Table Of ContentCoulomb drag in anisotropic systems: a theoretical study on a double-layer
phosphorene
S. Saberi-Pouya,1 T. Vazifehshenas,1,∗ M. Farmanbar,2 and T. Salavati-fard3
1Department of Physics, Shahid Beheshti University, G. C., Evin, Tehran 1983969411, Iran
2Faculty of Science and Technology and MESA+ Institute for Nanotechnology,
University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
3Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA
(Dated: May 27, 2016)
6 WetheoreticallystudytheCoulombdragresistivityinadouble-layerelectronsystemwithhighly
1 anisotropicparabolicbandstructureusingBoltzmanntransporttheory. Asanexample,weconsider
0 a double-layer phosphorene on which we apply our formalism. This approach, in principle, can be
2 tuned for other double-layered systems with paraboloidal band structures. Our calculations show
y the rotation of one layer with respect to another layer can be considered a way of controlling the
a dragresistivityinsuchsystems. Asaresultofrotation,theoff-diagonalelementsofdragresistivity
M tensor have non-zero values at any temperature. In addition, we show that the anisotropic drag
resistivityisverysensitivetothedirectionofmomentumtransferbetweentwolayersduetohighly
6 anisotropic inter-layer electron-electron interaction and also the plasmon modes. In particular, the
2 drag anisotropy ratio, ρyy/ρxx, can reach up to ∼ 3 by changing the temperature. Furthermore,
our calculations suggest that including the local field correction in dielectric function changes the
] results significantly. Finally, We examine the dependence of drag resistivity and its anisotropy
l
l ratioonvariousparameterslikeinter-layerseparation,electrondensity,short-rangeinteractionand
a insulating substrate/spacer.
h
-
s PACSnumbers: 73.20.Mf,72.10.-d,73.61.Cw
e
m
I. INTRODUCTION
.
t
a
m The advent of two dimensional (2D) materials has
- sparkled a considerable scientific interest due to their
d unique properties and their potential for applications
n
in electronic devices. Atomically thin 2D materials,
o
such as graphene [1], monolayer black phosphorous
c
[ (phosphorene)[2], hexagonal boron-nitride[3], and the
transition-metal dichalcogenides (TMDs)[4] represent a
3
particularly interesting class of 2D materials including
v
both semiconductors and metals. Phosphorene which
8
FIG. 1. (Color online)(a) Side view of a double-layer phos-
is an interesting monatomic layered crystalline material,
3
phorene system with the separation of d in the drag setup.
3 canbemechanicallyexfoliatedfromthebulkblackphos-
(b) Top view of phosphorene.
0 phorus due to the weak van der Waals interaction be-
0 tween layers[5].Unlike in group IV elemental materials
2. such as graphene, silicene[6] , or germanene [7, 8], phos- demand special attention due to many-body and trans-
0 phorene is a semiconductor with puckered orthorhom-
port properties [18–21]. The inter-layer Coulomb inter-
6 bic structure. This semiconductor has a nearly direct
actionplaysasignificantroleinthesecorrelatedsystems.
1 bandgapanditsbandstructureshowsalargeanisotropy
: InCoulombdragphenomenon,momentumcanbetrans-
v and high sensitivity to deformation [9]. This fact sug- ferredfrominteractingelectronsinonelayertoelectrons
i gests that crystallographic properties play an important
X in the adjacent layer[22–27]. The momentum transfer
role in the electronic behavior of this system. Recent
takes place through inter-layer Coulomb interaction, but
r studies reveal a high degree of anisotropic electronic
a doesnotinvolveanycarrierexchanges. Thisphenomenon
and optical properties for the phosphorene, which fur-
hasbeenpreviouslystudiedinafewnanostructuressuch
ther confirms the importance of this new 2D semicon-
as double quantum wells (DQW)[22, 28–37].
ductor as a promising candidate for electronic[10, 11],
Van der Waals bonding of 2D heterostructures makes
thermoelectric[12], and plasmonic applications. [13–17]
it possible to spatially separate two layers of graphene
Double-layered2Dstructuresconsistingoftwoparallel
(or any other 2D materials) down to several nanome-
electronorholesystemswhicharekeptinaclosevicinity
ters, by inserting a few atomic layers of a 2D insula-
tor, for instance h-BN, to isolate the layers from one an-
other. As shown by Gorbachev[38], et.al., a double-layer
∗ [email protected] graphene system has a strong Coulomb drag resistivity
2
with respect to GaAlAs heterostructures[38–41]. Like- [43]:
wise,Phosphoreneisa2Dmaterial,however,withhighly
(cid:88)
anisotropicenergydispersionincontrasttographeneand ραβJ =Ξ , (1)
21 1,α 2,β
other 2D materials. Therefore, it is particularly worth-
α=x,y
whiletoexaminetheeffectofanisotropyontheCoulomb
drag [42]. where J and Ξ are the current density and electric field,
In this paper we investigate the effect of band respectively. α and β indexes label x and y components
anisotropy on the Coulomb drag resistivity in a double- and ραβ =ρβα. The drag resistivity in the linear regime
21 12
layer electron system, consisting of two individual iso- has been studied in a variety of theoretical approaches
lated layers which are coupled via Coulomb interaction. assuming parabolic and non-parabolic band structures
We start from the expression for drag resistivity based (particularly graphene with a linear energy spectrum)
upon the semiclassical Boltzmann transport equation and considering both momentum-dependent and inde-
theory in the relaxation time approximation, and de- pendent intra-layer relaxation times[29, 30, 41, 44–46].
velop a general formalism, which includes the effect of Using different theoretical approaches such as the mem-
anisotropicenergydispersionandrotationallymisaligned ory function formalism[34], Kubo formula based on the
2D systems. As an example we apply this formalism leading-orderdiagrammaticperturbationtheory[47]and
to calculate Coulomb drag resistivity in a double-layer the linear response Boltzmann transport equation, the
phosphorene system(see figure 1). Numerical results drag resistivity matrix has been obtained with the as-
show a strong drag resistivity dependence on the band sumption of momentum-independent intra-layer trans-
anisotropy. It indicates that the drag resistivity along port time (see Appendix):
bigger mass, i.e., m , has a larger value. Furthermore,
we discuss how theydrag resistivity and its anisotropy ραβ = (cid:126) (cid:90) d2q q q
ratio depend on the carrier density, inter-layer separa- 21 2πe2n n k T (2π)2 α β
1 2 B
(2)
tion, rotation of layers and the choice of insulating sub- (cid:90) ∞ |U (q,ω)|2(cid:61)Π (q,ω)(cid:61)Π (q,ω)
strate/spacer. The drag resistivity is enhanced substan- × dω 21 2 1
sinh2((cid:126)ω/2k T)
tiallyinthey directionwhentheHubbardlocalfieldcor- 0 B
rection (LFC) is added to our formalism; LFC includes
Here, U (q,ω) is the temperature-dependent dynami-
21
the short-range exchange effect between electrons in the
callyscreenedinter-layerinteraction,Π (q,ω)andn be-
i i
same the layer.
ingthe2Dnon-interactingpolarizationfunctionandelec-
The rest of paper is organized as follows. Section
tronic density of ith layer, and k is the Boltzman con-
B
IIA, describes the model, theoretical formalism, and the
stant. InSec. IIC,wewillrewriteEq.(2)tomakeitmore
Coulomb drag resistivity in 2D anisotropic systems. In
convenient to use in a double-layer electron gas system
Sec. IIB, the dynamically screened inter-layer potential
with anisotropic band structure.
andalsotemperature-dependentanisotropicpolarization
function are presented. Section IIC contains results for
a double-layer phosphorene and the summary and con-
clusions are presented in Sec. III. B. Inter-layer potential and
temperature-dependent anisotropic polarization
function
II. COULOMB DRAG RESISTIVITY IN 2D
ANISOTROPIC SYSTEMS The dynamically screened inter-layer potential can be
obtained by solving the corresponding Dyson equation
[48]:
A. Model
V (q)
We consider a system composed of two parallel spa- Uij(q,ω)= det|(cid:15)ij(q,ω)|, (3)
tially separated 2D electron gases with anisotropic ij
parabolic-likebandstructures. Inthissystemthecarriers
where V (q) = ν(q)exp(−qd(1−δ )) is the unscreened
ij ij
arecoupledthroughCoulombinteractionandthereisno
2D Coulomb interaction with d being the layer spac-
tunnelingbetweenlayerssotheFermienergiesandchem- ing. ν(q) = 2πe2/qκ, with κ being the average dielec-
ical potentials can be considered independently. The
tric constant. Finally (cid:15) (q,ω) is the dynamic dielectric
ij
inter-layer Coulomb interaction can cause momentum
matrix of the system. For systems with high electron
transfer from the electrons in the drive layer, layer 1, to
density it is reasonable to employ the RPA to calculate
the carriers in the drag layer, layer 2. In doing so it gen-
(cid:15) (q,ω)[49, 50]:
ij
eratesapotentialdifferenceacrossthelayers. InFig.1(a)
weshowaschematicofthedragsetupwithphosphorene (cid:15) (q,ω)=δ +V (q)Π (q,ω) (4)
ij ij ij i
layers as anisotropic 2D electron gases. Also, a top view
of phosphorene monolayer can be seen in Fig.1(b). At low electron densities, the short-range local field ef-
The drag (inter-layer) resistivity, ρ, can be defined as fects are not negligible and must be included in the di-
3
electric matrix by replacing (1−G (q))V (q) for, V (q)
ij ij ij
where G (q) denotes the static intra- (i=j ) and inter-
ij
layer(i(cid:54)=j )elementsofLFCmatrix, respectively. Here
we incorporate only the intra-layer components of the
LFC factor because of their stronger effect on the drag
resistivity[51]:
q
G (q)= , (5)
ii (cid:112)
2 q2+k2
F
√
where G(q) and k = 2πn is the Hubbard LFC factor
F
andtheFermiwavevector,respectively,withnbeingthe
FIG.2. (Coloronline)Lossfunction,|(cid:61)(1/det(cid:15)(q,ω,T)|,for
electron density. For an electron gas system, the non- two crystallographic directions (a) θ =0 and (b) θ =π/2 at
interacting polarization function can be obtained from T=100 K with d=5 nm , n=3×1012cm−2 and η=1 meV.
the following equation[52]:
Π (q,ω)=−gs (cid:88) f0(Eqi)−f0(Eki+q) (6) Here,weconsiderq=q(cosθ,sinθ),inaccordancetothe
i ν Ei −Ei +(cid:126)ω+iη notationinRef. 54,tointroducerotationalparameterfor
k q k+q
the layers. Rotational angle, τ , is defined as the angle
i
Here f0(Eqi) is the Fermi distribution function in layer between x-axis in the laboratory frame an√d x direction
i at energy E corresponding to the wave vector q, of the ith layer. So, we can write Q = q mdRi/kF in
gs = 2 is spin degeneracy and η is the broadening which the orientation factor, Ri , is expressed as:
parameter, which accounts for disorder in the system.
The temperature-dependent dynamic polarization func- R =(cid:18)cos2(θ−τi) + sin2(θ−τi)(cid:19) (11)
tion for intra-band transition in an anisotropic 2D ma- i m m
x y
terial can be calculated by making use of the following
anisotropic parabolic energy dispersion relation In case of double-layer phosphorene, we have mx ≈
0.15m and m ≈ 0.7m where m is the free electron
0 y 0 0
(cid:126)2 k2 k2 mass[17]. Asitiswellknown,electroniccollectivemodes
Ei = ( x + y )−µ , (7)
k 2 m m i ofadouble-layersystemareobtainedfromzerosofthedi-
x y
electric function determinant, Eq.(4). In the presence of
in Eq.(6) for the polarization function: intra-band single particle excitations, there are two plas-
mon modes: the so-called acoustic and optical modes,
(cid:34) (cid:112)
Π (q,ω) (cid:90) Φ (K,T) 1 which show linear ωac(q) ∼ (R1R2/(R1+R2)dq and
i = − dK i sgn((cid:60)(Z )) (cid:112)
g2d Q − (cid:113)Z2 −K2 square-root ωop ∼ (R1+R2)q behavior at small wave
− vectors, respectively, and dependence on the orientation
(cid:35) factors [54].
1
−sgn((cid:60)(Z+))(cid:113) Tomaketheabovediscussionclearer,weshowtheloss
Z2 −K2 function of a system comprising two parallel monolayer
+
(8) phosphoreneseparatedbyd=5nmatT=100Kfortwo
In the above symmetric form of temperature-dependent main crystallographic directions: θ = 0 and θ = π/2, in
anisotropic polarization function, we define Q = Fig.2, respectively. One may notice that the acoustic
(cid:113) (cid:113) plasmon mode calculated here is weaker than the optical
m /Mˆ(q/k ), K = m /Mˆ(k/k ) where Mˆ is the
d F d F one. Thiscanbeexplainedbythefactthatthecoherence
mass tensor with diagonal element m and m along
√ x y of acoustic mode is quickly vanished due to the thermal
x and y direction, and m = m m is the 2D den-
d x y anddisorderbroadeningeffectsbecausetheω-qspectrum
sity of state mass. Moreover, g2d = md/π(cid:126)2 and of this mode is very close to the single particle excita-
Z± = (((cid:126)ω + iη)/(cid:126)QνF) ± (Q/2) with νF = (cid:126)kF/md tion region. Additionally, it can be recognized that the
and Φi(K,T) is given by: lower-energy acoustic plasmon and higher-energy opti-
calplasmonmodesfollowingdifferentasymptoticbehav-
K
Φ (K,T)= (9) ior at small wave vectors in both panels of Fig.2. Due
i 1+exp[(K2EFi −µi)/kBT] to the anisotropic band structure, the long-lived plas-
mon modes disperse differently in such a way that the
where µ is the chemical potential of layer i, which is de-
i larger effective mass along y leads to smaller resonance
termined by the particle number conservation condition
frequencies[13, 54, 55].
[53]:
As we mentioned earlier, the RPA is reliable for sys-
µ +k T ln[1+exp(−µ /k T)]=Ei (10) tems with√high electron densities. The density param-
i B i B F eter r = 2/(k a∗) with effective Bohr radius a∗ =
s F B B
4
C. Drag effect in a double-layer phosphorene
In this section, we first derive a formula for the drag
resistivity of a 2D anisotropic double-layer system with
parabolicbandstructureandthensolveitbymakinguse
of numerical methods. Eq.(2) is the general formula for
dragresistivitybasedonthelinearizedBoltzmanntrans-
portequation. Inthecaseoftwocoupledanisotropiclay-
ers,theoff-diagonalcomponentsofdragresistivitytensor
mayhavenon-zerovaluesasaresultoffiniteτ,unlikethe
isotropicsystemssuchasdouble-layergrapheneandcon-
ventional 2D electron gas. To make the difference more
FIG. 3. (Color online) (a) Fxx(q,ω,T) and (b) Fyy(q,ω,T) explicit, we rewrite Eq.(2) as follows:
fortwoalignedphosphorenemonolayerssandwichedbyAl O
2 3
layersatT=100Kwithd=5nm,n=3×1012cm−2 andη=1 (cid:126)2 (cid:90) (cid:90) ∞
meV . ραβ = dq dωFαβ(q,ω,T) (12)
D (2π)3e2n n k T
1 2 B 0
κ/(e2m ), which is defined as the average distance be-
d
tween electrons in a non-interacting 2D electron gas,
gives a measure for reliability of the RPA. In this fig-
ure, we consider the same electron density in layers,
n = n = 3 × 1012cm−2, the substrate and spacer
1 2
to be Al O with κ ≈ 12 that leads to an r ≈ 1.7
2 3 s
forwhichRPApredictselectronicscreeningqualitatively
well. However, Hubbard LFC can improve the result of
calculations by including exchange hole around interact-
ing electrons. Later on, we will employ LFC on top of
RPA to calculate the drag resistivity.
It is worth pointing out that considering a very thin
double-layerphosphorenesystemwhen,atthesametime,
FIG. 4. (Color online) The integrand of Eq.(13) at q = k
we assume there is no tunneling between the two layers, F
for two aligned monolayers sandwiched by Al O layers with
2 3
is not actually a problematic consideration because the n=3×1012cm−2, and η=1 meV, d=5 nm, along (a) x and
space between layers is filled by a slim dielectric mate-
(b) y directions at T=100 K and along y direction at (c) T=
rial. Al2O3 and h-BN have been successfully used as a 50Kand(d)T=10K.Theradialandazimuthalcoordinates
substrate and spacer to make such thin heterostructures are ω/ω and the angular orientation of q, respectively.
F
with no inter-layer tunneling[56–59]. Throughout this
paper, we assume the substrate is a thick layer of the
same material as spacer. where ραβ =ραβ and Fαβ(q,ω,T) is defined as:
D 21
(cid:90) 2π q3
Fαβ(q,ω,T)= dθψαβ(θ,τ ,τ ) |U (q,ω,T;θ,τ ,τ )|2(cid:61)Π (q,ω,T;θ,τ )(cid:61)Π (q,ω,T;θ,τ ) (13)
1 2 sinh2((cid:126)ω/2k T) 21 1 2 1 1 2 2
0 B
with ψαβ given by In order to understand how anisotropy affects the
drag resistivity, it is worth looking into the integrand
cos(θ−τ1)cos(θ−τ2), α=β =x of Eq.(12), Fαβ(q,ω,T) in more depth. In Fig.3, we
ψαβ(θ,τ ,τ )= sin(θ−τ )sin(θ−τ ), α=β =y show Fαβ(q,ω,T) for a coupled system composed of two
1 2 1 2
cos(θ−τ )sin(θ−τ ), α=x,β =y aligned phosphorene monolayers separated by 5nm at
1 2
T=100 K. We use the dimensionless variables q/k and
(14) F
ω/ω , where ω =(cid:126)−1E . Note that the integrand has
F F F
5
FIG. 6. (Color online) Scaled drag resistivity ρ T−2 as a
FIG. 5. (Color online) Anisotropic drag resistivity compo- D
function of temperature for different angels with n = 3×
nents, ραDβ calculated within RPA as a function of tempera- 1012cm−2, andη=1meV,d=5nm. Thephosphorenelayers
ture for two aligned phosphorene monolayers sandwiched by
are sandwiched by Al O layers.
Al O layers with n=3×1012cm−2 at η =1 meV and d=5 2 3
2 3
nm. The inset graph shows the anisotropy ratio ρyy/ρxx.
xdirection,ρxx,issmallerthanthedragresistivityalong
theydirection,ρyy,atanytemperaturesofinterestwith
significant weight in the 0 < q < kF interval, as is the adraganisotropyratio(seeinsetgraph), ρyy/ρxx, which
case in conventional 2D electron gas [47] but its values approximatelychangesfrom2uptoabout3. Webelieve
arelargeralongtheydirection. Thisisduetothegreater that a higher-energy resonance along x direction result-
effectivemassofelectronsiny direction,whichresultsin ing from the smaller effective mass, as discussed before,
lowerenergiesofthecollectivemodes,andinthismanner accounts for this behavior. Moreover, as it is expected
enhances the plasmons contributions [28, 47]. from general symmetry arguments, the off-diagonal ele-
The angular orientation of q impacts the drag resis- ments have zero values for aligned layers. In order to
tivity behavior considerably. We depict the integrand of understand how rotation of one layer with respect to the
Eq.(13) along x(α = β = x) Fig.4(a) and y(α = β = y) other about the normal direction to the layers (z direc-
Fig.4(b) directions in an aligned-layers system. At in- tion) impacts the behavior of drag resistivity, we present
termediate temperature T∼100 K, both modes (acous- calculations of the diagonal and off-diagonal elements of
tic and optical) take part and the single particle exci- the drag resistivity matrix for a couple of rotational an-
tation spectrum is sufficiently broadened to contribute gles in Fig. 6. Here, we set τ = 0 and τ = τ. It
1 2
effectively. As can be observed in the figure, the larger can be seen that as the angle of rotation increases, both
magnitude of the integrand occurs around θ = 0 and diagonal elements of drag resistivity decrease consider-
180◦ along the x direction and around θ =90◦ and 270◦ ably. This observation can be rationalized through the
for y direction, respectively. The integrand of Eq.(13) fact that by increasing the angle of rotation, one of the
along the y direction is plotted in Fig.4(c) and (d) for plasmonicbranchesisforcedintothedampedregime. As
two different temperatures: (c) T=50 K and (d) T=10 a result, the contribution of plasmons to the Coulomb
K. According to this figure, at T=10 K the drag resis- drag phenomenon, which is known to be significant, de-
tivity is mainly influenced by the acoustic mode which creases. When the angle of rotation is π/2, the diagonal
is lower in energy (ω < 0.5ω ) and the optical mode components of the drag resistivity tensor have zero val-
F
contribution starts to appears at 50 K. Here, the radial ues. In this configuration where the x-axis of one layer
andazimuthalcoordinatesdenoteω/ω andtheangular liesalongthey-axisoftheotherlayer,thevaluesofdiag-
F
orientation of q, respectively. The first set of calcula- onalelements,ραα ,becomeexactlyequaltothoseofthe
tionsofthedragresistivityinadouble-layerphosphorene off-diagonalelementsforasystemwithnorotation. Fur-
is presented in Fig.5. Here we show the diagonal and thermore,ourcalculationsshowthatwhenthelayersare
off-diagonal elements of the drag resistivity tensor calcu- rotatedwithrespecttooneanothertheanisotropiceffects
lated within the RPA, versus temperature for two paral- can create an interesting non-zero transversal drag resis-
lelalignedphosphorenemonolayerssandwichedbyAl O tivity, ρxy, which is absent in isotropic materials at zero
2 3
layers and separated by a distance of d =5 nm. While magnetic field. Thisobservation canbe fullyunderstood
the diagonal drag resistivity matrix elements increase in by Eq.(12-14) in which a misalignment of the laboratory
similar manner with temperature, there are significant and the layer axes gives rise to a non-zero value for the
differencesbetweenthevalues. Dragresistivityalongthe off-diagonal elements. This effect, however, may exist in
6
FIG. 7. (Color online) Scaled drag resistivity calculated within RPA (a) along y direction as a function of temperature and
the distance between two layers and (b) along x and y directions as a function of temperature with two different inter-layer
separations. Here,n=3×1012cm−2 andη=1meVandsystemcomprisingoftwoalignedphosphorenemonolayerssandwiched
by Al O .
2 3
FIG. 8. (Color online) Scaled drag resistivity, ρ T−2, as a function of temperature for two aligned phosphorene monolayers
D
sandwiched by Al O and calculated (a) within RPA at two different electron densities n = 3×1012cm−2 (solid line) and
2 3
n=1×1013cm−2(dashedline)(b)withinRPAandHubbardlocalfieldapproximationatelectrondensityn=3×1012cm−2and
(c)sandwichedbyAl O andh-BNcalculatedwithinHubbardlocalfieldapproximationatelectrondensityn=3×1012cm−2.
2 3
Here we set d= 5 nm and η= 1 meV.
a double-layer structures subjected to a perpendicularly and ρyy as a functions of temperature for two different
applied magnetic field [36, 38]. layer spacings: d = 2 nm and 5 nm in Fig.7(b), which
shows that the anisotropy ratio is less dependent on the
Anotherinterestinggeometricaleffectinadouble-layer
inter-layer distance.
phosphorene structure comes from changing the inter-
layer distance, which is presented in Fig.7. Fig.7(a) is a The effect of electron density on the drag resistivity is
3D plot showing the variation of ρyyT−2 as a function of alsoofinterest; hence, weillustrateitinFig.8(a). Asex-
layer spacing and temperature. It can be observed that pected for double-layers systems, for which the electron
the peaks occur at intermediate temperatures where the density increases, the Coulomb drag decreases and the
plasmoncontributiontothedragresistivityissignificant, resistivity peak moves toward higher temperature[28].
over the whole range of inter-layer distances. Also the Moreover, it is worth mentioning that the anisotropic
scaled drag resistivity decreases strongly when increas- effect is more pronounced at lower electron density.
ingtheseparationbetweentwolayersatalltemperatures. By including the Hubbard zero-temperature LFC, im-
One can attribute this behavior to the inter-layer inter- provements on the RPA results are studied in Fig.8(b).
action, which decays exponentially with the increasing Here, we employ the intra-layer local field factor, Eq.(5),
distance between layers, and decreases due to acoustic which is responsible for most of the drag resistivity en-
modesshiftingtowardhigherenergies. Havingsaidthat, hancement by the short-range interaction effects [60].
itisworthpointingoutthatthechanginginter-layerdis- Exchange interaction, which is taken into account by
tance does not significantly change the drag anisotropy the Hubbard LFC, impacts the inter-layer interaction
ratio. To trace this behavior, we plot both scaled ρxx throughthedielectricfunctionofthesystem(seeEq.(3)).
7
III. CONCLUSION
To summarize, we have derived a formula for the drag
resistivity in a structure composed of two spatially sepa-
rated2Delectrongassystemswithanisotropicparabolic
bandstructures. Wehaveassumedtheelectrongasesare
sandwiched by insulators so that there is no tunneling
between the layers. We have chosen double-layer phos-
phorene as an example system on which we apply our
anisotropic drag theory. Our numerical results confirm
thatthedragresistivitydependsnotonlyonthetypically
considered parameters such as temperature, inter-layer
separation, carrier density and nature of elementary ex-
citations,butalsoonthedirectionofmomentumtransfer
between the two layers in addition to the rotational pa-
rameter. Our calculations also show that while the diag-
onal elements of anisotropic drag resistivity tensor have
FIG. 9. (Color online) The anisotropy ratio of drag resis- different values due to different electron effective masses
tivity, ρyy/ρxx, as a function of temperature for two aligned along x and y directions at any temperatures of interest,
phosphorenemonolayerssandwichedbyAl2O3(solidline)and there are non-zero off-diagonal elements for the rotated
h-BN(dashedline)calculatedwithinHubbardlocalfieldap- structure. The non-zero off-diagonal elements have not
proximation at electron density n = 3×1012cm−2. Here we
been reported before in a 2D coupled system without
set d= 5 nm and η= 1 meV.
an applied magnetic field. According to the numerical
results, both diagonal elements of anisotropic drag resis-
tivity tensor increase with decreasing inter-layer separa-
tion and electron density. We show that, the anisotropic
Our calculations show that including the LFC factor en-
ratio varies effectively with the change of temperature
hancesthedragresistivityresultsnotablybystrengthen-
and electronic density. To improve on RPA results at
ing the inter-layer interaction [60]. Furthermore, for the
low electron density, we have included the zero tempera-
parameters used here one can see that the values of the
ture Hubbard LFC factor in our calculations and shown
anisotropy ratio are slightly different for both approxi-
that the inclusion of LFC enhances the drag resistivity
mations. We also investigate the effects of substrate and
values by almost a factor of 2. Moreover, we have stud-
spacer dielectric materials on anisotropic drag resistivity
iedtheeffectsofsubstrate/spacerontheanisotropicdrag
by considering two already experimentally used insula-
resistivity. We show that a substrate/spacers with high
tors, namely Al O [59] and h-BN [57], in phosphorene
2 3 dielectricconstantisabletoincreaseanisotropicdragre-
systems. Here,weassumen=3×1012cm−2 correspond-
sistivity considerably.
ing in h-BN case to density parameter r ∼ 5 which
s These results provide qualitative insight into the im-
makesitnecessarytoconsidertheLFCfactorinourcal-
pact that anisotropic band structure can have on drag
culation. The results indicate that including the LFC
resistivity, as an important transport quantity in a cou-
in drag resistivity calculations is important for the both
pled 2D structure. The present work also suggests that
studied substrates. Significant quantitative differences
the rotational parameter between layers can be consid-
betweentheresultsofRPAandlocalfieldfactorapprox-
eredasanextradegreeoffreedomfortheapplicationsof
imationsuggestastrongsensitivityofthedragresistivity
momentum transfer between coupled layers.
to the effective intra-layer electron-electron interactions.
Our results indicate that the anisotropic drag resistiv-
ity has higher values at all temperatures, when Al O
2 3 ACKNOWLEDGMENTS
is used as spacer/substrate compared with the case in
which h-BN is used (see Fig.8(c)), due to a larger dielec-
ItisapleasuretothankD.Otalvaroformakinguseful
tric constant of Al O . Employing the high- κ materials
2 3 comments on this manuscript.
as substrates/spacers enhances the inter-layer electron-
electron interaction due to the reduced screening effects
between two layers.
Appendix A: Drag Resistivity Tensor In A
Finally, we present the anisotropy ratio for two dif- Double-Layer System With Anisotropic Parabolic
ferent substrates in Fig. 9. We have employed Hubbard Band Structure
LFCtothedielectricfunctiontoaccountfortheexchange
short-rangeeffects. Calculationsshowthatdifferentsub- Here, we present a derivation of Eq.(2) for the drag
strates have slightly different effects on the anisotropy resistivity tensor in a rotationally misaligned double-
ratio and shift the maximum expected anisotropy ratio. layerelectrongassystemwithanisotropicparabolicband
8
structure by following the approach of Ref. [43], closely.
We suppose that the intra-layer transport time is inde-
∂f0
pendent of the wave vector. So, in a 2D system with en- e ( 1)(v )t.Ξ =−Hˆ [g ](k ) (A7)
ergy dispersion of the form of Eq.(7), the effective mass, 1 ∂E 1 1 1 1 1
Mˆ, transport time, τˆ, and mobility, µˆ , are symmetric
t t and
andmomentum-independent2×2tensorsandrelatedby:
∂f0
µˆt =eMˆ−1τˆt (A1) e2(∂E2)(v2)t.Ξ2 =−Hˆ2[g2](k2)+S[g1,g2 =0](k2)
(A8)
We assume one layer (layer 1) is fixed and take its lat- where the superscript t means the transpose and Ξ and
i
tice principal axes along the laboratory coordinate sys- Hˆ are the electric field and negative of the linearized
i
tembuttheotherlayer(layer2)isrotatedbyanangleτ. intra-layer collision operator in layer i, respectively. In a
Hence, in the laboratory frame, Mˆ, τˆ, and µˆ have zero 2Dsemiconductorwithanisotropicparabolicbandstruc-
t t
and non-zero off-diagonal elements in the fixed and ro- ture, the electron velocity is simply related to the wave
tated layers, respectively. In layer 2, these non-diagonal vector
matrices can be expressed in terms of the diagonal ones
by introducing rotation matrix, Rˆ(τ), e.g. for effective vi(ki)=(cid:126)Mˆi−1ki (A9)
mass tensor we have
From the above equations g and g can be obtained as:
1 2
Mˆ2 =Rˆ(−τ)Mˆ1Rˆ(τ) (A2) g1(k1)=−e1Hˆ1−1(cid:2)(∂∂fE10)(v1)t(cid:3)(k1).Ξ1 (A10)
Since we are dealing with the symmetric effective mass and
andtransporttimetensors,eachoneisequaltoitstrans-
pose. Moreover, due to the diagonal representation of g (k )=−e Hˆ−1(cid:2)(∂f20)(v )t(cid:3)(k ).Ξ +Hˆ−1[S](k )
these tensors in the laboratory frame, we have: 2 2 2 2 ∂E 2 2 2 2 2
(A11)
Mˆ−1τˆ =τˆ Mˆ−1 (A3) Since the current in layer 2 is equal to zero
i t,i t,i i
In the Boltzmann transport equation framework, we (cid:90) dk (cid:18) ∂f0 (cid:19)
define a deviation function g(k) as J =−2e k T 2 ( 2)v g (k )=0 (A12)
2 2 B (2π)2 ∂E 2 2 2
one finds the following relation:
(cid:18)∂f0(k)(cid:19)
δf ≡f(k)−f0(k)=−kBT ∂Ek g(k) (A4) 2e2k T (cid:90) dk2 (cid:18)(∂f20)v (cid:19)Hˆ−1(cid:2)(∂f20)(v )t(cid:3)(k ).Ξ
2 B (2π)2 ∂E 2 2 ∂E 2 2 2
where f(k) is the non-equilibrium Fermi distribution (cid:90) dk (cid:18) ∂f0 (cid:19)
function and f0(k) = f0(E ). The linearized inter-layer =−2e k T 2 ( 2)v Hˆ−1[S](k )
k 2 B (2π)2 ∂E 2 2 2
collision integral is given by:
(A13)
The left hand side of the above equation is equals to
(cid:90) dk (cid:90) dq
S[g1,g2](k2)=2 (2π1)2 (2π)2w(q,Ek1+q−Ek1) En2qe.(2Aµˆt5,2).Ξ2. By employing the following identities to
×f0(k )f0(k )(cid:2)1−f0(k +q)(cid:3)
1 1 2 2 1 1
×(cid:2)1−f0(k −q)(cid:3)(cid:2)g (k )+g (k )
2 2 1 1 2 2 δ(E +E −E −E )
−g (k +q)−g (k −q)(cid:3) k1 k2 k1+q k2−q
1 1 2 2 (cid:90) ∞
×δ(Ek1 +Ek2 −Ek1+q−Ek2−q) =(cid:126) 0 dωδ(Ek1 −Ek1+q−(cid:126)ω)δ(Ek2 −Ek2−q+(cid:126)ω)
(A5)
with
(A14)
and
w(q,ω)=4π(cid:126)−1|U (q,ω)|2 (A6)
21
f0(E )[1−f0(E )]=[f0(E )−f0(E )]n (E −E )
1 2 2 1 B 1 2
(A15)
Considering weak inter-layer interaction, the coupled
Boltzmann equations are given by: Eq.(A13) can be rewritten as:
9
Hene, Eq.(A16) can be rewritten as:
4e k T (cid:90) dq (cid:90) ∞
n2e2µˆt,2.Ξ2 =− 2πB (2π)2 0 dω|U21(q,ω)|2 n e µˆ .Ξ =−4e2kBT (cid:90) dq (cid:90) ∞dω|U (q,ω)|2
×nB((cid:126)ω)nB(−(cid:126)ω) 2 2 t,2 2 π (2π)2 0 21
(cid:20)(cid:90) dk ×nB((cid:126)ω)nB(−(cid:126)ω)
× (2π2)2[f20(k2)−f20(k2+q)] ×(cid:20) 1 (cid:90) dk2 [f0(k )−f0(k +q)]
×δ(Ek2 −Ek2+q−(cid:126)ω)Hˆ2−1(cid:2)(∂∂fE20)v2(cid:3)(k2)(cid:21) ×δ(2EkkB2T−Ek(22+πq)2−(cid:126)2ω)τ2ˆt,2[v22(k22+q)
×(cid:20)(cid:90) (d2kπ1)2[f10(k1)−f10(k1+q)] −v2(k2)](cid:21)(cid:20)keB1T (cid:90) (d2kπ1)2[f10(k1)
(cid:21) −f0(k +q)]δ(E −E −(cid:126)ω)
×δ(E −E −(cid:126)ω)[g (k )−g (k +q)] 1 1 k1 k1+q
k1 k1+q 1 1 1 1 (cid:18) (cid:19)t (cid:21)
(A16) × τˆt,1[v1(k1+q)−v1(k1)] .Ξ1
(A18)
where n ((cid:126)ω) is the Bose distribution function and in
B The DC electric fields in two layers are related by:
the second line we use the Hermitian property of H−1.
2
Using relaxation time approximation, one can define:
Ξ =ρˆ J =n e ρˆ µˆ .Ξ (A19)
2 21 1 1 1 21 t,1 1
(cid:0) (cid:1)t
g (k )=−e Hˆ−1(cid:2)(∂fi0)(v )t(cid:3)(k ).Ξ ≡ ei τˆt,ivi(ki) .Ξi Inserting Eqs.(A9) and (A19) into Eq.(A18) and consid-
i i i i ∂E i i i k T ering the properties of effective mass and transport time
B
(A17) tensors, one gets:
4e k T (cid:90) dq (cid:90) ∞
n e n e µˆ ρˆ µˆ .Ξ =− 2 B dω|U (q,ω)|2n ((cid:126)ω)n (−(cid:126)ω)
1 1 2 2 t,2 21 t,1 1 π (2π)2 21 B B
0
(cid:20)τˆ Mˆ−1Rˆ(−τ)q(cid:90) dk (cid:21)
× t,2 2 2 [f0(k )−f0(k +q)]δ(E −E −(cid:126)ω) (A20)
2k T (2π)2 2 2 2 2 k2 k2+q
B
(cid:20)qte Mˆ−1τˆ .Ξ (cid:90) dk (cid:21)
× 1 1 t,1 1 1 [f0(k )−f0(k +q)]δ(E −E −(cid:126)ω)
k T (2π)2 1 1 1 1 k1 k1+q
B
By multiplying both sides of above equation by µˆ−1 −sinh−2((cid:126)ω/2k T) , Eq.(A20) can be written as:
t,2 B
andusingmobilityrelation,Eq.(A1),commutativeprop-
erty, Eq.(A17) and the equality 4nB((cid:126)ω)nB(−(cid:126)ω) = ραβ = (cid:126)2 (cid:90) dq q q (cid:90) ∞dω |U21(q,ω)|2
21 2πn1e1n2e2kBT (2π)2 α β 0 sinh2((cid:126)ω/2kBT)
(cid:20)(cid:90) dk (cid:21)
× 1 [f0(k )−f0(k +q)]δ(E −E −(cid:126)ω)
(2π)2 1 1 1 1 k1 k1+q
(cid:20)(cid:90) dk (cid:21)
× 2 [f0(k )−f0(k +q)]δ(E −E −(cid:126)ω)
(2π)2 2 2 2 2 k2 k2+q
(A21)
whereq andq aretheα andβ componentsoftrans-
α β
ferredwavevectorcorrespondingtothelayer1andlayer
2 in the laboratory frame, respectively. Finally by using
thepolarizabilionfunctiondefinition,Eq.(6),oneobtains
the Eq.(2).
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