1 Observation of Giant Orbital Magnetic Moment s and Paramagnetic Shift in Artificial Relativistic Atom s and Molecules Zhehao Ge1 Sergey Slizovskiy23 Peter Polizogopoulos1 Toyanath Joshi1 Takashi Taniguchi4

2025-04-30 0 0 1.64MB 24 页 10玖币

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Observation of Giant Orbital Magnetic Moments and Paramagnetic

Shift in Artificial Relativistic Atoms and Molecules

Zhehao Ge1,*, Sergey Slizovskiy2,3,*, Peter Polizogopoulos1, Toyanath Joshi1, Takashi Taniguchi4,

Kenji Watanabe5, David Lederman1, Vladimir I. Fal’ko2,3,6, †, Jairo Velasco Jr.1, †

1Department of Physics, University of California, Santa Cruz, California, USA

2Department of Physics and Astronomy, University of Manchester, Oxford Road, Manchester, M13

9PL, UK

3National Graphene Institute, University of Manchester, Booth Street East, Manchester, M13 9PL

4International Center for Materials Nanoarchitectronics National Institute for Materials Science,

1-1 Namiki, Tsukuba, 305-0044, Japan

5 Research Center for Functional Materials National Institute for Materials Science, 1-1 Namiki,

Tsukuba, 305-0044, Japan

6 Henry Royce Institute for Advanced Materials, Manchester, M13 9PL, UK

*These authors contributed equally to this manuscript.

†Email: jvelasc5@ucsc.edu, Vladimir.Falko@manchester.ac.uk

Abstract:

Massless Dirac fermions have been observed in various materials such as graphene and

topological insulators in recent years, thus offering a solid-state platform to study relativistic

quantum phenomena. Single quantum dots (QDs) and coupled QDs formed with massless Dirac

fermions can be viewed as artificial relativistic atoms and molecules, respectively. Such structures

offer a unique platform to study atomic and molecular physics in the ultra-relativistic regime. Here,

we use a scanning tunneling microscope to create and probe single and coupled electrostatically

defined graphene QDs to unravel the unique magnetic field responses of artificial relativistic

nanostructures. Giant orbital Zeeman splitting and orbital magnetic moment up to ~ and

~ are observed in single graphene QDs. While for coupled graphene QDs, Aharonov–Bohm

oscillations and strong Van Vleck paramagnetic shift ( are observed. Such

properties of artificial relativistic atoms and molecules can be leveraged for novel magnetic field

sensing modalities.

Quantum dots (QDs) are often referred to as artificial atoms because of their atomic-like

electronic structure1,2. They have been widely studied over the last 40 years in semiconductors and

have provided immense fundamental insight3-5. Recently, the confinement of massless Dirac

fermions in electrostatically defined QDs has been achieved in graphene6-15 and topological

insulators16. Different from semiconductor QDs formed with massive Schrödinger fermions, QDs

populated by massless Dirac fermions can be viewed as artificial relativistic atoms, thus offering

a unique opportunity to study atomic properties in the ultra-relativistic regime.

Graphene is an ideal platform for studying relativistic quantum phenomena because it hosts

massless Dirac fermions17 and has high tunability via electrostatic gating. As a result, multiple

relativistic quantum phenomena have been demonstrated with graphene such as Klein

tunneling18,19 and atomic collapse20,21. Such phenomena are important not only for fundamental

research but also for technological applications. For example, Klein tunneling renders graphene

pn junctions highly transparent, which makes graphene an outstanding platform for electron optics

applications such as negative refraction22, Veselago lensing23, and beam collimation24,25.

When graphene massless Dirac fermions are confined into a quantum dot (QD), another

intriguing platform for relativistic physics is realized, an artificial relativistic atom. For such a

system, the usual relationship between orbital magnetic moment () and angular momentum (





)

for atomic states (  





,  is the Bohr magneton) is invalid. This is because massless Dirac

fermions disobey the non-relativistic relationship between velocity and momentum,   .

Instead,  is given by the area of the atomic orbit  multiplied by the electrical current (

 ),

which results in    . Because of this, the large and constant Dirac velocity 

together with a sizable atomic orbital radius  can produce extremely large  for artificial

relativistic atoms. One direct consequence of this large  is a giant Zeeman splitting for artificial

atomic orbital states in a magnetic field (), which can potentially be useful for sensing. Such

properties of artificial relativistic atoms, however, have not been experimentally demonstrated to

date. In this article we investigate single and coupled graphene QDs that are subjected to an

external perpendicular B. These structures function as artificial relativistic atoms and molecules,

and reveal intriguing  responses that originate from the relativistic nature of these nanostructures

such as giant orbital Zeeman splitting and strong paramagnetic shift.

Observation of linear orbital Zeeman splitting

We study graphene QDs defined by electrostatically induced circular pn junctions with a

scanning tunneling microscope (STM) as schematized in Fig. 1a. Although Klein tunneling18,19

makes it difficult to confine massless Dirac fermions, their oblique incidence onto the circular pn

junction boundary (schematized in Fig. 1b) avoids the 100% transmission occurring at normal

incidence. This allows for the formation of quasi-bound states in graphene QDs, which have been

confirmed in previous experiements6,8-12,14,15. In zero B, the clockwise and counterclockwise quasi-

bound states possessing the same radial quantum number () and angular quantum numbers ()

are degenerate due to time reversal symmetry. The directions of their , however, are opposite

(Fig. 1c). Thus, by applying an external B, the degeneracy between the clockwise and

counterclockwise quasi-bound states is lifted through an orbital Zeeman effect (Fig. 1d), leading

to a splitting energy     





. This linear orbital Zeeman splitting can be used to measure 

of graphene QD states.

Importantly, the Berry phase change of graphene QD states10,26 in  precludes the

measurement of . To avoid this, we create graphene QDs with unprecedently sharp potential wells

(further discussion in SI section S2). Our method involves using a two-step tip voltage pulsing

technique based on prior works9,27 (details in SI section S3) on samples with reduced hexagonal

boron nitride (hBN) thickness. Figure 1e shows a typical 

 spectra along a line across the

center of a circular pn junction created with this technique on a large angle () twisted bilayer

graphene (tBLG)/hBN sample. By tracking the graphene charge neutrality point (marked by white

dots in Fig. 1e), we estimate the potential variation to be 200~ across . This is

2~3 times sharper than previous works that utilized a related tip pulsing technique9,10,14,15. Figure

1f shows a comparison of 

 point spectra at    and  of the graphene QD shown

in Fig. 1e. Evidently, the 

 peaks are much sharper off center than at the QD center. This is

because near the QD boundary, states with larger m are concentrated, which correspond to Dirac

fermions propagating tangentially to the pn junction, resulting in a stronger reflection and hence

better confinement8,9. For the remainder of this work, we will focus on these large  states.

We now study the response of our graphene QDs to a perpendicular . Figure 2a shows

the comparison of 



  measured across the center of another graphene QD with a sharp

potential well in    and . Splitting patterns are clearly seen in    as dimples

near the QD boundary where high  states concentrate. Figure 2b shows the evolution of 



point spectra at    in various , the splitting and merging of graphene QD states can be

seen as B increases. To visualize this behavior more clearly, 



  with high  resolution

was acquired and is shown in Figure 2c. These data were taken from the same graphene QD shown

in Fig. 2a at   , here 





  are presented to enhance the visibility of 



peaks (raw 



  data in Extended Data Fig. 1). We observe a clear linear splitting for

each QD state. We attribute this behavior to orbital Zeeman splitting and find it is present at

locations off the QD center but absent near the center where low m states concentrate (see SI

section S4). These experimental findings are all in good agreement with simulations based on a

tight binding (TB) model for a graphene QD (methods in SI section S5) with a quadratic potential

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1ObservationofGiantOrbitalMagneticMomentsandParamagneticShiftinArtificialRelativisticAtomsandMoleculesZhehaoGe1,*,SergeySlizovskiy2,3,*,PeterPolizogopoulos1,ToyanathJoshi1,TakashiTaniguchi4,KenjiWatanabe5,DavidLederman1,VladimirI.Fal’ko2,3,6,†,JairoVelascoJr.1,†1DepartmentofPhysics,UniversityofCalif...

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1 Observation of Giant Orbital Magnetic Moment s and Paramagnetic Shift in Artificial Relativistic Atom s and Molecules Zhehao Ge1 Sergey Slizovskiy23 Peter Polizogopoulos1 Toyanath Joshi1 Takashi Taniguchi4

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