Friday, October 21, 2016

Measuring temperature at the milliKelvin scale

How do we tell the temperature of some piece of material?  I've written about temperature and thermometry a couple of times before (here, here, here).  For ordinary, every-day thermometry, we measure some physical property of a material or system where we have previously mapped out its response as a function of temperature.  For example, near room temperature liquid mercury expands slightly with increasing \(T\).  Confined in a thin glass tube, the length of a mercury column varies approximately linearly with changes in temperature, \(\delta \ell \sim \delta T\).  To do primary thermometry, we don't want to have some empirical calibration - rather, we want to measure some physical property for which we think we have a complete understanding of the underlying physics, so that \(T\) can be inferred directly from the measured quantity and our theoretical expressions, with no adjustable parameters.  This is particularly important at very low temperatures, thousandths of a Kelvin above absolute zero, where the number of things that we can measure is comparatively limited, and tiny flows of power (from our measurements, say) can actually produce large percentage temperature changes.

This recent paper shows a nice example of applying three different primary thermometry techniques to a single system, a puddle of electrons confined in 2d at a semiconductor interface, at about 6 mK.  This is all the more impressive because of how easy it is to inadvertently heat up electrons in such 2d layers.  All three techniques rely on our understanding of how electrons behave at low temperatures.  According to our theory of electrons in metals (which these 2d electrons are, as far as physicists are concerned), as a function of energy, electrons are spread out in a characteristic way, the Fermi-Dirac distribution.  From the theory side, we know this functional form exactly (figure from that wikipedia link).  At low temperatures, all of the electronic states below a highest-filled-state are full, and all above are empty.  As \(T\) is increased, the electrons smear out into higher energy states, as shown.  The three effects measured in the experiment all depend on \(T\) through this electronic distribution:
Fig. 2 from the paper, showing excellent, consistent agreement
between experiment and theory, showing electron temperatures 
of ~ 6 mK. 
  • Current noise in a quantum point contact, the fluctuations in the average current.  For this particular device, where conduction takes place through a small, controllable number of quantum channels, we think we understand the situation completely.  There is a closed-form expression for what the noise should do as a function of average current, with temperature as the only adjustable parameter (once the conduction has been measured).
  • "Coulomb blockade" in a quantum dot.  Conduction through a puddle of electrons connected to input and output electrodes by tunneling barriers ("pinched off" versions of the point contacts) shows a very particular form of current-voltage characteristic that is tunable by a nearby gate electrode.   The physics here is that, because of the mutual repulsion of electrons, it takes energy (supplied by either a voltage source or temperature) to get charge to flow through the puddle.  Again, once the conduction has been measured, there is a closed-form expression for what the conductance should do as a function of that gate voltage.
  • "Environmental" Coulomb blockade in a quantum dot.  This is like the situation above, but with one of the tunnel barriers replaced by a controlled resistor.  Again, there is an expression for the particular shape of the \(I-V\) curve where the adjustable parameter is \(T\).  
As shown in the figure (Fig. 2 from the paper - open access and also available on the arxiv), the theoretical expressions do a great job of fitting the data, and give very consistent electron temperatures down to 0.006 K.  It's a very impressive piece of work, and I encourage you to read it - look at Fig. 4 if you're interested in how challenging it is to cool electrons in these kinds of devices down to this level.

Saturday, October 08, 2016

What do LBL's 1 nm transistors mean?

In the spirit of this post, it seems like it would be a good idea to write something about this paper (accompanying LBL press release), particularly when popular sites are going a bit overboard with their headlines ("The world's smallest transistor is 1nm long, physics be damned").  (I discuss most of the background in my book, if you're interested.)

What is a (field effect) transistor and how does it work?  A transistor is an electronic switch, the essential building block of modern digital electronics.  A field-effect transistor (FET) has three terminals - a "source" (an input), a "drain" (an output) on either side of a semiconductor "channel", and a "gate" (a control knob).  If you think of electrical current like fluid flow, this is like a pipe with an inlet, and outlet, and a valve in the middle, and the gate controls the valve.  In a "depletion mode" FET, the gate electrode repels away charges in the channel to turn off current between the source and drain.  In an "accumulation mode" FET, the gate attracts mobile charges into the channel to turn on current between the source and drain.   Bottom line:  the gate uses the electrostatic interaction with charges to control current in the channel.  There has to be a thin insulating layer between the gate and the channel to keep current from "leaking" from the gate.   People have had to get very clever in their geometric designs to maximize the influence of the gate on the charges in the channel.

What's the big deal about making smaller transistors?  We've gotten where we are by cramming more devices on a chip at an absurdly increasing rate, by making transistors smaller and smaller.  One key length scale is the separation between source and drain electrode.  If that separation is too small, there are at least two issues:  Current can leak from source to drain even when the device is supposed to be off because the charge can tunnel; and because of the way electric fields actually work, it is increasingly difficult to come up with a geometry where the gate electrode can efficiently (that is, with a small swing in voltage, to minimize power) turn the FET off and on.

What did the LBL team do?  The investigators built a very technically impressive device, using atomically thin MoS2 as the semiconductor layer, source and drain electrodes separated by only seven nm or so, a ZrO2 dielectric layer only a couple of nm thick, and using an individual metallic carbon nanotube (about 1 nm in diameter) as the gate electrode.  The resulting device functions quite well as a transistor, which is pretty damn cool, considering the constraints involved.   This fabrication is a tour de force piece of work.

Does this device really defy physics in some way, as implied by the headline on that news article?  No.  That headline alludes to the issue of direct tunneling between source and drain, and a sense that this is expected to be a problem in silicon devices below the 5 nm node (where that number is not the actual physical length of the channel).   This device acts as expected by physics - indeed, the authors simulate the performance and the results agree very nicely with experiment.

If you read the actual LBL press release, you'll see that the authors are very careful to point out that this is a proof-of-concept device.  It is exceedingly unlikely (in my opinion, completely not going to happen) that we will have chips with billions of MoS2 transistors with nanotube gates - the Si industry is incredibly conservative about adopting new materials.  If I had to bet, I'd say it's going to be Si and Si/Ge all the way down.   (You will very likely need to go away from Si if you want to see this kind of performance at such length scales, though.)   Still, this work does show that with proper fabrication and electrostatic design, you can make some really tiny transistors that work very well!

Monday, October 03, 2016

This year's Nobel in physics - Thouless, Kosterlitz, Haldane

Update:  well, I was completely wrong!  Topology ruled the day.  I will write more later about this, but Congratulations to Thouless, Kosterlitz, and Haldane!

Real life is making this week very busy, so it will be hard for me to write much in a timely way about this, and the brief popular version by the Nobel Foundation is pretty good if you're looking for an accessible intro to the work that led to this.  Their more technical background document (clearly written in LaTeX) is also nice if you want greater mathematical sophistication.

Here is the super short version.  Thouless, Kosterlitz, and Haldane had major roles to play in showing the importance of topology in understanding some key model problems in condensed matter physics.

Kosterlitz and Thouless (and independently Berezinskii) were looking at the problem of phase transitions in two dimensions of a certain type.  As an example, imagine a huge 2d array of compass needles, each free to rotate in the plane, but interacting with their neighbors, so that neighbors tend to want to point the same direction.  In the low temperature limit, the whole array will be ordered (pointing all the same way).  In the very high temperature limit, when thermal energy is big compared to the interaction between needles, the whole array will be disordered, with needles at any moment randomly oriented.  The question is, as temperature is increased, how does the system get from ordered to disordered?  Is it just a gradual thing, or does it happen suddenly in a particular way?  It turns out that the right way to think about this problem is in terms of vorticity, a concept that comes up in fluid mechanics as well (see this wiki page with mesmerizing animations).  It's energetically expensive to flip individual needles - better to rotate needles gradually relative to their neighbors.  The symmetry of the system says that you can't spontaneously create a pattern to the needles that has some net swirliness ("winding number", if you like).  However, it's relatively energetically cheap to create pairs of vortices with opposite handedness (vortex/antivortex pairs).  Kosterlitz, Thouless, and Berezinskii showed that these V/AV pairs "unbind" collectively at some finite temperature in a characteristic way, with testable consequences.  This leads to a particular kind of phase transition in a bunch of different 2d systems that, deep down, are mathematically similar.  2d xy magnetism and superconductivity in 2d are examples.  This generality is very cool - the microscopic details of the systems may be different, but the underlying math is the same, and leads to testable quantitative predictions.

Thouless also realized that topological ideas are critically important in 2d electronic systems in large magnetic fields, and this work led to understanding of the quantum Hall effect.  Here is a nice Physics Today article on this topic.   (Added bonus:  Thouless also did groundbreaking work in the theory of localization, what happens to electrons in disordered systems and how it depends on the disorder and the temperature.)

Haldane, another brilliant person who is still very active, made a big impact on the topology front studying another "model" system, so-called spin chains - 1d arrangements of quantum mechanical spins that interact with each other.  This isn't just a toy model - there are real materials with magnetic properties that are well described by spin chain models.  Again, the questions were, can we understand the lowest energy states of such a system, and how those ordered states go away as temperature is increased.  He found that it really mattered in a very fundamental way whether the spins were integer or half-integer, and that the end points of the chains reveal important topological information about the system.  Haldane has long contributed important insights in quantum Hall physics as well, and in all kinds of weird states of matter that result in systems where topology is critically important.  (Another added bonus:  Haldane also did very impactful work on the Kondo problem, how a single local spin interacts with conduction electrons.)

Given how important topological ideas are to physics these days, it is not surprising that these three have been recognized.   In a sense, this work is a big part of the foundation on which the topological insulators and other such systems are built.

Original post:  The announcement this morning of the Nobel in Medicine took me by surprise - I guess I'd assumed the announcements were next week.  I don't have much to say this year; like many people in my field I assume that the prize will go to the LIGO gravitational wave discovery, most likely to Rainer Weiss, Kip Thorne, and Ronald Drever (though Drever is reportedly gravely ill).    I guess we'll find out tomorrow morning!

Sunday, October 02, 2016

Mapping current at the nanoscale - part 1 - scanning gates

Inspired by a metaphor made by our colloquium speaker, Prof. Silke Paschen, this past week, I'd like to try to explain to a general audience a couple of ways that people have developed for mapping out the flow of charge in materials on small scales.

Eric Heller's art piece "Dendrite", based
on visualization of branching current flow.
Often we are interested in understanding how charge flows through some material or device.  The simplest picture taught in courses is an analogy with water flowing through a pipe.  The idea is that there is some input for current, some output for current, and that in the material or device, you can think of charge moving like a fluid flowing uniformly along.  Of course, you could imagine a more complicated situation - perhaps the material or device doesn't have uniform properties; in the analogy, maybe there are obstacles that block or redirect the fluid flow.  Prof. Eric Heller of Harvard is someone who has thought hard about this situation, and how to visualize it.  (He's also a talented artist, and the image at right is an example of artwork based on exactly this issue - how the flow of electrons in a solid can branch and split because of disorder in the material.)

There's a different analogy that might be more useful in thinking about how people actually map out the flow of current in real systems, though.  Suppose you wanted to map out the roads in a city.  These days, one option would be to track all GPS devices (especially mobile phones) moving faster than, say, a few km/h.  If you did that you would pretty quickly resolve a decent map of the streets of a city, and you'd find where the traffic is flowing in high volume and at what speed.  Unfortunately, with electronic materials and devices, we generally don't have the option of tracking each individual mobile electron.  

Some condensed matter experimentalists (like Bob Westervelt, for example) have developed a strategy, however.  Here's the traffic analogy: You would set up traffic cameras to monitor the flow of cars into and out of the city.  Then you would set up road construction barrels (lanes blocked off, road closures) in known locations in the city, and see how that affected the traffic flow in and out of town.  By systematically recording the in/out traffic flow as a function of where you put in road closures, you could develop a rough map of the important routes.  If you temporarily close a road that hardly carries any cars, there won't be any effect on the net traffice, but if you close a major highway, you'd see a big effect.  

The experimental technique is called scanning gate microscopy.  Rather than setting up traffic cones, the experimentalists take a nanoscale-sharp conductive tip and scan it across the sample in question, mapping the sample's end-to-end conduction as a function of where the tip is and what it's doing.  One approach is to set the tip at a negative potential relative to the sample, which would tend to repel nearby electrons just from the usual like-charges-repel Coulomb interaction.  If there is no current flowing near the tip, this doesn't do much of anything.  If the tip is right on top of a major current path, though, this can strongly affect the end-to-end conduction.   It's a neat idea, and it can produce some impressive and informative images.  I'll write further about another technique for current mapping soon.