Julien Barrier

At the end of last year, I receive the prize of the engineer for the Future from the French magazine l’Usine Nouvelle, foreshadowing some of the work I’m carrying out for my thesis.

That was for me the occasion to talk about Moiré lattices. I had planned to give a similar talk for the three minute thesis challenge but because of the covid-19 outbreak, this been cancelled. I now think this is a good occasion to develop this here.

I started the presentation in December by giving the example of a screen picture: if you take a photography of your screen, you’ll see a wavy pattern, due to the fact the pixels of your phone are misaligned with the pixels of the pictured screen. This is called a moiré, and there are lots of other occurrences of such a pattern in real life.

In condensed matter physics, moiré have skyrocketed over the past two years, as technology now enables to stack incommensurate layers of different 2D materials. For example, combining graphene with hexagonal boron nitride (that has the same honeycomb pattern than graphene, but 2% larger), create another periodicity, with a unit cell on the order of 10nm, that is 100 times larger than the graphene unit cell. We aim at characterising the behaviour of phonons, electrons, excitons and other quasiparticles in this kind of superlattices, and eventually discover new states of matter.

Second-generation Dirac fermions and the Hofsdtater’s butterfly.

The creation of moiré superlattice increases the lattice constant hence reduces the size of the Brillouin zone. For the Dirac spectrum of graphene, it duplicates the structure at higher filling, that we call second-generation Dirac fermions (Yankowitz 2012). In a magnetic field, superlattices made with aligned hBN enabled the observation the fractal Hofstadter’s butterfly spectrum, via the formation of third-generation neutrality points at high magnetic fields. This fractal pattern can notably be described by the Diophantine equation n/n₀ = t 𝚽/𝝓₀ + s, where n/n₀ is the normalised carrier density, with n₀ = 1/A_sl, A_sl being the superlattice unit cell area, and 𝚽 the magnetic flux per unit cell, 𝝓₀ the magnetic flux quantum, t and s integers. (Ponomarenko 2013, Dean 2013, Hunt 2013, Yu 2014)

Brown-Zak oscillations, or magnetic Bloch states

A more recent examples include the discovery of so-called Brown-Zak oscillations, that are conductivity oscillations occurring at flux fractions 𝚽/𝝓₀ = p/q, where p and q are integers. This is analogous to the Bloch theorem, as whenever the magnetic length has the size of the unit cell, the electronic wavefunction accommodates the periodicity of the lattice and carriers behave as if there were no magnetic field (B_eff = 0). In superlattices made of graphene and hBN, the lattice constant of the Moiré is on the order of 10nm, allowing the observation of such states at magnetic fiels on the order of 20T. (Krishna-Kumar 2017 & 2018) One of the project I have been working on recently aims at characterising the new metallic states created in these Brown-Zak oscillations.

Graphene with a twist

For graphene-graphene twisted bilayers, other effects have been reported. The most famous is now the presence of superconducting domes between Mott insulating states, discovered by Cao et al in the MIT group. If the angle between the two graphene layers is much reduced, the size of insulating domains increases and allows the current to flow in 1D channels. Xu, Berdyugin et al, in our group have notably observed an Aharonov-Bohm effect, where carriers are transported through a triangular network and can interfere together, as in the case of the optical Young slits experiment. This leads to oscillations, with a periodicity in magnetic field intensity, that is a function of the angle between the two layers.

These are only a few examples of what becomes possible by stacking two atomically-thin layers together; the future may be even brighter in this field of physics.

References:

• Yankowitz et al 2012, Emergence of superlattice Dirac points in graphene on hexagonal boron nitride, Nature Physics
• Ponomarenko et al 2013, Cloning of Dirac fermions in graphene superlattices, Nature
• Dean et al 2013, Hofstadter’s butterfly and the fractal qunatum Hall effect in moiré superlattices, Nature
• Hunt et al 2013, Massive Dirac Fermions and the Hofstadter Butterfly in a van der Waals heterostructure, Science
• Yu et al 2014, Hierarchy of Hofstadter states and replica quantum Hall ferromagnetism in graphene superlattices, Nature Physics
• Krishna-Kumar et al 2017, High-temperature quantum oscillations caused by recurring Bloch states in graphene superlattices, Science
• Krishna-Kumar et al 2018, High-order fractal states in graphene superlattices, PNAS
• Xu, Berdyugin et al, 2019, Giant oscillations in a triangular network of one-dimensional states in marginally twisted graphene, Nature Communications

During Summer 2018, I had a research project at Stanford Synchrotron Radiation Lightsource (SSRL) that was wrapping up my masters degree in chemical physics at ESPCI Paris. I used to work on alloys of halide perovskites, aimed at photovoltaic applications. Some part of the work I did there has been published this week in Matter. I take this opportunity to blog about hybrid perovskites for photovoltaics and how we address one of the current limitations this system faces.

Hybrid halide perovskites for photovoltaics

It is no secret on this blog that the perovskite structure is composed of a BX₆ octahedral framework, in the interstices of which A cations are placed. The band gap of the halide perovskite structure largely relies on the length of the B-X bond, as the valence band is an hybrid of metal (B) p and halide (X) p orbital and the conduction band is an anti-bonding combination of metal s and halide p orbital. Thus, a lengthening of the B-X bond would decrease the orbital overlap, stabilise the valence band, hence reducing the band gap.

As a result, compositional adjustments of the A and X lattice have shown abilities to distort the perovskite structure and tune the band gap of the material to target specific applications. Particularly, one of the applications that has the potential to reach the market soon is the production of tandem solar cells, that are silicon cells on top of which a perovskite layer is spun. The key requirement for maximising the efficiency of such a cell is to have the perovskite absorbing low photon energies, and the silicon beneath absorbing the high energy photons that would pass through the perovskite layer. To do this, a number of studies have shown that compositions combining formamidinium (FA) and cesium (Cs) on the A site, lead (Pb) on the B site and a mixture of bromine (Br) and iodine (I) on the X site would allow the production of tandem cells with an optimised band gap.

Photo-induced degradation

However, the perovskites of the form (FA_yCs_1-y) Pb (Br_xI_1-x)₃ are subject to photo-induced phase segregation. In fact, under illumination, thin films made of these alloyed halide perovskites segregate between iodine-rich regions and bromine-enriched domains. Iodine-rich regions have a lower band-gap, that funnels recombination of charge-carriers through energetically-favourable pathways, limiting the voltage of devices, thus reducing the efficiency of solar cells.

Results

However, if the influence of the composition on optical stability is known, the impact of the crystal structure and the changes in morphology had not been isolated from optical instabilities. In this paper, we explored the phase segregation behaviour of perovskites of the form (FA_yCs_1-y) Pb (Br_xI_1-x)₃, processed as thin films, with controlled morphology.

We used synchrotron x-ray diffraction to map these alloys across the cubic-tetragonal solvus. We coupled these measurements with time-dependent photoluminescence spectroscopy to assess stability under illumination. In the paper, we report that there is an imperfect correlation between the crystallographic phase and the phase segregation behaviour. Notably, we find an island of optical stability onto the phase diagram, that lies in the tetragonal phase. Our conclusion is that the phase is not the sole determinant for optical stability. Then, we proposed several phase segregation mechanisms by eliminating some of the current beliefs with our measurements.

In short, phase segregation is currently limiting mass production of perovskite tandem solar cells. Understanding it and getting over this limit may equip perovskite technology with impressive abilities to target climate change. We proposed here a conceptual framework within which optical stability can be explored. More work is obviously necessary to fully understand both the thermodynamics and kinetics of photo-induced phase segregation, everything of this may guide the design of mixed halide perovskite with optimised band-gaps.

Reference:

Structural origins of light-induced phase segregation in organic-inorganic halide perovskite photovoltaic materials

Rachel E. Beal, Nanna Zhou Hagström, Julien Barrier, Aryeh Gold-Parker Rohit Prasanna, Kevin A. Bush, Donata Passarello, Laura T. Schelhas, Karsten Brüning, Christopher J. Tassone, Hans-Georg Steinrück, Michael D. McGehee, Michael F. Toney, and Ana Flávia Nogueira.
Matter. Volume 02. Issue 01. pages 1-13. January 2020
DOI: 10.1016/j.matt.2019.11.001

Since last year, I’ve been a teaching assistant in the Introduction to programming for physicists course at The University of Manchester. This is taught in the first semester of Y2 and we introduce them to Python. While marking the students’ assignments, we have noticed that most of the deductions were made in style. All these are common mistake, but when accumulated, they make the code hard to read. Here are a few notes on coding style. More can be find from PEP8 or Google Python Style Guide.

Structure your code

A good code structure would be the following:

1. header
2. import statements
3. constant definitions
4. function definitions
5. main code

Header.

It should contain a title (e.g. the name of the programme), an author and a date. This allows the reader to quickly understand the purpose of the code, without having to read all the lines in the file. Including the author name serves for copyright, but importantly, indicates who to contact for a question or any request about the code. The date serves to check whether the code is up-to-date.

Example:

# -*- coding: utf-8 -*-
"""
example_header.py
The purpose of this code is to show an example of a header.
Note the first line about encoding. It is good practice to keep it as
it allows some IDE to enforce the encoding and prevent some bugs.
This is mostly a Python 2 feature.

Last updated: 31oct 2019
Julien Barrier - julien.barrier@manchester.ac.uk

"""

Constant definitions.

In python, we call constants global variables. In other languages (e.g. JavaScript, C), constants cannot be changed throughout the document. This is not the case in python, so we place them at the top of the document to state that they are not supposed to be changed. They are useful because it allows us, in this course, to set values specific to the problem we want to solve. When evaluating, we — the examiner — may change these constants and check the program result. It is much easier to have them early than trawling through the code to update every occurrence. In python, we define constant names with the following convention: CAPITALIZED_WITH_UNDERSCORES.

Function definitions.

It is good practice to define a comment inside the function, to describe quickly what the function does and tell the variable type for the input and output arguments.

Example:

def joule_to_ev(value_joule):
    '''
    Converts an energy value from Joule to eV
    args: value_joule (float)
    '''
    value_ev = value_joule/1.60e-19
    return value_ev

Name your functions/variables

All the names for functions/variables should be self explanatory. They should be written in full English. For example, it is always better to write mass = 3.0 rather than m = 3.0. Function names should be written using snake_case in Python. In other languages, different conventions are used (like camelCase in JavaScript, for example).

Ideally, you should look for self explanatory names, meaningful and concise.

Line breaks

maximum line length

It is common to have no longer than 79 characters on a line, so that all the code is visible without having to scroll horizontally, on any screen.

break a line within a function

The convention used when writing a function is to include an extra level of indentation for the variable names.

Example:

def function(
        variable1, variable2, variable_with_a_very_long_name, variable3,
        variable5):
    # default indentation level for the function

break a line in an expression

The convention is to break the line before the mathematical operator, and align it to the innermost parenthesis. When you call a function, break the line at the coma, and the next line should start at the level of the opening parenthesis. When breaking the line in a middle of mathematical operation, it should be done such as the next line starts with the operator at the same level as the previous term. Most of the time, the whole expression should be wrapped up within parenthesis (or eventually a backslash \ ), to ensure Python understands.

Blank lines

The amount of blank lines should be adjusted carefully. If there are too much, the code looks sparse and is hard to read. If there are too few and you have only one block of code, it becomes hard to analyse. Function definitions should be surrounded by two blank lines above and below. Otherwise, use blank lines to show the structure of the code and separate different logics.

Spacing

Whitespaces should be included as in standard typography rules. That means they should not be included:

• before a coma, semicolon or colon;
• before or after a parenthesis, bracket or brace;
• before the open parenthesis/bracket starting a function, an argument list a list indexing or slicing.
• when passing a kwarg (keyword argument, e.g. plt.plot(x, y, ls='-', lw=.5, c='r'));
• at the end of a line;
• to align tokens on consecutive lines, such as =,:,#;

On the other hand, they should be included:

• around assignment operations (=, +=, *=, etc.)
• around comparisons operators (==, <, >, !=, <>, <=, >=, in, not in, is, is not)
• around Booleans (and, or, not);
• after a hashmark in a comment (#)
• when a mathematical operation contains different levels of priority, include whitespaces to show priority (e.g. y = x*2 - 1)

Example:

This code calculates a value of π using a Monte Carlo method: This code is not the most efficient but shows the general structure we expect in the Introduction to programming for physicists course.

# -*- coding: utf-8 -*-
"""
Monte Carlo pi
Program that calculates a value for pi using a Monte Carlo method

Last updated: 31oct 2019
Julien Barrier - julien.barrier@manchester.ac.uk
"""

import random
import math
import matplotlib.pyplot as plt

ITERATIONS = 1000

def check_unit_norm(x,y):
    """
    Checks if inputs are inside the unit circle
    x (float)
    y (float)
    returns Boolean
    """
    if math.sqrt(x**2 + y**2) <= 1:
        return True

counter = 0

for i in range(1,ITERATIONS):

    # we pick a random X,Y position
    X = random.random()
    Y = random.random()

    if check_unit_norm(X, Y):
        COUNTER += 1;
        plt.plot(X, Y, 'bo')
    else:
        plt.plot(X, Y, 'ro')

    # we print out the current value of pi after every 100th point
    if i > 0 and i % 100 == 0:
        print ("Currently %9.7f\n" % (4 * counter / i))

pi = 4 * counter / ITERATIONS
print("A value for pi is %9.7f\n" % pi)

As part of my PhD, I’m operating a dilution fridge 24/7. Recently, I found myself explaining to lot of colleagues or friends how it works, so I think a blog post about this may be quite useful.

Good electron transport measurements necessitate low temperature measurements for several reasons:

• quantum effects do not persist at high temperatures;
• the Fermi edge is sharp at low temperatures;
• lattice vibrations (phonons) are much weaker at low temperatures, inducing low electron-phonon interactions.

Ultra low temperatures

To reach and maintain low temperatures, liquid helium inside cryostat have been the mainstream technique. However, it is difficult and expensive to achieve temperatures significantly below 1K only by pumping ⁴He to low pressures. To get to lower temperatures, H. London proposed in 1951 an idea of the ³He/⁴He dilution refrigerator, that has been realised ten years later at Leiden University by H. London, G.R. Clarke and E. Mendoza, and quickly improved by BS. Neganov in Dubna and H.E. Hall at the University of Manchester⁽¹⁾. The mechanism consists in mixing two stable isotopes of helium, that cools down when it separates in two phases.

The phase diagram of ³He/⁴He is shown below:

The x-T phase diagram shows different interesting features. Let’s consider pure liquids. Liquid ⁴He becomes superfluid below 2.17K, and pure ³He does not show such a phase transition in the temperature range we can consider for a dilution fridge. We note that ⁴He superfluidity terminates for ³He concentrations higher than 67.5%. At this point, the λ-line meets the phase separation line, below which a forbidden region (two-phases) exits. This means that below this line, the two isotopes are only miscible for some particular concentrations. On the side of this two-phase region, there is a light concentrated phase that is rich in ³He, and a heavy dilute phase, rich in ⁴He. The concentration of the ³He in those two phases depends upon the temperature. Because the enthalpy of ³He in the two phases is different, it is possible to cool down the mixture by evaporating the ³He from the concentrated phase into the dilute phase. We may note on the ⁴He-rich side, that the concentration of the dilute isotope does not rich zero when decreasing the temperature close to 0K. The ³He concentration rather reaches the finite concentration of 6.6%, even for T=0K. This property is of paramount importance for the design of a dilution fridge.

The following figure shows schematics of the ³He/⁴He dilution refrigerator. When the refrigerator is started, the ³He/⁴He mixture is condensed into the dilution unit. This allows the cooling down of the mixture down to 1.2K. The tricritical point is at 0.86K so it is the temperature needed to get phase separation. Hence the still is being cooled down to this point with a ⁴He flow, resulting in cooling down the incoming ³He before it enters the heat exchangers and mixing chamber. This later results in phase separation. Gradually, the rest of the dilution unit is cooled down to the point where the phase separation occurs.

Operating Principle

In the operating system, the ³He is extracted from the dilute phase, and returned to the concentrated phase, keeping the system in a dynamic equilibrium. This is done by pumping away ³He from the liquid surface in the still, at temperature at which the pressure of ³He is an order of magnitude higher than the one of ⁴He.

As a result the ³He concentration in the dilute phase in the still becomes lower than in the still, that allows the osmotic pressure to drive a flow of ³He to the still.

Wiring and Connections

The device that needs to be cooled down is fixed over the mixing chamber, ensuring a good thermal contact with the dilute phase. All the connections to the room temperature equipment are anchored at different points of the fridge to reduce the heat load on the mixing chamber and limit heat exchanges with outside. Within the cryostat, it is important to minimise the head load. One way to conduct heat within the measured sample is through the wires that are linked to room temperature connections. For this reason, it is needed to chose materials that have a low thermal conductivity. Unfortunately, a good electrical conductivity often comes with a high thermal conductivity, except for superconducting materials that have no electrical resistance and negligible thermal conductivity. For this reason, conventional superconducting wires can be used below 8K to minimise head load on the sample stage.

Other wires can also be used. They are thermally anchored (we call this thermally sunk) to thermalised plates to ensure the wire is at the required temperature.

Limitations

There is no fundamental limitation on how low can a dilution fridge operate. However, most of the commercial refrigerators are limited above 2mK, because both the viscosity and the thermal conductivity of the mixture would increase as the temperature is lowered. To prevent this, the diameter of the tubes should be reduced and their lengths increased. This would make the volume of the fridge increase madly.

References

1. Pobell F 2007 Matter and Methods at Low Temperatures (Springer)
2. Hall HE, Ford PJ, and Thomson K 1966 A helium-3 dilution refrigerator Cryogenics. 6 (2): 80–88.

As a PhD student at the University of Manchester, I attended this morning a workshop presenting early career fellowships. I’m sharing here some notes about it as it may be valuable to my readers.

The workshop was organised by Dr. Radha Boya, who currently holds a Royal Society Fellowship at UoM. In a first part, different fellowship awardees gave us advice to craft a fellowship application, and this was followed by UoM research support teams presenting the help they can provide. In here, I aim at presenting the main advice from part one presentations.

Fellowships?

In the European research system, tenure-track positions are not common. A few years ago, it used to be possible to get a permanent position directly after a postdoc (or a PhD), but that has never become easier with time. To compensate for this, many funding institutions (ERC, Research Councils UK, European Commission, Wellcome Trust, The Royal Society, The Bill & Melinda Gates foundation, etc.) as well as research Institutes/Universities award early career fellowship for researchers that have enough experience to carry out their own independent research, but lack recognition to secure a permanent position. These institutions usually look for future research leaders with promised world authority on specific topics. The fellowships are usually tenable for 3 to 5 years, after which the fellow ideally obtains a post in the institution in he/she was awarded the fellowship. In this sense, this is comparable to the US tenure track system (with the exception one is not entitled “Assistant Professor” but just “Fellow”).

Applying

During the workshop, some guidance was given by successful fellowship candidates. A common ground for fellowships include a career summary (CV, personal statement, awards, career history, qualifications, list of publications), a research proposal and financial details about the proposed project that aims at selecting candidates for interviews. One of the recurring advice to craft the application was preparation. Before starting to write anything, it has been advice to think about the main idea and the structure of the application package. It has notably said that finding a good idea is almost getting half way. However, the research proposal should show adequacy with the candidate’s background: you may not propose something you’re not familiar with, or would not be able to justify preliminary experience. The personal statement should be oriented the same way. Never forget it is a strictly personal document, that usually writes where the writer’s interest stem from. The workshop panel also highlighted that applicants should feature independent research with a novel and individual idea. In no way this can be an extension of the candidate’s supervisor/P.I. research topics.

Overall, these two documents (along with the curriculum vitæ), should show the panel that the candidate is the best do carry the proposed research from submission date (not five years after the proposal!). It has also been noted that some funding agencies care a lot about hot topics. This does not especially mean that you should select your project depending on what hits the headline of top academic journals, but rather try to uncover and highlight the hidden value of your idea. Thinking about the bigger picture may help: for example highlighting the energy applications of some unknown photo-electric materials may be a hot value you’re not thinking in the first place. It was also reminded that a candidate that loves what he/she does would certainly be more successful that a candidate choosing hot topics to merely be read.

Based on all of this, devising a research proposal can be summarised in addressing the following questions:

• is the proposed work novel?
• is the proposed work feasible? i.e. has the applicant done related research before, has he/she a clear plan to achieve it?
• why does the project matter in the global picture?
• is the applicant the adequate researcher to carry it out?

Finally, when designing the financial details, it was noted that funding agencies would not select the cheaper projects, but rather based on the envisaged success of the project. Remember that your institution may provide guidance for this part.

What happens after?

The successful applicants would be offered an interviews. Keys for success again includes preparation. The panel told us to get feedbacks from previous colleagues that may be the best to judge our work and ask adequate questions. Having a mock interview organised with researchers outside of your field can also be valuable, as all the judging panel may not directly work in your research area.

If success is not in the cards, remember that the fellowship applications usually have very low success rates (10-15%). This means you should not discount yourself based on first rejections. In this case, you may apply again and again, by always building on feedback from past applications. Don’t be salty about a rejection for a very selective fellowship, but rather try to get feedback that will help you to strengthen future ones.

Would you have other advice for early career researrchers? Let’s discuss this on twitter!