Deep learning extraction of band structure parameters from density of states a case study on trilayer graphene Paul Henderson1Areg Ghazaryan2Alexander A. Zibrov3 4Andrea F. Young4and Maksym Serbyn2

2025-04-26 0 0 2.15MB 12 页 10玖币

侵权投诉

Deep learning extraction of band structure parameters from density of states: a case

study on trilayer graphene

Paul Henderson,1Areg Ghazaryan,2Alexander A. Zibrov,3, 4 Andrea F. Young,4and Maksym Serbyn2

1School of Computing Science, University of Glasgow, Scotland

2Institute of Science and Technology Austria (ISTA), Am Campus 1, 3400 Klosterneuburg, Austria

3Department of Physics, Harvard University, Cambridge, MA, USA

4Department of Physics, University of California, Santa Barbara, CA, USA

(Dated: September 19, 2023)

The development of two-dimensional materials has resulted in a diverse range of novel, high-quality

compounds with increasing complexity. A key requirement for a comprehensive quantitative theory

is the accurate determination of these materials’ band structure parameters. However, this task is

challenging due to the intricate band structures and the indirect nature of experimental probes. In

this work, we introduce a general framework to derive band structure parameters from experimental

data using deep neural networks. We applied our method to the penetration ﬁeld capacitance

measurement of trilayer graphene, an eﬀective probe of its density of states. First, we demonstrate

that a trained deep network gives accurate predictions for the penetration ﬁeld capacitance as a

function of tight-binding parameters. Next, we use the fast and accurate predictions from the

trained network to automatically determine tight-binding parameters directly from experimental

data, with extracted parameters being in a good agreement with values in the literature. We

conclude by discussing potential applications of our method to other materials and experimental

techniques beyond penetration ﬁeld capacitance.

I. INTRODUCTION

Electronic band structure of crystalline solids repre-

sents a simple yet very rich example of emergence. Under

the inﬂuence of scattering from the lattice potential, the

electron may acquire a diﬀerent value of eﬀective mass,

became massless, and acquire additional quantum num-

bers such as pseudospin. In addition, electronic band

structure determines basic properties of materials, pro-

vided interactions are weak enough [1]. Therefore, identi-

fying material parameters that determine the band struc-

ture is of crucial importance. From a theoretical point

of view ab initio methods such as density functional the-

ory have achieved enormous success in this direction [2].

Nevertheless one typically relies on experimental data to

quantitatively extract band structure properties. Exper-

imentally there exist numerous ways to access the elec-

tronic structure, such as angle resolved photoemission

[3] and X-ray absorption spectroscopy [4], de Haas-van

Alphen eﬀect based on magnetic oscillations [5], analyz-

ing reﬂection and absorption spectra [6], and electronic

transport measurement [7], to name just a few. Despite

such a wealth of measurement techniques, matching ex-

perimental results with theoretical predictions remains a

challenging problem due to the complexity of the band

structure, which translates into a large number of in-

volved parameters.

The recent surge of two-dimensional (2D) materials [8]

brings new aspects to the problem of determining band

structure. First, often the complexity and the number

of parameters in 2D materials is considerably lower com-

pared to their three-dimensional counterparts. Besides,

2D materials feature additional level control such as mod-

ifying charge density by gating, and may have an ex-

tremely high crystal quality. This opens access to high

resolution experimental data, which may potentially be

used for precise determination of band structure param-

eters. A particular example of such data is provided by

so-called penetration ﬁeld capacitance measurements [9],

that eﬀectively probe the density of states (DOS) of the

material as a function of carrier density and transverse

electric ﬁeld. Such experimental data has been used to

determine material parameters such as hopping matrix

elements in several 2D systems [10, 11].

Typically, extraction of band structure parameters

based on experimental data relies on an eﬃcient solution

to what we term the forward problem. In the speciﬁc

example of penetration ﬁeld capacitance measurements

sensitive to the DOS, this means simulating the DOS for

speciﬁc values of material parameters such as hopping

matrix elements entering tight-binding model of the band

structure. However, the existence of an eﬃcient solution

for this forward problem does not guarantee a fast solu-

tion to the inverse problem—identiﬁcation of the physi-

cal parameters corresponding to a set of empirical data.

The inverse problem is challenging because (i) solving

for the best-ﬁtting parameters is a high-dimensional op-

timization problem that requires numerous simulations

of the forward problem at each step that can quickly

become very costly numerically; and (ii) experimental

measurements are typically aﬀected by additional factors

not easily accounted for in simulation (e.g. geometric and

parasitic capacitance, disorder), meaning that an exact

match between the data simulated in the forward prob-

lem and that obtained from experiments is not possible.

The typical approach is therefore manual comparison of

an experimental dataset with a large number of simulated

ones, relying on physical intuition of which features are

important. This process is laborious and computation-

ally expensive [12], calling for the development of more

arXiv:2210.06310v2 [cond-mat.mes-hall] 18 Sep 2023

eﬃcient and systematic approaches.

In this work we present a machine learning based

method that automates the process of comparing numer-

ical simulation and experimental data, so the physical

parameters of the band structure that gave rise to a par-

ticular experimental dataset can be determined with min-

imal human eﬀort. Recently machine learning and arti-

ﬁcial neural network techniques have seen various appli-

cations in the realm of physical sciences [13]. In con-

densed matter physics, artiﬁcial neural networks have

been used to represent quantum states [14, 15] and learn

these states from available data [16, 17]. In a diﬀerent

direction, recently machine learning models were use for

photonic crystals band diagram prediction and gap opti-

misation [18–20]. Despite a large number of more the-

oretical applications, machine learning approaches are

only starting to be employed in analysis of experimental

data. Recent examples include identiﬁcation of quantum

phase transitions [21] and hidden orders from experimen-

tal images [22]. These few examples highlight the strong

potential of machine learning based approaches on ex-

perimental data, that we further exploit in the present

work.

A conceptual overview of our approach is shown in

Fig. 1. To extract the band structure parameters from

experimental data, we ﬁrst train a deep neural net-

work (DNN) [23, 24] that solves the forward problem by

replicating the numerical calculation of the DOS (Sec-

tion II A). To this end we use the simulation of the ex-

perimental data shown in Fig. 1(a). In the particular

example of penetration ﬁeld capacitance data considered

here, the simulator uses the band structure parameters,

the asymmetry potential between two edges of the system

(physically equivalent to transverse electric ﬁeld) and the

chemical potential as input parameters. As an output we

get charge density and from that determine the DOS by

diﬀerentiating density with respect to chemical poten-

tial. A set of simulated data is used to train the DNN in

Fig. 1(b). Constructed in a way to eﬃciently replace the

data simulator, the DNN acts as a function that takes the

band structure parameters, the asymmetry potential and

directly charge density as input, and outputs the corre-

sponding DOS. It is constructed by learning from a large

dataset of simulation results, optimising its output to al-

ways match that of the simulator. The resulting DNN

represents a fast and diﬀerentiable replacement for the

physical simulation. It can therefore be used to eﬃciently

solve the inverse problem (Section II B). In particular, the

values of parameters that gave rise to a given dataset are

extracted using gradient-based optimisation in Fig. 1(c),

where we iteratively modify the band structure parame-

ters until the DNN’s output matches the provided DOS

values.

The task of mapping a vector of inputs (e.g. band

structure parameters) to a continuous output (e.g. DOS)

is known as regression in machine learning [24]. A DNN

implements such a mapping as a series of chained ma-

trix multiplications (‘layers’) interleaved with element-

wise non-linear functions (‘activations’). Each layer mul-

tiplies the vector of outputs from the previous by some

weight matrix, to give an updated vector [23]; the ﬁnal

layer typical yields a single value. The weight matri-

ces are optimized (‘trained’) using ﬁrst-order optimiza-

tion (e.g. gradient descent), such that the overall map-

ping from inputs to output approximates some func-

tion deﬁned by a training dataset of inputs and desired

outputs. The celebrated universal approximation the-

orem [25] proves that a neural network with just two

layers (but unbounded width) can represent any smooth

function. More recently, it has been shown that this is

also true for a neural network of bounded width (but un-

bounded depth) [26]. In practice, even ﬁnite-sized DNNs

have proven very successful in approximating complex

functions in many domains of science. One of our contri-

butions is to show a DNN can also provide an accurate

estimate of DOS given band-structure parameters, ﬁeld

strength, and chemical potential.

Since our ﬁnal goal is to determine the band-structure

parameters from experimental measurements of penetra-

tion capacitance, it might seem natural to train the DNN

for exactly this task (the inverse problem), instead of

the forward problem. However, this is not possible in

practice. We have only a single experimental dataset,

for which the parameters are unknown, whereas machine

learning techniques require a large dataset of training ex-

amples (with the true output known) to learn the desired

mapping from. If instead we trained on easy-to-obtain

simulated data, the resulting model would not work on

experimental data since the latter is signiﬁcantly diﬀerent

from the former both in terms of the relative magnitude

of features in the data, and the locations of features such

as edges. These diﬀerences may arise since the simula-

tor uses a simpliﬁed eﬀective model of the material and

does not account for screening at the microscopic level,

disorder, strain, experimental uncertainties, and possibly

other ingredients. In contrast, statistical learning theory

requires that the training and test data be drawn from

the same distributions if a trained model is to work on

the latter [27].

As a speciﬁc application, we demonstrate the frame-

work outlined above on Bernal stacked (ABA) tri-

layer graphene. For this material both band structure

parametrization [28, 29] and high quality experimental

measurements are readily available [11], calling for an

accurate extraction of the band structure parameters.

The determination of the band structure was performed

by tour de force manual ﬁtting in an earlier work [11],

thus allowing us to benchmark our approach. First, we

train the DNN and show it gives an eﬃcient and accu-

rate surrogate for numerical calculation of the DOS for

this system, for a wide range of band structure param-

eters (Section III C). Next, we use the DNN for auto-

matically solving the inverse problem of determining the

physical parameters giving rise to certain values of the

penetration capacitance (Section III D). Finally, we ap-

ply this to experimental data from Ref. [11] by exploit-

neural

network

measure

difference

iteratively

update γ

predicted

training

data

physical

simulator 

Δ1

empirical Cp

transform

predicted

to Cp

(a) data generation

γ ,

...

Δ1

FIG. 1. Our approach to the direct extraction of the band structure parameters uses datasets obtained with a physical

simulator shown in panel (a) to train a DNN in panel (b), thus providing a more eﬃcient solution to the forward problem. The

DNN-based simulator is used for a gradient optimization of the band structure parameters, allowing to extract their values

from experimental data in panel (c).

ing techniques from computer vision, that allow match-

ing important features of the measurements (e.g. Van

Hove singularity peaks and jumps of DOS), while ignor-

ing features that diﬀer between experimental and simu-

lated data (e.g. measurement noise and discreteness of

calculation grid) (Section III E). The resulting values of

parameters agree within error bars with the estimates

from the literature, thus providing a particular bench-

mark for our approach.

The paper is organized as follows: Section II describes

the general structure of the DNN for the forward problem

and our minimization approach for inverse problem. Sec-

tion III applies the framework constructed in Section II to

ABA graphene simulation results for the forward and in-

verse problems, eventually utilizing it for extracting band

parameters from experimental results. Finally Section IV

is devoted to discussion of the main results and general-

ization of the model to other systems.

II. METHOD

We assume access to a simulator S(see Fig. 1) that

in the particular example of the penetration ﬁeld capaci-

tance calculates the charge density nand density of states

ν=∂n

∂µ given band structure parameters γ, interlayer

asymmetry ∆1, and chemical potential µ. We assume

γ∈Γ, where Γ deﬁnes a physically-plausible range for

those parameters, and similarly that (∆1, n)∈P. The

approach presented here is general, while the speciﬁc

physical meaning of these parameters will be discussed

in Section III. Typically the simulator will be slow to

evaluate, making it diﬃcult to use ‘in the loop’ for solv-

ing the inverse problem, of ﬁnding the physical parame-

ters γcorresponding to observed data. We shall instead

use Sto generate training data represented by tuples

(γ,∆1, n, ν) for a machine learning regression model—a

deep neural network—Fω, that will be trained to approx-

imate Sin Section II A. We shall then use the result-

ing DNN Fωwhen solving the inverse problem in Sec-

tion II B.

A. Forward problem

We introduce a function Fωthat maps γ, ∆1and n

to the DOS. We choose Fωto be a deep neural network

(DNN) [23], with weights ω; these weights are free pa-

rameters that determine the function it represents. Our

goal is that Fωmatches Sas closely as possible for all

relevant values of γ, ∆1and µ, i.e. if the simulator re-

turns nand νfor given (γ,∆1, µ) and if (∆1, n)∈P

are within the domain of physically realistic parameters,

then DNN approximates well the DOS, Fω(γ,∆1, n)≈ν.

The network weights ωwould ideally be set to minimise

the absolute diﬀerence between the network’s predicted

values and those νfrom the simulator, over the entire

parameter space Γ ×P:

ω= argmin ω′Z(∆1, n)∈P,γ∈ΓFω′(γ,∆1, n)−νdγd∆1dn

In practice we instead minimise the mean error over a

ﬁnite training set [24] of points T ⊂ Γ×Pat which we

have precomputed nand νusing the simulator S, i.e.

ω= argmin ω′1

|T | X

(γ,∆1, n, ν)∈T Fω′(γ,∆1, n)−ν.(1)

To solve this optimisation problem, the weights ωare

initialised using the heuristic of Ref. [30], then itera-

tively updated using the ﬁrst-order stochastic-gradient

optimiser Adam [31] with a minibatch size of 512 and

learning rate (step size) of 10−3. We use a DNN with

ﬁve fully-connected layers of 512 units each, with ELU

nonlinearities [32], layer normalisation [33], and residual

connections [34]. For the input layer, we use Fourier fea-

ture embedding with four octaves [35]; for the output

layer, we use a single linear unit, see Appendix A for

details. We select these architectural parameters, and

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

Deeplearningextractionofbandstructureparametersfromdensityofstates:acasestudyontrilayergraphenePaulHenderson,1AregGhazaryan,2AlexanderA.Zibrov,3,4AndreaF.Young,4andMaksymSerbyn21SchoolofComputingScience,UniversityofGlasgow,Scotland2InstituteofScienceandTechnologyAustria(ISTA),AmCampus1,3400Klosterne...

展开>> 收起<<

Deep learning extraction of band structure parameters from density of states a case study on trilayer graphene Paul Henderson1Areg Ghazaryan2Alexander A. Zibrov3 4Andrea F. Young4and Maksym Serbyn2.pdf

共12页,预览3页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Deep learning extraction of band structure parameters from density of states a case study on trilayer graphene Paul Henderson1Areg Ghazaryan2Alexander A. Zibrov3 4Andrea F. Young4and Maksym Serbyn2

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: