Supermoir e low-energy eective theory of twisted trilayer graphene Yuncheng Mao1Daniele Guerci2and Christophe Mora1 1Universit e Paris Cit e CNRS Laboratoire Mat eriaux et Ph enom enes Quantiques 75013 Paris France

2025-05-02 0 0 3.34MB 17 页 10玖币
侵权投诉
Supermoir´e low-energy effective theory of twisted trilayer graphene
Yuncheng Mao,1Daniele Guerci,2and Christophe Mora1
1Universit´e Paris Cit´e, CNRS, Laboratoire Mat´eriaux et Ph´enom`enes Quantiques, 75013 Paris, France
2Center for Computational Quantum Physics, Flatiron Institute, New York, New York 10010, USA
Stacking three monolayers of graphene with a twist generally produces two moir´e patterns. A
moir´e of moir´e structure then emerges at larger distance where the three layers periodically realign.
We devise here an effective low-energy theory to describe the spectrum at distances larger than the
moir´e lengthscale. In each valley of the underlying graphene, the theory comprises one Dirac cone
at the ΓMpoint of the moir´e Brillouin zone and two weakly gapped points at KMand K0
M. The
velocities and small gaps exhibit a spatial dependence in the moir´e-of-moir´e unit cell, entailing a
non-abelian connection potential which ensures gauge invariance. The resulting model is numerically
solved and a fully connected spectrum is obtained, which is protected by the combination of time-
reversal and twofold-rotation symmetries.
I. INTRODUCTION
Following the theoretical prediction and experimental
discovery of tantalizing physical properties in twisted bi-
layer graphene (TBG) [1–9], different multilayer stack-
ings of individual graphene sheets or graphene sheets
with hexagonal boron nitride layer have been explored
where exotic electronic properties may emerge with or
without the relative rotations between the layers [10–40].
With two layers, a geometrical moir´e pattern emerges
with a periodicity largely exceeding the original unit
cell of graphene and kinetic energy is quenched in the
vicinity of the magic angle ∼1.05◦. Experiments have
revealed numerous appealing properties such as super-
conductivity [6,38,41–44], correlated insulator phase
[7,23,38,41,42], nematicity [45–48], integer and frac-
tional Chern insulators [14,49–54], spontaneous flavor
polarization [35,37,55], orbital ferromagnetism [56–59]
or strange-metal behavior [60–63].
FIG. 1. (a) Real-space arrangement of the staircase tri-
layer graphene monolayers. (b) The Brillouin zones of the
3 graphene sheets. (c) Illustration of the decomposition of
native moir´e vectors into approximate moir´e vectors and su-
permoir´e vectors.
Adding a third graphene sheet on top of TBG with a
relative rotation generates the twisted trilayer graphene
(TTG) configuration. Experiments in TTG have shown
evidence of correlated insulating phases and robust su-
perconductivity with unconventional pairing [27,28,39,
64–67]. In comparison with TBG, TTG exhibits better
tunability of the electronic structure in mainly three as-
pects: (1) the two independent twist angles of TTG allow
for an extra degree of freedom of manipulation [68]; (2)
in contrast with TBG where an horizontal shift of one
layer can be canceled by an appropriate choice of origin
or equivalently, by a unitary transform, such invariance
does not always exist in TTG; (3) the band structure of
TTG is largely tunable by applying a perpendicular elec-
tric field [69,70]. Experimentally so far, the most suc-
cessful implementation of TTG [64–67] is the symmetric
stacking where the top and bottom layers are rotated in
the same direction with respect to the middle layer. Such
an arrangement has exactly the same moir´e periodicity
as TBG but depends on the relative horizontal shift of
one layer. Its success lies in a remarkable band structure
occurring at a magic angle ∼1.48◦(√2 times the magic
angle of TBG) [13] where an almost flat band coexists
with a large velocity Dirac cone [13,29,71,72]. The cor-
related phases of symmetric TTG have been explored in
a number of theory papers [73–76].
In this paper we focus on yet another configuration:
the staircase TTG [77] where the top and the bottom lay-
ers are rotated oppositely with respect to the middle layer
as illustrated in Fig.1. In this case, the two moir´e pat-
terns resulting from pairs of subsequent layers combine
to form a supermoir´e or moir´e of moir´e structure [68],
even for equal twist angles. The unit cell of supermoir´e
is parametrically larger than the individual moir´es. The
corresponding Brillouin zone is in turn much smaller with
fewer electron densities than TBG or symmetric TTG.
Remarkably, an experimental realization [28] of staircase
TTG has revealed correlated insulating states at elec-
tronic densities as small as 1010 cm−2in agreement with
the formation of supermoir´e mini-bands which sponta-
neously choose a spin/valley polarization. The same ex-
periment [28] also measured zero-resistance states sugges-
tive of superconductivity. The superconducting instabil-
arXiv:2210.11507v2 [cond-mat.mes-hall] 20 Mar 2023
2
ity is generally favored by van Hove singularities or by an
accumulation of states at certain energies as for instance
with flat bands. A strongly peaked density of states at
charge neutrality was predicted theoretically [68] in the
parameter regime of the experiment [28]. The physical
origin behind the accumulation of low-energy states in
this model is however unclear.
The aim of our work is to provide an effective low-
energy theory for the moir´e of moir´e band structure.
The direct numerical solution to the moir´e continuum
(or Bistritzer-MacDonald) model extended to TTG can
be achieved [68,78] but it requires the diagonalization of
large matrices and poses a challenge, for instance with the
inclusion of interactions. Our approach is a low-energy
limit to the Bistritzer-MacDonald (BM) theory. Whereas
the graphene lattice is coarse-grained in BM to achieve
an effective model valid at moir´e scales, we further per-
form a low-energy coarse-graining at the moir´e scale and
derive an effective, but asymptotically exact, theory for
moir´e of moir´e mini-bands.
At moir´e length scales, staircase TTG has a moir´e spec-
trum [77], in fact two with the time-reversed valleys K
and K0of the Graphene Brillouin zone. In each valley,
we identify three low-energy sectors in the vicinity of the
ΓM,KMand K0
Mpoints, which makes six sectors in total.
Each sector presents a Dirac cone (ΓM) or weakly gapped
Dirac cone (KMand K0
M) where velocities and gaps vary
over the supermoir´e unit cell. Importantly, this spatial
dependence, akin to a motion in a curved space, imposes
the emergence of a non-abelian real space connection to
ensure the gauge invariance of the model. The impact
of the connection on the band structure is crucial: the
density of states of the resulting model differs strongly
from an average over the different moir´e hamiltonians.
The connection, in deep analogy with a magnetic field,
redistributes weights and tends to regroup energies.
Our approach can be seen as a generalization of the k·p
theory to a smooth change of band structure, a method
pioneered by Kohn and Luttinger [79] for semiconductors
(see also Ref. [80]) and later adapted, under the name
of envelope functions, to solve the spectrum of semicon-
ductor heterostructures and superlattices [81–85]. The
specificity of graphene material studied in this work is
the presence of Dirac cones as opposed to the quadratic
dispersion of semiconductors, and the strong role played
by the non-abelian connection amplified by the proximity
of the conduction and valence bands. Our work has also
several analogies with conical intersections in the Born-
Oppenheimer approximation for molecules [86,87].
In this paper, we derive an effective low-energy model
for the supermoir´e length scale, which consists generi-
cally of a space-dependent anisotropric Dirac cone and
a non-abelian connection, valid for arbitrary twist an-
gles θ1and θ2provided that θ1'θ21. The ap-
proach also provides a general framework to derive a low-
energy theory problem featuring perturbed periodicities.
The low-energy spectrum is drastically modified by the
supermoir´e scale which introduces mini-bands close to
charge neutrality making the system susceptible to col-
lective instabilities at carriers densities much lower than
the moir´e ones [28].
We structure the paper as follows. In Sec. II we in-
troduce the continuum model for the staircase trilayer
graphene. This is followed by the definition in Sec. III of
the local Hamiltonian which depends parametrically on
the supermoir´e position. The derivation of the effective
model is discussed in Sec. IV. We discuss the k·ptheory
in Sec. IV A, the low-energy effective model in Sec. IV B
and the symmetries of the effective model in Sec. IV C.
Details on the properties of the anisotropic Dirac cone
and of the electronic spectrum are given in Appendix A
and C-E, respectively. We then present our results for
the low-energy electronic spectrum obtained in the case
of equal twist angles in Sec. V. The spectrum for the
case θ1/θ2= 1/2 is given in Sec. VI. Appendix Fcon-
tains a useful 1D toy model which details the origin of
the wavefunction singularity when the effective Dirac ve-
locity changes sign. Finally, we conclude the discussion
in Sec. VII by briefly summarizing the scope of the work
and setting the context for future studies.
II. STAIRCASE TRILAYER GRAPHENE
We model three overlaid monolayers of graphene
slightly twisted with respect to each other. We consider a
staircase configuration (all twist angles are positive when
measured from bottom to top) with an angle θ1between
the middle and top layers and θ2between the bottom
and middle. Twisted trilayer graphene shown in Fig. 1(a)
develops two moir´e structures modulated by θ1|K|and
θ2|K|with Kthe Dirac point of graphene. The mismatch
between these two modulations displayed in Fig. 1(b) cre-
ates an additional supermoir´e lattice with a weaker mod-
ulation. Without loss of generality, assuming θ1/θ2'p/q
with pand qcoprime integers, the two moir´e wave vectors
q1and q0
1in Fig. 1(c) can be expressed as:
q1=p¯
q1+δq1/(p+q),
q0
1=q¯
q1−δq1/(p+q)(1)
where ¯
q1is the wave vector of the moir´e pattern while
δq1is the supermoir´e modulation which takes into ac-
count deviations from perfect commensuration and the
lack of collinearity between q1and q0
1. As long as the
two twist angles are small the two wave vectors are well
separated |δq1||¯
q1|. Eq. (1) becomes particularly sim-
ple for equal twist angle θ1/θ2'1. In this case the
moir´e interference pattern develops with a space modu-
lation ¯
q1=¯
θuz∧Kwith uzaxis perpendicular to the
TBG plane, ¯
θ= (θ1+θ2)/2 and the supermoir´e wave
vector is
δq1=¯
θ2uz∧uz∧K+δθ uz∧K(2)
where δθ =θ2−θ1is the angle deviation. Themoir´e and
supermoir´e lattices are both triangular. Their principal
3
axis are rotated by 90◦with respect to each other in the
equal-angle case δθ = 0 and parallel in the opposite lim-
iting case of ¯
θ2δθ. It is worth stressing that, as the
vectors q1and q0
1can never be arranged to be collinear,
the supermoir´e pattern develops even when the twist an-
gles are exactly equal θ1=θ2.
The single-particle band spectrum is described within
a continuum model where the Dirac cones in each layer
are coupled by the transverse tunneling of electrons. In
the K valley, it reads
HK=
ˆ
k·σθαP3
j=1 Tje−iqj·r0
h.c. ˆ
k·σαP3
j=1 Tje−iq0
j·r
0h.c. ˆ
k·σ−θ
,
(3)
whereas the other valley K0is the time-reversal part-
ner HK0=H∗
K. We have introduced the differential op-
erator ˆ
k=−i(∂x, ∂y), σ= (σx, σy) and the matrices
(j= 0,1,2)
Tj+1 =rσ0+ cos(2πj/3) σx+ sin(2πj/3) σy,(4)
σθ=eiσzθ/2σe−iσzθ/2,α=wAB /Eθ, with Eθ=~v0kDθ
measuring the dimensionless coupling between the lay-
ers. ris the corrugation parameter [88]. The model is
characterized by a set of symmetries. It is invariant un-
der a 2π/3 rotations around the zaxis noted C3z, as
well as under πrotations around the xaxis C2zwhen
θ1'θ2. In addition, the symmetry C2zTcombining
time-reversal and πrotation around zis also satisfied.
In contrast, the particle-hole symmetry, which emerges
at the level of the Dirac cones of monolayer graphene,
is broken by the Hamiltonian in Eq. (3) even when the
θ-dependence of the matrices σθis neglected. We use
the approximation σθ'σin the rest of this paper, we
remark that this does not qualitatively change our result
and simplifies the formulation of our approach.
III. LOCAL HAMILTONIANS
Without loss of generality but for simplicity, in this
Section and in the next Sec. IV and Vwe focus on the
case of almost equal angles θ1'θ2to construct our
low-energy approach. The case of unequal angles will
be treated in Sec. VI.
We set the stage for a low-energy approach by iden-
tifying a slowly varying potential in the Hamiltonian.
We decompose the moir´e momenta qj=¯
qj+δqj/2,
q0
j=¯
qj−δqj/2 and rewrite the Hamiltonian as
HK=
ˆ
k·σαV1(r, φ(r)) 0
h.c. ˆ
k·σαV1(r,−φ(r))
0h.c. ˆ
k·σ
(5)
with the interlayer hopping potential V1(r, φ(r)) =
P3
j=1 Tje−iφj(r)e−i¯
qj·r. The slow variables are the
phases φj(r) = δqj·r/2 which vary over the super-
moir´e superlattice lengthscale. On shorter scales, we can
approximate them to be constant φj(r)'φj. We use
in fact the notation φj(R) to indicate that they are lo-
cally constant but still depend on the position Rin the
supermoir´e superlattice. With this approximation, the
Hamiltonian Eq. (5) becomes a set of local Hamiltonians
HM(φ) =
ˆ
k·σαV1(r,φ) 0
h.c. ˆ
k·σαV1(r,−φ)
0h.c. ˆ
k·σ
(6)
where the three phases φ= (φ1, φ2, φ3) depend on the
position Rand evolve over the supermoir´e superlattice.
In contrast with the exact original Hamiltonian Eq. (3),
the local Hamiltonian HM(φ), with fixed phases φ, is ex-
actly periodic on the moir´e real-space lattice. Similarly to
the twisted bilayer case, it is readily solved numerically in
Fourier space [3]. Each eigenstate wavefunction expands
over an hexagonal lattice in moir´e momentum space and
the convergence is exponentially fast with the number of
lattice points [89]. HM(φ) also possesses a gauge invari-
ance: its spectrum is invariant under a global shift of all
phases φj→φj+φ0with arbitrary φ0. In other words,
the first phase φ1can always be chosen to be zero. Physi-
cally, the phases (φ2, φ3) can be interpreted as indicators
of the local relative position of the middle layer with re-
spect to the top and bottom layers. It is always possible
to find a location, then defined as the origin r= 0, where
the top and bottom layers align along an AA stacking.
We denote two particular cases of interest: (i) AAA local
stacking for (φ2, φ3) = (0,0) and (ii) ABA local stacking
for (φ2, φ3) = ±(2π/3,−2π/3).
FIG. 2. Local spectra for (staircase-wise) twisted trilayer
graphene with θ1'θ2= 2.0◦calculated for different stack-
ing configurations. The phases φ2,3map the shifted position
of the middle layer before the twisting. (a) (φ2, φ3) = (0,0),
AAA stacking (b) (φ2, φ3) = (2π/3,−2π/3), ABA stacking
(c) (φ2, φ3) = (−2π/3,2π/3), ABA stacking (d) (φ2, φ3) =
(0.4π, −0.4π), generic stacking.
4
A symmetry analysis reveals that the local Hamilto-
nian HM(φ) generally breaks all symmetries (C3z,C2x
and C2zT) except at fine-tuned values of φcorresponding
to the local AAA and ABA stacking mentioned above.
In contrast with the exact Hamiltonian Eq. (3), the lo-
cal Hamiltonian HM(φ) exhibits a particle-hole symme-
try protecting a Dirac cone at ΓM, the Γpoint of the
moir´e Brillouin zone, as proven in Appendix E. In the two
cases of AAA and ABA stacking, the symmetries C3zand
C2xC2zTare preserved protecting one Dirac cone at KM
and another at K0
M. Aside from these symmetric configu-
rations, a general value of φweakly gaps the Dirac points
at KMand K0
M, the larger the angle ¯
θthe smaller the
gaps. In Fig. 2the band structures of the local Hamilto-
nians with different typical phase factors are shown. The
case where φ2=φ3= 0 (Fig. 2(a)) corresponds to the
model previously discussed in Ref. [77].
IV. EFFECTIVE MODEL
What we learned from the above analysis in Sec. III is
that low-energy states, i.e. states with energies close to
charge neutrality, are found in the vicinities of the ΓM,
the KMand K0
Mpoints of the moir´e Brillouin zone. We
have moreover obtained a set of local Hamiltonians and,
in an incoherent regime, the spectrum would be com-
posed by the additions of all the corresponding eigen-
values. We wish however to describe states that span
coherently the entire supermoir´e space. We need to con-
struct states interpolating continuously the solutions of
the local Hamiltonians of Sec. III. As we demonstrate
below, this can be achieved with a quasi-local approach,
similar in spirit to the conventional k·pmethod but ex-
tending it to a spatially varying perturbation. The issue
of gauge invariance is crucial in fixing the structure of
the resulting low-energy theory.
A. k·pmethod
Before presenting our theory, let us briefly review the
k·papproach relevant for the case where the phases
φare spatially homogeneous and we look for the band
spectrum of the local Hamiltonian HM(φ) in the vinicity
of ΓM. There are two zero-energy states at ΓM, solutions
of the coupled differential equations
HM(φ)uΓ,β(r) = 0 (7)
where β= 1,2 labels them and the twofold degeneracy
is protected by particle-hole symmetry, see Appendix E.
Assuming a small enough momentum |k|aM1, where
aMdenotes the moir´e lattice constant, we use a simple
ansatz for the low-energy eigenstates of HM(φ)
ψk(r) = [f1uΓ,1(r) + f2uΓ,2(r)] eik·r.(8)
Using that the plane wave eik·ris almost constant over a
moir´e unit cell, we project HM(φ)ψk(r) = εkψk(r) over
the zero-energy states uΓ,1/2(r) and integrate over the
unit cell to find
Heff (k)f1
f2=εkf1
f2(9)
with the low-energy effective Hamiltonian Heff (k) = Γ·k
and Γβ,β0=huΓ,β|σ|uΓ,β0i, or more precisely the inte-
grals over the moir´e unit cell
Γβ,β0=ZMUC
dru∗
Γ,β(r)σuΓ,β0(r).(10)
Interestingly, we have shown here that the vinicity of
ΓMis described by a Dirac cone, generally anisotropic,
whose properties are entirely determined by the matrix
elements between the two zero-energy solutions. In fact,
Heff (k) = Γ·k
can be seen as the (leading) first order term of a Taylor
expansion. The second order term vanishes to satisfy
particle-hole symmetry and the next non-vanishing term
is the third order trigonal wrapping contribution that we
will not further explore.
There is another important feature related to gauge
invariance. The choice of the two zero-energy states
uΓ,1/2(r) is somewhat arbritrary as any unitary trans-
formation U
u0
Γ,1(r)
u0
Γ,2(r)=UuΓ,1(r)
uΓ,2(r)(11)
provides another admissible set of orthogonal zero-energy
states but a different low-energy Hamiltonian H0
eff (k) =
Γ0·kwith Γ0=UΓU+. A simple gauge transform
nevertheless retrieves the original effective Hamiltonian
Heff (k) = U+H0
eff (k)Uand the spectrum is clearly in-
variant on the particular choice of zero-energy states as
expected.
B. Low-energy effective model
With this construction in mind, we turn to the case of
interest where the phases φvary over the supermoir´e pat-
tern and introduce again the zero-energy states at ΓM
given in Eq. (7), obtained at fixed values of the phases φ,
HM(φ)uΓ,φ,β(r) = 0.(12)
We have reinstated here the subscript φto stress the
fact that a different pair of zero-energy states is defined
for each φ. We then use the following ansatz for the
eigenstates of the original continuum Hamiltonian Eq. (5)
(not the local one)
ψ(r) = f1(r)uΓ,φ(r),1(r) + f2(r)uΓ,φ(r),2(r) (13)
where the functions f1,2(r) are slowly varying on the
moir´e scale. The functions uΓ,φ(r),1(r) vary rapidly on
摘要：

Supermoirelow-energye ectivetheoryoftwistedtrilayergrapheneYunchengMao,1DanieleGuerci,2andChristopheMora11UniversiteParisCite,CNRS,LaboratoireMateriauxetPhenomenesQuantiques,75013Paris,France2CenterforComputationalQuantumPhysics,FlatironInstitute,NewYork,NewYork10010,USAStackingthreemonolayers...

展开>> 收起<<
Supermoir e low-energy eective theory of twisted trilayer graphene Yuncheng Mao1Daniele Guerci2and Christophe Mora1 1Universit e Paris Cit e CNRS Laboratoire Mat eriaux et Ph enom enes Quantiques 75013 Paris France.pdf

共17页,预览4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！

相关推荐

更多
立即下载
分类：图书资源 价格：10玖币 属性：17 页 大小：3.34MB 格式：PDF 时间：2025-05-02

开通VIP享超值会员特权

  • 多端同步记录
  • 高速下载文档
  • 免费文档工具
  • 分享文档赚钱
  • 每日登录抽奖
  • 优质衍生服务

作者详情

相关内容

更多

热门标签

人际关系 动力学连接体 力的合成 配电装置 高考理综 全宋诗作者索引 公务员考试
/ 17
评分 收藏
分享
立即下载
客服
关注