the transport or it can
enable the transport so the concept of
gate is very important in throughout this
this
course and the gate has become so small
that just the number of dopant that
induce electrons in this channel uh is
in the is in the T in this simulation it
is 36 dopin yeah so we are really
dealing with the discrete number of uh
Quantum particles atoms electrons in uh
in the modern
transistors and so not only uh the
quantum mechanical nature of these
particles is important but also uh the
imperfections that come from the
quantized nature of the number of these
particles so for example in this
transistor it's 36 dopin in the one next
to it it could be 30 dopin and in the
one next to it it could be 45 dopin uh
so these uh
fluctuations uh are a big problem in the
field and uh they're also often
encountered in uh Quantum devices that
we will
consider yeah so the description for
such uh for such devices is uh is very
different from the classical approach
properties now this is a example taking
this uh to the extreme uh this
especially the upper one this is a
transistor that consists of a single
atom this uh uh this is a single uh
impurity of phosphorus on the surface of
silicon and this one is conducting so
what you do is you uh come with your
electrons from here you hop onto here
then you hop off and you continue so
this transistor consists of a single
atom need L to say electrons also travel
through this transistor one by
one so the what is also important here
is the charging energy of an electron
yeah so the reason why the two cannot go
is because it costs so much energy to
put an electron on this island that if
you put a second electron on this island
you just don't have the available energy
energy
yeah so this uh device is obtained by uh
uh with a s TM technique so scanning
scanning teling microscope uh it also
provides this image with an atomic
resolution but uh it is also used to
implant the phosphorus uh here uh I will
not go into details but you can look up
the research of this excellent group um
and so by this they can Define single
atom transistors so of course this is
not scalable to billions of
transistors yet you cannot send a needle
and implant 1 billion of these atoms but
as a powerful demonstration that you can
scale the transistor down to the
ultimate Quantum limit and uh this uh
this is a not a real picture by the
cartoon but it is from pit uh and this
is a a similar concept uh these green
atoms designate the points uh for the
island of a transistor this is not a
single atom transistor but this is a few
atom transistor um that we have access
okay so what I told you is that uh the
people in the semiconductor industry are
uh interested in uh quantum mechanical
descriptions of uh electrical devices in
order to better understand the circuits
that they already working with yeah
uh but they're also looking at devices
and that is to enhance the functionality
of their circuits beyond what they can do
do
now yeah so for example uh currently the
backbone of all this technology is a
transistor like I explained it is a gate
source and a drain but what if uh a gate
is also spin
dependent so you're adding another
degree of Freedom into the functionality
of this transistor which can allow you
to maybe Implement a certain logic gate
with these
transistors uh with fewer operations
yeah so new functionality that is not
there in Coss is one of the focuses of
this International technology road map
for semiconductors itrs this is a very
important document that drives the
development of not just industry but also
also
research in particular they have a whole
sub division which is designated to
Beyond seamos ideas so the spin gate is
one idea of something where you go
beyond seos and you can build computers
that work on fundamentally different principles
principles
inside uh and this uh um ladder of uh uh
concept summarizes the 2011 vision of
itrs Beyond seos and these are kind of a
scatter of keywords
but if you just stare at them you will
find many of the words that are relevant
to to this course of course Quantum
Quantum State
Quantum uh but not only Nano structured
materials spin orientation strongly
correlated electron
State um carbon yeah
spintronics um so trying to do different
uh new materials um and also uh
Quantum is um prominently featured in this
this
itrs so this is uh this shows you what
they're thinking about going
seos okay so once again in this course
we will uh study transport so passing of
charged particles through certain samples
samples
um and we will study them uh from the
quantum mechanical point of view and the
concepts that uh I'm hoping to make firm
uh in this course are first of all
quantization of various electrical
properties for example conductance
charge magnetic
flux in uh low dimensional uh systems in
particular yeah so the I will tell you
that a lot of these experiments are done
in systems where electrons are heavily
dimensions then phenomena related to
Quantum interference remember I showed
you this cavity with a wave function
that looked rather chaotic well that is
uh due to interference of electron with
itself and so many of the quantum
transport phenomena are interference
phenomena and finally uh Quantum t Ing
and blockade the example of a
transistor where in the center you have
one atom and the source and the drain
are disconnected how does the electron
get from the source to the island and
out to the drain it tunnels yeah so
quantum mechanics comes in in a
transport of electrons through
disconnected parts of a circuit and you
can still cross them with
tunneling and um these are the Core
Concepts and then we will develop uh uh
towards uh new experiments that are done
these days and those involve super
conductivity Quantum computation and uh
maybe a subset of that or maybe an
independent field at the end of the
course we will touch on the new topological
effects okay so to
um this slide shows where um Concepts
like itrs road map for Semiconductor
industry and the fundamental physics uh
meet together to form a happy marriage
yeah so
the a number of um fundamental fields
that we study in the lab may find
applications in uh in real computers or
other devices in the future um and there
therefore we get extra support for
working on these ideas even if we just
interested in their fundamental value so
first of all probably all of you heard
about uh at least Quantum Computing but
maybe the other ones so Quantum
Computing a way to uh store and process
information and quantum mechanical
degrees of freedom using Concepts like
superposition and entanglement to boost
computational power for certain
algorithms right not for everything
spintronics like the example of a gate
that is spin dependent uh or transport
resistance of a channel which is spin
dependent embedding extra degrees of
freedom into transistors and elements
like that or using spin for memory is
another example of spintronics but from
a fundamental point of view studying the
behavior of spins in solid state down to
the single spin
level um and controlling single
individual spins with external controls
that is also in the real realm of
spintronics opto electronics is one
example of a bigger field of hybrid
Technologies where different degrees of
freedom and combined in a single device
here we might send in a single electron
and get out a single Photon for example
but uh what if we now take a electron
with a well- defined Spin and get out a
well defined polarization of that Photon
or what if we take an electron which has
a super position of spin States and
transfer that superposition into a
superposition of polarizations of
photon yeah so this is one example other
examples of hybrid techniques are
something like Nano mechanical Nano
electromechanical systems where
transport of electrons drives a
mechanical mode and maybe down to the
quantum level and there are other examples
okay so um classical transport um the
best known example is of course ohms law
right and ohms law is just a linear
proportion between voltage and current
is rather boring but rather Universal
and it takes a lot of effort to make
something that deviates from OHS
law and we will almost entirely study
devices that completely violate this law
in every possible way and um the
quantities that will govern whether
devices violate M's law and also that
will uh just show up in the measurement
uh uh like steps or interference
patterns in uh in the data are these
three Quantum values yeah they all
consist of fundamental Quantum constants
and they're very important so for of
course the Quantum of charge if we um
can observe transport of electrons one
by one through a certain structure we
will immediately see uh that the
transport properties such as
conductance um current voltage are
quantized in the units of
charge uh this one is a little bit less
trivial um apparently resistance or
better yet conductance can also be
quantized yeah the Quantum of
conductance uh this should be G I
apologize uh is just e squar over H so
if you work out these uh units this will
be one/ ohm and it is this messy value
so it's much easier to remember a
Quantum of resistance which is about uh
26 kiloohms 25.8
kilohms yeah so for
example um if a resistance of a system
is smaller than the Quantum of
resistance then maybe the system will
behave classically if the resistance is
higher some kind of quantum effects
might show
up and finally Quantum of magnetic flux
just e over H this governs the various
interference patterns of electrons as
they travel through the sample and it is
uh uh in these units it's 4 to 10 minus5
Weber now I I told you that you can
directly see all these three values in
experiments uh here's a proof of that
this is from Thomas in's book uh ON
Semiconductor Nano structures um these
are three examples of real experiments
this is data that each show a
quantization of one of these quantities
so maybe
let's go through them together uh
because maybe some of you already know
what these are or maybe we can
guess so for let's start uh here uh who
is you can you don't have to know you
can also
guess so we have to look at the axis and
device I need any you guess uh which
one yeah so resistance is oscillating as
we change magnetic field and the sample
looks like
a like a ring
ring
not L levels don't require a ring they
ideas okay I will I will explain uh so
this first one is uh okay let's at least
uh can you at least someone tell me what
is quantized
here charge conductance or flux flux
flux exactly magnetic flux is quanti uh
the quantization of magnetic flux can be
read out from here um
so the sample is a ring and you can send
the electrons from here to here and you
apply magnetic field like that yeah and
uh the thing is this ring this entire
ring is smaller than the wavelength of
electrons yeah so it is a Quantum regime
where electron can go through the entire
ring but it it is one wave function it
it is it is uh in other words you can
say that the def phasing length of the
electron is longer than this
structure yeah so then what does this
electron do well it can go via this path
through the ring or it can go through
this path through the ring yeah well
which path will it go
through both both yeah it is a quantum
mechanical object so it will go through
both parts and at the end at the drain
it will reunite with itself and it
will interfere interfere with itself
that's right so then uh we have to uh
for the interference the important value
is the quantum mechanical phase yeah and
in general electron will acquire a
different phase going through this side
of the loop or going through this side
of the loop if there is magnetic field
present so if the geometry is perfectly
symmetric maybe it will get the same
phase and we will get constructive
interference uh at at the end but if the
if there is magnetic field present then
we have to uh
include uh the gauge Vector potential in
the evolution of this phase yeah and
that is called the Aron of bomb effect
some did you hear about the Aron of bomb effect
effect
some of you not everyone most it's
important for me to know because then I
can uh you know adjust the course uh so
that we cover the right
Concepts yeah so then
uh um this is a vector potential is a
gauge field and so it has to be periodic
and it turns out to be periodic in the
units of quantized magnetic flux yeah so
for whatever the area of this
Loop um as soon as we put in one unit of
magnetic flux the system should remain
uh should come to the initial state
where it started and that's uh uh why we
get these uh oscillating patterns now another
another
important uh concept that this device
that this experiment transmits is that
resistance yeah something you measure
with a voltmeter or current
meter is
dependent on the interference of
electrons yeah so for example maybe if
the interference is destructive at the
at the drain will the resistance become
lower resistance will become higher
that's right so uh because uh the wave
function will have a a minimum here and
it will be more difficult for the
electrons to come out so if the
interference is
constructive maybe it is easier for them
to come
out and so the details of this pattern
uh are a separate field of study why do
why are these wiggles have this
magnitude why do they show another
modulation um what is the role of these
other exits we can just dedicate a whole
lecture to a Ron of bom effect but just
for the beginning for the first lecture
just wanted you to get this flavor of this
this
effect um as an exercise Let's uh
estimate the area of this Loop yeah so
there is no length scale we cannot see
the length scale but we know that
about 5
mesla put in one flux Quantum in this
Loop and the flux Quantum
Simplicity this numbers work out really well
actually the
way so F divid by 10 milon Tesla well
let's say five
five be
10 -
is flux
F0 is B *
B so this is 10-
3 so this is 10 Theus
Theus
12 which is 1 Micron squared
squared
yeah so the dimension the characteristic
dimension of this Loop is one
micron so this must be a semiconductor
because in a metal there's
uh ah you can still get a coherence
length this long if you cool down the
system very low yeah so we don't know
what this system is but it looks like a
semiconductor to me
anyways all right so quantized
flux now uh who knows what this effect
is quantum hle effect Quantum hole
effect that's right so the here we have
um as a function of now very large
magnetic field let's not go into why
this happens for now but we tune with a
with a control knob which is magnetic
field through this looks like a
staircase of
resistance yeah now let's not look at
this this is the actual whole part of
the quantum hole effect which shows
little spikes in whole voltage but let's
not look at that let's just look at this
staircase of
yeah so the question is what is this
staircase okay this is a bad way to ask
this question
uh what other what are the values of
resistance that correspond to this uh staircase
and so it doesn't it it uh it doesn't
look like uh the resistance is quantized
actually huh so we we should if
resistance were quantized we should see
steps of 26
kiloohms here we have uh steps first of
all they are steps of different size and
they don't correspond to any 10 kiloohm
scale they they correspond to my
actually just a 1 kilm scale
here well
um this is uh somewhat tricky if you
haven't thought about it long enough but
basically there is another value which
is quantized and if you repot this graph
equal exactly yeah so the what is
quantize this conductance in this case
not the
resistance yeah so the quite often you
see that the conductance is quantized
and the resistance is not quantized and
the reason for that is fairly simple uh
and I don't expect you to
um explain this to me because I didn't
tell you what Quantum whole effect is
a um conductance or electrons that flow
through a number of quantum mechanical
one-dimensional channels and each
not but electrons can flow through them in
in
parallel so to get the total resistance
of these channels we have to add them like
that like parallel resistance yeah but
this is actually a Quantum of
conductance so in a in the language of
conductance it is much easier to deal
with this effect we just have to add up
the conductances of all the channels and
we will get the total
inductance so basically if you look at
the quantum whole
trace and you invert this scale then
these will become perfect steps of
conductance but you can estimate how
many channels are here for example at
the top of the graph yeah because here
you have something which is uh of order
a kiloohm and uh conductance Quantum is 20
20
kiloohms yeah so that means we have a on
the order of 20 a little bit less
graph okay
finally this is this graph shows well uh
I will not tease you this is charge
quantization it's the only one
left so but uh this is a very uh
interesting and a very important effect
which is called kolon blade
no and it is related to the fact that um
if you have an island disconnected
island of where electrons can live it
costs you a certain energy to put
electrons on this island and it is the
capacitive energy is e^ s over C that's
the capacitive
energy so what this sample is it is a
again a semiconductor structure shaped
by Gates so these are metal gates on the
top and they electrons that we care
about they live underneath the surface
and so we can create an pinched off
Island here by adjusting voltages on
these Gates so what you have to look
kind of through this pattern and think
of a little island again of the size of
a one
micron which is disconnected from these
contacts so then these will be source
and drain and the connection will be via
tunnel barriers and the tunnel barriers
are just created by these Gates by
making negative voltages and preventing
through and now here the control knob is
a another gate voltage and this gate
voltage it it is a plunger it it
actually the it is a jargon for plunging
electrons out of the dot so what you see
here is as you make the plunger more
negative you kick the electrons out of
the of this island one by
one and the way you do this is by uh you
can think of it two ways either by
making it more negative you shrink the
size of the islands and electrons just
don't fit so they come out or you can
think of it in terms of energy you
change the energy of the electrons the
capacitive energy or the chemical
potential by adding electrostatic energy
from the
gate yeah so you we will come back to
such structure many times uh uh in this
course so you don't have to understand
it fully uh but let's just look at the
data and it shows Peaks and in between
the Peaks there is no
conductance and um Peaks are in
conductance yeah so that means that at
each Peak it cost you equal energy to
have n electrons and N plus one
electrons on this island and therefore
uh you can freely add electrons and they
will come out and that means you can
have trans
Port yeah so certain energies are
equalized um and that gives you a peak
in conductance and then when you tune
away from the peak uh that means that it
costs you extra energy to put electron
on the island and the transport becomes
blocked current becomes zero conductance
becomes zero and that is called coolum
blockade and so a presence of a single
electron on this
island can completely block the current
and this is very against ohms law right
whereas in in OHS law you can put as
many electrons as you want through the
island and they will just come out after
bouncing inside the inside the
island so quantized flux quantized
conductance and quantized charge can all
be seen in uh semiconductor Nano
structures and of course they're
semiconductor because they're from a
[Music] structures
okay the field is um 20 to 30 years old
and this is about the time scale going
back in time when uh people started
observing quantum mechanical effects in
uh electrical measurements effects like
interference and quantization so for
example Quantum whole effect was
discovered in
1981 exactly 30 years ago so that is the
time scale uh for this field and uh
initially um and quite naturally uh the
field has to establish the basic effects
such as okay electrons are waves they
have mass they have spin they have
charge what happens yeah so those are
examples of these very basic effects
which just tell you that electrons can
interfere conductance in one dimensional
channels can be quantized and uh
charging energy is an important energy
scale and uh let's call these basic
electrons that have just these
properties and uh and they can be used
to describe these very important
Concepts and this will be roughly the
first part of the course throughout
which we will study the foundational
experiments in transport such as the
ones I showed in the previous
slide but 30 years is a long time and it
can also get a little bit boring uh
studying such basic electrons now how
many times can you see interference over
and over again and So lately a lot of
effort has been dedicated
to these more amazing electrons yeah
they actually the same electrons but you
kind of pull out extra blades from the
knife and uh you
include various interactions and uh band
structure effects into
consideration and you also use these
extra um extra tools as as
functionalities in your Quantum devices
so this is a again a happy marriage
where we can understand more about the
condensed matter systems that we're
studying by including these effects and
these are only some examples though very
important ones but also we can build
devices that exploit these effects in their
their
see okay we have 15 minutes um I want to
go through them uh one by one because I
think they're quite important um so this
one is actually still borderline basic
electrons yeah but I think it is a
little bit non-trivial and so worth
mentioning um so in uh in free space
electrons are fundamental particles
described by D equation and included in
the standard model and those uh
electrons have uh uh the well defined
mass mass of an electron a certain
charge certain spin does not have to be
the case in semiconductors and
primarily uh
the mass of an electron can be very different from the for
different from the for example in these important
example in these important semiconductors like Gallum arsenide
semiconductors like Gallum arsenide which is um actually the semiconductor
which is um actually the semiconductor which probably give rise to all of these
which probably give rise to all of these three
three curves a mass of an electron is much
curves a mass of an electron is much smaller than the mass of a free
smaller than the mass of a free electron and one if one
electron and one if one uh thing that it gives you is actually I
uh thing that it gives you is actually I think higher Mobility for these
think higher Mobility for these electrons so it's good for electronics
electrons so it's good for electronics to use Gallum arenite but another thing
to use Gallum arenite but another thing it gives you is
it gives you is that mass smaller Mass makes the
that mass smaller Mass makes the wavelength larger these are lighter
wavelength larger these are lighter particles so it's easier to make them
particles so it's easier to make them Quantum yeah so we can think of quantum
Quantum yeah so we can think of quantum devices which are bigger in this
devices which are bigger in this semiconductor and this idea is kind of
semiconductor and this idea is kind of taken to the extreme in graphine
taken to the extreme in graphine uh where I'm sure many of you know that
uh where I'm sure many of you know that the mass is
the mass is zero yeah so the these electrons are in
zero yeah so the these electrons are in graphine they're sometimes called
graphine they're sometimes called relativistic particles because uh uh
relativistic particles because uh uh this is the relativistic Der equation
this is the relativistic Der equation that gives you a zero mass for these
that gives you a zero mass for these particles and so some of the quantum
particles and so some of the quantum effects like Quantum whole effect in
effects like Quantum whole effect in graphine you can see it at room
graphine you can see it at room temperature
now electron electron interactions that is a very large area and in principle
is a very large area and in principle coolum blockade is an example of
coolum blockade is an example of electron electron interaction because
electron electron interaction because electron is sitting on an island and it
electron is sitting on an island and it is not letting other electrons on so
is not letting other electrons on so that's an electron electron interaction
that's an electron electron interaction but um um it is quite natural to
but um um it is quite natural to consider this interaction especially at
consider this interaction especially at low temperatures and it gives rise to
low temperatures and it gives rise to many interesting uh effects for example
many interesting uh effects for example uh sometimes the quantization of
uh sometimes the quantization of conductance uh is not in the units of
conductance uh is not in the units of one Quantum of conductance but in the
one Quantum of conductance but in the units of
units of 1/3 so this is uh kind of easy to
1/3 so this is uh kind of easy to explain uh by having particles of charge
explain uh by having particles of charge e travel in U independent channels and
e travel in U independent channels and then you get uh this quantization of
then you get uh this quantization of conductance but in fractional Quantum
conductance but in fractional Quantum whole effect there are fractions like
whole effect there are fractions like 1/3
1/3 uh 17th uh 1/ 15th uh these kind of
uh 17th uh 1/ 15th uh these kind of fractions and they uh correspond to
fractions and they uh correspond to actually fractional
actually fractional charges so the by by including electron
charges so the by by including electron electron interactions you can even take
electron interactions you can even take one electron and kind of split it apart
one electron and kind of split it apart which you cannot do with elementary
which you cannot do with elementary particle electrons in free space so this
particle electrons in free space so this is quite fascinating condo effect is
is quite fascinating condo effect is another interesting example of
another interesting example of interactions where if you have a
interactions where if you have a electron with a spin sitting on an
electron with a spin sitting on an island
island electrons around their Island gather
electrons around their Island gather together and Screen the
together and Screen the spin so this is a strongly interacting
effect hyperfine coupling uh this relates to a very important area in
relates to a very important area in condensed matter physics uh which is a
condensed matter physics uh which is a interaction of one particle with many
interaction of one particle with many interaction with a bath of degrees of
interaction with a bath of degrees of freedom in this case it's a spin bath of
freedom in this case it's a spin bath of nuclear spins uh hyperfine interaction
nuclear spins uh hyperfine interaction in condensed matter is a little bit
in condensed matter is a little bit different from Atomic physics where
different from Atomic physics where electron in an atom interacts with a
electron in an atom interacts with a nuclear spin of that
nuclear spin of that atom yeah in condensed met physics
atom yeah in condensed met physics because electron overlaps many
because electron overlaps many atoms then one electron can interact
atoms then one electron can interact with 1 million
with 1 million nuclei which are within the wave
nuclei which are within the wave function of an
function of an electron so then uh you have to
electron so then uh you have to calculate the
calculate the Ensemble averages of coupling of an
Ensemble averages of coupling of an electron spin of single spin to many
electron spin of single spin to many many spins of the nuclei so this is this
many spins of the nuclei so this is this interaction is very important in the
interaction is very important in the context of uh spin cubits because
context of uh spin cubits because coupling to these various uh embedded
coupling to these various uh embedded spins uh makes the Cubit decohere
spins uh makes the Cubit decohere perform poorly so this is an important
perform poorly so this is an important problem to understand but maybe it can
problem to understand but maybe it can also be used as a spin memory so for
also be used as a spin memory so for example if you can use
example if you can use one spin of one electron to control one
one spin of one electron to control one million nuclear spins you can transfer
million nuclear spins you can transfer information into these nuclear spins
information into these nuclear spins store it there and then extract It
store it there and then extract It Back Spin orbit coupling this is a again
Back Spin orbit coupling this is a again sort of a band structure effect but I
sort of a band structure effect but I put it separately because it is becoming
put it separately because it is becoming very important recently especially with
very important recently especially with the topological phases uh but basically
the topological phases uh but basically it is a very simple effect uh a spin of
it is a very simple effect uh a spin of an electron is coupled to the motion of
an electron is coupled to the motion of an
an electron so if if the electron flies
electron so if if the electron flies this way uh the spin of an electron sees
this way uh the spin of an electron sees an magnetic field may be this way but if
an magnetic field may be this way but if it flies back the magnetic field is this
it flies back the magnetic field is this way so then by controlling how electrons
way so then by controlling how electrons move the spin can be controlled or by
move the spin can be controlled or by controlling the spin we can control how
controlling the spin we can control how electrons
move um and this is a like I said important interaction in for cubits and
important interaction in for cubits and for topological phases from topological
for topological phases from topological insulators myana fermion and
insulators myana fermion and uh spin hole effect and all many of the
uh spin hole effect and all many of the recent developments in the quantum
recent developments in the quantum transport and superc conductivity is
transport and superc conductivity is another type of
another type of interaction uh where electrons interact
interaction uh where electrons interact pairwise to form spin singlets called
pairwise to form spin singlets called Cooper Pairs and when you go from one
Cooper Pairs and when you go from one electron to two electrons coupled then
electron to two electrons coupled then these Cooper pairs become bons and so
these Cooper pairs become bons and so single electrons are Fons but two
single electrons are Fons but two electrons then then the spin becomes
electrons then then the spin becomes one uh and they become bons so what they
one uh and they become bons so what they do when you lower the temperature they
do when you lower the temperature they undergo B Einstein condensation and that
undergo B Einstein condensation and that is called super conductivity so these
is called super conductivity so these electrons can flow through crystals
electrons can flow through crystals without
without resistance which is already very
resistance which is already very remarkable but this effect is actually
remarkable but this effect is actually 101 years old so that is not so hot
101 years old so that is not so hot anymore but then taking superc
anymore but then taking superc conductivity and putting it into Quantum
conductivity and putting it into Quantum devices which are small nanoscale can
devices which are small nanoscale can give you very interesting effects and
give you very interesting effects and some of them we will cover in this
some of them we will cover in this course for example you can build a super
course for example you can build a super conducting Cubit which I think uh is one
conducting Cubit which I think uh is one of the next
of the next slides
slides okay think we discussed this very uh
okay think we discussed this very uh um well detailed enough for now but I
um well detailed enough for now but I just wanted to say that this effect
just wanted to say that this effect Still Remains one of the foundational
Still Remains one of the foundational one of the most clear dramatic
one of the most clear dramatic these dramatic steps and important
these dramatic steps and important effects in the field of quantum
effects in the field of quantum transport even though it's already 30
transport even though it's already 30 years old U and I will show you a couple
years old U and I will show you a couple more
more examples of uh experiments that really
examples of uh experiments that really fascinate me or interest me because I
fascinate me or interest me because I also work on them um just to set the
also work on them um just to set the scope of the of the
field so this one is uh somewhat similar and also uh not so new it is from
and also uh not so new it is from 1998 H and uh it is uh very important
1998 H and uh it is uh very important because it shows again quantization of
because it shows again quantization of conductance as a function of gate in
conductance as a function of gate in this case uh but now it is uh in a
this case uh but now it is uh in a one-dimensional system yeah this system
one-dimensional system yeah this system is called a Quantum Point contact and it
is called a Quantum Point contact and it is a narrow channel in a
is a narrow channel in a semiconductor here you can see the side
semiconductor here you can see the side view electrons actually live 100
view electrons actually live 100 nanometers underneath the surface and
nanometers underneath the surface and there are Gates on top uh with which you
there are Gates on top uh with which you can uh if you apply negative voltages to
can uh if you apply negative voltages to them you can deplete electrons here and
them you can deplete electrons here and here so where yellow parts are there are
here so where yellow parts are there are no electrons underneath and so electrons
no electrons underneath and so electrons are forced to go through this channel so
are forced to go through this channel so you apply a source and drain bias here
you apply a source and drain bias here it turns out that if you measure
it turns out that if you measure conductance it is
conductance it is quantized and that has to do with
quantized and that has to do with electrons propagating through this as
electrons propagating through this as completely uninterrupted ballistic
completely uninterrupted ballistic waves and then uh the conductance uh per
waves and then uh the conductance uh per wave is a Quantum of conductance and so
wave is a Quantum of conductance and so if two waves can go through it's two
if two waves can go through it's two Quant of conductance three waves it's
Quant of conductance three waves it's three and this is a beautiful staircase
three and this is a beautiful staircase of
of conductance um this one this data is
conductance um this one this data is from uh Del where I worked I'm very um
from uh Del where I worked I'm very um proud to have worked for the for the
proud to have worked for the for the people who have measured it so Leo
people who have measured it so Leo cowenhoven
cowenhoven uh my my boss from Del he was a master
uh my my boss from Del he was a master student uh on this project and according
student uh on this project and according to him he was the one uh late in the
to him he was the one uh late in the night uh setting up some scan and oh
night uh setting up some scan and oh this kind of staircase showed up so it
this kind of staircase showed up so it must have been amazing to see something
must have been amazing to see something like that early on in the
like that early on in the career um and this remains one of the uh
career um and this remains one of the uh most important Quantum transport effects
most important Quantum transport effects to
to date now let's jump uh 20 years to the
date now let's jump uh 20 years to the Future and um go from
Future and um go from foundational uh effects to uh recent
foundational uh effects to uh recent developments and this is
developments and this is a I would say a set of T the force
a I would say a set of T the force experiments that
experiments that show ultimate Quantum control over spins
show ultimate Quantum control over spins of single particles yeah
of single particles yeah so again I did not explain these
so again I did not explain these semiconductors yet but uh just think
semiconductors yet but uh just think that in this island defined by these
that in this island defined by these Gates you can confine a single
Gates you can confine a single electron so what they've done here is
electron so what they've done here is they've uh measured the spin of that
they've uh measured the spin of that single electron using a transport
measurement so what the the data is it is a certain time dependent pulse like
is a certain time dependent pulse like this um applied to a certain gate and uh
this um applied to a certain gate and uh maybe it's this gate and then they
maybe it's this gate and then they measure current through this uh through
measure current through this uh through this sensor and uh depending on whether
this sensor and uh depending on whether electrons is there or not the current uh
electrons is there or not the current uh is higher or lower and that is what
is higher or lower and that is what gives you these
gives you these pulses um and this little blip here this
pulses um and this little blip here this little blip is a detection of a single
little blip is a detection of a single spin of a single electron so basically
spin of a single electron so basically if the spin is um in a certain direction
if the spin is um in a certain direction let's say up then it can escape from the
let's say up then it can escape from the dot and that gives you a blip and if
dot and that gives you a blip and if it's spined down it cannot Escape uh and
it's spined down it cannot Escape uh and then there's no blip so whether there is
then there's no blip so whether there is a blip or not tells you what the spin
a blip or not tells you what the spin was of that
was of that electron then uh this uh second
electron then uh this uh second experiment is a uh not just a readout
experiment is a uh not just a readout but it's a control over single spin in
but it's a control over single spin in this case they took this kind of
this case they took this kind of structure and they put over it a little
structure and they put over it a little coil to create create a magnetic
coil to create create a magnetic field and they have applied a microwave
field and they have applied a microwave frequency at the Lor frequency of the
frequency at the Lor frequency of the spin and so they have induced rotations
spin and so they have induced rotations of the spin and so you can rotate from
of the spin and so you can rotate from uh the spin being uh down so cannot
uh the spin being uh down so cannot escape to spin being up and can escape
escape to spin being up and can escape and so when you measure a current as the
and so when you measure a current as the spin rotates You observe these
spin rotates You observe these oscillations These are called rby
oscillations These are called rby oscillations I will explain what they
oscillations I will explain what they are later in the course for now I just
are later in the course for now I just want to show you that they were able to
want to show you that they were able to control the orientation of a single Spin
control the orientation of a single Spin and actually in between these Wiggles
and actually in between these Wiggles what they are creating is a Quantum
what they are creating is a Quantum superposition of spin up and spin
superposition of spin up and spin down so this single spin is not just
down so this single spin is not just spin it's a single spin Quantum bit can
spin it's a single spin Quantum bit can be used for a quantum
be used for a quantum computer and finally this experiment
computer and finally this experiment they have made two copies of this one to
they have made two copies of this one to isolate two spins coupled together and
isolate two spins coupled together and by bringing them close and far
by bringing them close and far apart uh they were able to uh rotate
apart uh they were able to uh rotate from both spins pointing up to spins
from both spins pointing up to spins pointing in the opposite direction so go
pointing in the opposite direction so go from singlet to
from singlet to Triplet and that shows that they can not
Triplet and that shows that they can not only control a single spin but they can
only control a single spin but they can couple and control the quantum state of
couple and control the quantum state of two spins yeah so if you can if you
two spins yeah so if you can if you continue going this road uh you can
continue going this road uh you can build a whole Quantum register of spins
build a whole Quantum register of spins which you can control with voltages on
which you can control with voltages on little
little Gates and by now they I know that
Gates and by now they I know that they're already at four and they're
they're already at four and they're going forward with
this um yeah so this is a example of an experiment where we again
example of an experiment where we again scale like in that example with a
scale like in that example with a transistor right we scale down to a
transistor right we scale down to a single particle single electron and we
single particle single electron and we control a single degree of Freedom a
control a single degree of Freedom a single spin there is a very interesting
single spin there is a very interesting Other
Other Extreme can we control big objects yeah
Extreme can we control big objects yeah can we make big objects behave Quantum
can we make big objects behave Quantum mechanically and make superpositions of
mechanically and make superpositions of uh things like Statue of Liberty well no
uh things like Statue of Liberty well no but in the field of quantum transport
but in the field of quantum transport there is a another famous example this
there is a another famous example this is also from Del um where they have
is also from Del um where they have created a Quantum superposition of
created a Quantum superposition of an electrical current flowing clockwise
an electrical current flowing clockwise and counterclockwise in a
and counterclockwise in a loop so what you see here is a metal
loop so what you see here is a metal Loop it is actually a
Loop it is actually a superconductor uh with some uh with some
superconductor uh with some uh with some tunneling Junctions so this is a a
tunneling Junctions so this is a a little bit special Loop but in essence
little bit special Loop but in essence it is a loop with a certain inductance
it is a loop with a certain inductance and uh they've isolated it from the
and uh they've isolated it from the environment and cooled it down to low
environment and cooled it down to low temperatures uh such that the current
temperatures uh such that the current that circulates here
that circulates here can circulate in two directions at the
can circulate in two directions at the same time and the number of particles
same time and the number of particles that participate in this current is in
that participate in this current is in the billions billions of electrons
the billions billions of electrons flowing clockwise and counterclockwise
flowing clockwise and counterclockwise at the same time so some people consider
at the same time so some people consider this an example of macroscopic quantum
this an example of macroscopic quantum coherence although this is a disputed
coherence although this is a disputed point because again this Dimension is
point because again this Dimension is one micron and you have seen before that
one micron and you have seen before that there are many Quantum effects that uh
there are many Quantum effects that uh manifest themselves at this uh at this
manifest themselves at this uh at this length
length scale so again uh these
scale so again uh these oscillations uh is uh where they let the
oscillations uh is uh where they let the current evolve and then after some time
current evolve and then after some time they measure does it flow this way or
they measure does it flow this way or this way and uh they get a when it's red
this way and uh they get a when it's red it's flowing this way and when it's
it's flowing this way and when it's green it's flowing the other way as a
green it's flowing the other way as a function of time there is this Quantum
function of time there is this Quantum rby evolution of current and this
rby evolution of current and this circuit here is a detector so this is uh
circuit here is a detector so this is uh what measures this current and it is
what measures this current and it is called a squid super conducting Quantum
called a squid super conducting Quantum interferometer
okay I think we we should stop here um and we will continue on uh Wednesday
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