Table Of ContentVan der Waals Interactions Between Thin Metallic Wires and Layers
N. D. Drummond and R. J. Needs
TCM Group, Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom
Quantum Monte Carlo (QMC) methods have been used to obtain accurate binding-energy data
for pairs of parallel, thin, metallic wires and layers modelled by 1D and 2D homogeneous electron
gases. We compare our QMC binding energies with results obtained within the random phase
approximation, finding significant quantitative differences and disagreement over the asymptotic
behavior for bilayers at low densities. We have calculated pair-correlation functions for metallic
biwire and bilayer systems. Our QMC data could be used to investigate van der Waals energy
8
functionals.
0
0
PACSnumbers: 71.10.-w,73.21.Hb,73.20.-r,02.70.Ss
2
n
a One-dimensionalconductorssuchascarbonnanotubes (In a 2D HEG r is the radius of the circle that contains
s
J are essential components of many proposed nanotechno- one electron on average.) At very large separations the
2 logicaldevices,andarecurrentlythesubjectofnumerous vdW attraction is dominated by the Casimir effect,5,6 in
experimentalandtheoreticalstudies. Ithasrecentlybeen whichthezero-pointenergyofphotonmodesbetweenthe
i] demonstrated1thatthevanderWaals(vdW)interaction metallic layers gives rise to an attractive force. However
c
between pairs of distant, parallel, thin conducting wires we restrict our attention to the range of separations in
s
- assumedinmanycurrentmodelsofmetallic carbonnan- which vdW effects are dominant.
l
r otubes is qualitatively wrong. In this letter we provide WithintheRPA,thebindingenergymaybecalculated
mt the first accurate binding-energy data for pairs of thin asthe changeinthe zero-pointenergyofplasmonmodes
metallic wires and layers,which can be used as a bench- as a function of separation.1 However, the RPA is poor
.
t markforsubsequenttheoreticalstudiesortoparametrize in low-dimensional systems, and ceases to be valid at
a
model interactions between 1D and 2D conductors. lowdensities,wherecorrelationeffectsbecomedominant.
m
Thin, electrically neutral wires are attracted to one We have therefore performed quantum Monte Carlo7,8
-
d another by vdW forces. The standard method for cal- (QMC) calculations of the binding energies of pairs of
n culating the vdW interaction between objects is to as- thin, metallic wires and layers modelled by 1D and 2D
o sumetherearepairwiseinteractionsbetweenvolumeele- HEGs with neutralizing backgrounds. In particular we
c mentswithanattractivetailoftheformUPP(r) r−6, have used the variational and diffusion quantum Monte
[ ∝−
which is appropriate for the vdW interaction between Carlo (VMC and DMC) methods as implemented in the
1 molecules. Summing these interactions for a pair of 1D casino code.9 DMC is the most accurate method avail-
v parallel wires separated by a distance z gives a vdW able for studying quantum many-body systems such as
3 binding energy of U(z) z−5. Such pairwise vdW electron gases. We have also calculated pair-correlation
7 models have beenused in∝stu−dies ofsingle-walledcarbon functions(PCFs),enablingustoexaminethecorrelation
3 nanotubes.2,3 However, a recent investigation1 of the in- hole responsible for the vdW attractionbetween pairs of
0
teraction between pairs of thin, metallic wires modelled wires and layers.
.
1 by 1D homogeneous electron gases (HEGs) within the In our QMC calculations we use the full Coulomb
0
randomphaseapproximation(RPA)foundthatthebind- potential, so that for 1D HEGs the many-electron
8
ing energy falls off (approximately) as4 wave function must go to zero at both parallel- and
0
: antiparallel-spin coalescence points for electrons in the
v U(z) √rs , (1) same wire. The nodal surfaces for paramagnetic and
Xi ≈−16πz2[log(2.39z/b)]3/2 ferromagnetic 1D HEGs are therefore the same, and so
the fixed-nodeDMC energy—whichisequaltothe exact
r
a wherebisthewireradiusand2rsisthelengthofthewire ground-stateenergybecausethe nodalsurfaceisexact—
sectioncontainingoneelectrononaverage. The pairwise
is independent of the spin polarization. This conclusion
vdW model is clearly appropriate for an insulator or for does not violate the Lieb-Mattis theorem10 because the
ametallicwirewhoseradiusisgreaterthanthescreening
1D Coulomb interaction is pathological in the formal
length, but is inappropriate for a thin conductor such as sense of Lieb and Mattis.11 For convenience we choose
a single-walled carbon nanotube.1
to work with ferromagnetic 1D HEGs, because a Slater
Likewise, the binding energy per particle of a pair of
determinant wave function produces the correct nodal
thin, parallel metallic layers can be shown to decay as
surface in this case. Our QMC studies of 1D HEGs will
be published elsewhere.12
0.012562√πr
s
U(z)= − (2) Fermionic symmetry is imposed in 2D via the fixed-
2z5/2
nodeapproximation,13 inwhichthenodalsurfaceiscon-
within the RPA,1,5 compared with U(z) z−4 within strained to equal that of a trial wave function. We ex-
∝ −
thepairwisevdWtheory,wherez isthelayerseparation. pect a very high degree of cancellation of fixed-node er-
2
rorswhenthebindingenergyiscalculated. Ithasalready Method Energy (a.u. / el.) Var. (a.u.) %age corr. en.
beenshownthatfixed-nodeDMCisabletodescribevdW
HF −0.215943040112 — 0%
forces between helium14 and neon atoms.15
SJ-VMC −0.2319668(4) 0.0000360(2) 99.974(3)%
Wehaveverifiedthattime-stepandpopulation-control
SJB-VMC −0.2319710(3) 0.0000046(1) 100.000(3)%
biases in our DMC energies are negligible by repeating
SJB-DMC −0.2319709(3) — 100%
some of the calculations using different time steps and
populations. Finite-size errors are a more serious prob-
lem, although most of the bias cancels out when the
TABLE I: Energy, energy variance, and fraction of corre-
energy difference is taken to obtain the binding energy.
lation energy retrieved using different levels of theory and
Twistaveraging16ortheadditionoffinite-sizecorrections
wave function for a 15-electron 1D ferromagnetic HEG at
areunlikelytoreducethebiasinthe bindingenergy. We rs=15 a.u. “HF” stands for Hartree-Fock theory, “SJ” de-
expect our binding-energy results to be valid so long as notesaSlater-Jastrowtrialwavefunctionand“SJB”aSlater-
the wire or layer separation is small compared with the Jastrow-backflow trial wave function.
length of the 1D or 2D simulation cell; for larger sepa-
rations the systemresembles a pair of insulatorsand the
ulation cell is L = Nr , where N is the number of
binding energy is expected to fall off more steeply (in rs,N s
electrons per wire, and in 2D the size of the simulation
accordance with the pairwise vdW model).
cell is L √Nr . The binding energy falls off more
We have used Slater-Jastrow-backflow trial wave rs,N ≈ s
functions.8,17 For our 2D bilayer calculations the Slater steeply once z becomes a significantfractionof Lrs,N, as
expected. Clearly the binding energies enter the asymp-
part of the wave function consists of a product of four
totic regime when z r . We have therefore fitted the
determinantsofplane-waveorbitalsforspin-upandspin- ≫ s
RPAasymptoticbinding-energyformsto ourQMC data
down electrons in each of the two layers. For our 1D bi-
in the range r z L . We believe the errors in
wire calculations the Slater part consists of a product of s ≪ ≪ rs,N
the fitted exponents are about 0.1–0.2.
twodeterminantsofplane-waveorbitalsfortheelectrons
ThefitstotheDMCbiwirebinding-energydatashown
ineachwire(recallthateachwireisferromagneticinour
in Fig. 1 are
calculations). Slater determinants for a 1D HEG are of
Vandermonde form, and could therefore be rewritten as U (z) = 0.0815z−2.28[log(27000z)]−3/2 (3)
polynomials and evaluated in a time that scales linearly 1 −
with system size; however other parts of the QMC al- U (z) = 0.0225z−1.98[log(1.95z)]−3/2 (4)
3
−
gorithm such as the evaluation of the two-body Jastrow U (z) = 0.0967z−2.17[log(0.492z)]−3/2, (5)
10
terms and backflow functions take up a significant frac- −
tion of the computer time, so for convenience we have where U (z) is the binding energy at density parameter
rs
continued to employ the usual determinant-evaluating r . The DMC binding-energy data are clearly in much
s
and updating machinery of QMC calculations.18 better agreement with the RPA [Eq. (1)] than with the
Our Jastrow factors consist of polynomial and pairwise vdW theory [U(z) z−5]. It is not meaningful
plane-wave two-body terms19 satisfying the Kato cusp to comparethe prefactorsbe∝cause of the arbitrarinessof
conditions.20 Inspiteofthefactthatthenodalsurfaceis our choice of the wire radius b in the RPA theory.
exactin1D,two-bodybackflowcorrelations17werefound ThefitstotheDMCbilayerbinding-energydatashown
tomakeaverysignificantimprovementtothewavefunc- in Fig. 2 are
tion, as can be seen in Table I. Backflow functions were
used in all of our 1D calculations and in our 2D calcu- U1(z) = 0.00637z−2.58 (6)
−
lations at rs = 1 a.u. Free parameters in the trial wave U3(z) = 0.0388z−2.64 (7)
functionwereoptimizedbyminimizingtheunreweighted U (z) = −0.882z−3.16, (8)
variance of the local energy.21,22 10 −
Each wire or layer is accompanied by a neutralizing where U (z) is the binding energy at density parame-
rs
background. When studying biwires (bilayers) we need ter r . At high densities (r = 1 and 3 a.u.) our results
s s
toaddthe interactionbetweenthe electronsineachwire clearly show the z−2.5 behavior predicted by Eq. (2).
−
(layer) and the background of the opposite wire (layer), At low density (r = 10 a.u.) the binding energy falls
s
plustheinteractionofthe twobackgrounds. Thiscontri- off more steeply than predicted by the RPA, although
bution to the energy is Ecap = log(z)/(2rs) for a biwire the asymptotic behavior is clearly better described by
and Ecap =z/rs2 for a bilayer. the RPA than the pairwise vdW theory [U(z) z−4].
∝ −
The binding energies of pairs of 1D and 2D HEGs are DMCandtheRPAgivesimilarprefactorsfortheasymp-
shown in Figs. 1 and 2. The approximate RPA binding totic binding energyatr =3a.u.,but the DMC prefac-
s
energy shown in Fig. 1 was obtained using b = r /10 tor is somewhat lower at r =1 a.u.
s s
in Eq. (1). The wire radius b is therefore small com- PCFs were accumulated by binning the interparticle
pared with the other length scales in the system. We distances in the electron configurationsgeneratedby the
have included DMC results at several different system VMC and DMC algorithms. The error in the VMC and
sizes in Figs. 1 and 2. In 1D, half the length of the sim- DMC PCFs g andg is first orderin the errorin
VMC DMC
3
) ) -1
c. DMC (55 elec. / wire) c.10
e e DMC (114 elec. / layer)
u. / el10-3 DFRiPMt AtCo ( D(D1Mo0b1Cs eo dlnea ctea.t /a wl.;i rbe )= 0.1 a.u.) u. / el FRiPt Ato (DSeMrnCe ldiuasta & Bjork)
a. a.10-2
( (
y y
g -4 g
er10 er
n n
g e g e10-3
n n
di -5 di
n10 n
bi bi
of of 10-4
g. g.
Ma10-6 Ma
1 10 1 10
Wire separation (a.u.) Layer separation (a.u.)
) -2 )
c.10 DMC (35 elec. / wire) c.
e e
el DMC (55 elec. / wire) el -2
/ DMC (101 elec. / wire) / 10
u. 10-3 Fit to DMC data u.
a. RPA (Dobson et al.; b = 0.3 a.u.) a.
( (
y y 10-3
g -4 g
r10 r
e e
n n
e e
g g 10-4
in10-5 in DMC (18 elec. / layer)
d d
n n DMC (74 elec. / layer)
bi bi -5 DMC (114 elec. / layer)
f -6 f 10 Fit to DMC data
g. o10 g. o RPA (Sernelius & Bjork)
Ma Ma10-6
1 10 100 1 10
Wire separation (a.u.) Layer separation (a.u.)
) -3 ) -2
c.10 c.10
e DMC (55 elec. / wire) e DMC (74 elec. / layer)
l l
e DMC (75 elec. / wire) e DMC (114 elec. / layer)
u. / 10-4 FRiPt Ato (DDMobCso dna teat al.; b = 1 a.u.) u. / 10-3 FRiPt Ato (DSeMrnCe ldiuasta & Bjork)
a. a.
( (
y y
g10-5 g10-4
r r
e e
n n
e e
g -6 g -5
n10 n10
i i
d d
n n
i i
b -7 b -6
f 10 f 10
o o
g. g.
Ma10-8 Ma10-7
10 100 5 25
Wire separation (a.u.) Layer separation (a.u.)
FIG. 1: (Color online) Binding energy per particle of a 1D FIG. 2: (Color online) Binding energy per particle of a 2D
HEG biwire as a function of wire separation for rs = 1 a.u. HEG bilayer as a function of layer separation for rs =1 a.u.
(top panel), rs = 3 a.u. (middle panel), and rs = 10 a.u. (top panel), rs = 3 a.u. (middle panel), and rs = 10 a.u.
(bottom panel). The DMC time steps were 0.04, 0.2, and (bottom panel). The DMC time steps were 0.007, 0.05, and
2.5a.u.atrs=1,3,and10a.u.,andthetargetconfiguration 0.5a.u.atrs =1,3,and10a.u.,andthetargetconfiguration
population was 2048 in each case. populations were 1024, 320, and 1024.
4
the trialwavefunction, butthe errorinthe extrapolated
PCF g = 2g g is second order in the error
ext DMC VMC
on 1 in the wave function−.8 PCFs for biwires and bilayers are
i
ct shown in Figs. 3 and 4, respectively. The correlation
un0.8 1.02 holesbetweentheelectronsinoppositewiresorplanesare
f 1.00
n F exceedinglyshallow,althoughtheyextendoveradistance
o C0.98
ti0.6 P roughly equal to the separation. It can be seen that the
a 0.96
el interwire PCF has the long-ranged oscillatory behavior
r 0.94
or0.4 0 5 10 15 exhibited by 1D HEG PCFs.
c Electron-electron separation (a.u.)
-
r
i
Pa0.2 Same wire
Opp. wire
0
0 50 100 150
Electron-electron separation (a.u.)
In conclusion we have used QMC to obtain the first
FIG. 3: (Color online) VMC PCF for a 1D HEG biwire at accurate binding-energy data for pairs of thin, parallel,
rs = 3 a.u. and wire separation z = 3 a.u. The inset shows metallicwiresandlayers. Ourresultsareinbroadagree-
theinterwirecorrelation holeingreaterdetail. Itwasverified mentwithrecentRPAcalculationsofthebindingenergy
for smaller system sizes that theDMC and VMCPCFs were
and complete disagreement with the standard pairwise
in excellent agreement (as expected due to the accuracy of
vdW model. However there are significant differences
thetrial wave function illustrated in Table I).
between the DMC and RPA results for bilayers: at high
densities the asymptotic behavior of the binding energy
asafunctionofseparationisthesamebuttheprefactoris
1 different, and at low densities the DMC binding energy
n falls off more rapidly, implying that correlation effects
o
cti0.8 1.02 neglected in the RPA are important in this regime. Our
n datacanserveasabenchmarkforfuturetheoreticalstud-
on fu0.6 PCF1.00 iaelssoofbtehuesbedindtoinpgaernaemrgeitersizoefm1Dodaenldin2teDraHctEioGnss,baentdwceaenn
i
at 0.98 thinconductors. Togetherwithourresultsforbiwireand
l
re0.4 0 2 4 6 8 bilayer PCFs, our data could be used to investigate en-
or Electron-electron separation (a.u.)
c ergy functionals that incorporate vdW effects for use in
air-0.2 SSaammee llaayyeerr,, soapmp.e s sppinin density-functional calculations.
P
Opp. layer
0
0 2 4 6 8
Electron-electron separation (a.u.)
FIG. 4: (Color online) Extrapolated PCF for a 2D HEG bi-
layerat rs=1 a.u.and layerseparation z=2a.u. Theinset We acknowledge financial support from Jesus College,
shows the interwire correlation hole in greater detail. The Cambridge and the UK Engineering and Physical Sci-
VMC and DMC PCFs are in excellent agreement, implying ences Research Council. Computing resources were pro-
that the extrapolated PCF is reliable. vided by the Cambridge High Performance Computing
Service.
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5
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