Long-range Coulomb interactions in surface systems: a first principles description within self-consistently combined GW and dynamical mean field theory

0.51 MB·English

by  Philipp Hansmann

Long-range Coulomb interactions in surface systems: a first principles description
within self-consistently combined GW and dynamical mean field theory
P. Hansmann,1 T. Ayral,1,2 L. Vaugier,1 P. Werner,3 and S. Biermann1,4
1Centre de Physique Th´eorique, Ecole Polytechnique, CNRS-UMR7644, 91128 Palaiseau, France
2Institut de Physique Th´eorique (IPhT), CEA, CNRS, URA 2306, 91191 Gif-sur-Yvette, France
3Department of Physics, University of Fribourg, 1700 Fribourg, Switzerland
4Japan Science and Technology Agency, CREST, Kawaguchi 332-0012, Japan
(Dated: January 21, 2013)
Systemsof adatoms on semiconductor surfaces display competing ground statesand exotic spec-
3
tral properties typical of two-dimensional correlated electron materials which are dominated by a
1
complex interplay of spin and charge degrees of freedom. We report a fully ab initio derivation of
0
low energy Hamiltonians for the adatom systems Si(111):X, with X=Sn, Si, C, Pb, that we solve
2
withinself-consistent combinedGWanddynamicalmeanfieldtheory(“GW+DMFT”).Calculated

n photoemissionspectraareinagreementwithavailableexperimentaldata. Werationalizeexperimen-
a tally observed tendencies from Mott physics towards charge-ordering along the series as resulting
J from substantial long-range interactions.

8
1 PACSnumbers: 71.15.Mb,73.20.At,71.10.Fd,71.30.h

]
l
e Understanding the electronic properties of materials tems in [34]). While the first surprise are the relatively
- with strong electronic Coulombcorrelationsremains one largevaluesoftheonsiteinteractionswhichwefindtobe
r
t of the biggest challenges of modern condensed matter oftheorderofthebandwidth( 1eV),mostimportantly
s ≈
. physics. The interplay of delocalization and interactions we show that non-local interactions are large (nearest-
t
a is notonly at the originof exotic groundstates, but also neighborinteractionof 0.5eV)and,hence,anessential
m ≈
determines the excitation spectra of correlated materi- part of the resulting many-body Hamiltonians. This re-
- als. The “standard model” of correlated fermions, the sultconfirmspreviousspeculationsabouttheimportance
d
Hubbard model, in principle captures these phenomena. of non-local effects in these materials[21, 29]. We solve
n
o Yet, relating the model to the material on a microscopic these Hamiltonians within fully self-consistent combined
c footing remainsa challenge. Evenmore importantly,the GW and dynamical mean field theory (“GW+DMFT”)
[ approximation of purely local Coulomb interactions can [35], calculating in particular (single particle-) angular

1 becomesevereinrealisticmaterials,wherelong-rangein- resolved photoemission spectra (ARPES) and the (two
v teractions and charge fluctuation physics cannot be ne- particle-) charge susceptibility. We identify a clear-
5 glected. cut materials trend starting from Si(111):C deep in a
2
Mott phase to Si(111):Pb which shows tendencies to-
Systems of adatoms on semiconducting surfaces, such
3
wards metallicity and charge-ordered states driven by
4 as Si(111):X with X=Sn, C, Si, Pb, have been suggested
. [1] to be good candidates for observing low-dimensional non-local interaction terms. Comparing our results to
1
available experimental data yields encouraging insights:
0 correlated physics. Commonly considered to be realiza-
Without adjustable parameters we reproduce the experi-
3 tions of the one-band Hubbard model and toy systems
1 for investigating many-body physics on the triangular mentally measured gap size of insulating Si(111):Sn and
: its transition to a “bad-insulator” at elevated tempera-
v lattice, such surfaces have been explored experimentally
tures. Moreover, based on the charge susceptibility, we
i [2–18] and theoretically [19–31].These so-calledα-phases
X identify the electronic tendency of Si(111):Pb towards
show a remarkable variety of interesting physics includ-
r charge-ordering of the so-called 3 3 symmetry, which
a ing commensurate charge density wave (CDW) states ×
is indeed seen experimentally by scanning tunneling mi-
[5, 6, 9] and isostructural metal to insulator transitions
croscopy. Our work is the first one that addresses the
(MIT)[14]. However, while specific systems and/or phe-
electronicpropertiesofrealmaterialsonthebasisoffully
nomenahavebeeninvestigatedalsotheoretically,acom-
self-consistent GW+DMFT calculations (for a non-self-
prehensive understanding including materials trends is
consistentcalculationsee [36], for self-consistentcalcula-
still lacking. A central goal of our work is to present a
tions for models see [37–39]). [48]
unified picture that relates, within a single framework,
different materials (adatom systems), placing them in a
The single-particle part of the Hamiltonian is calcu-
common phase diagram.
lated in the local density approximationof density func-
We derive low-energy effective Hamiltonians ab initio tional theory. In Fig. 1 we present LDA bandstructures
fromacombineddensityfunctionalandconstrainedran- for the series Si(111): C,Si,Sn,Pb . For all considered
{ }
domphaseapproximation(cRPA)scheme[32]intheim- systemsthesurface-stateinthesemiconductinggapisin-
plementationof[33](seealsotheextensiontosurfacesys- deed responsible for a well-separated, single band around
2
1.0
V] TABLE I. Values of the bare (V) and static, screened (U0 =
e U(iν =0)) values for on- and intersite nearest neighbor (nn)
[ C Si Sn Pb interaction parameters. Also reported are the values of the
0.0 static component of the effective (ω = 0) calculated from
U
GW+DMFT, see text.
C Si Sn Pb
-1.0
t 38.0 50.0 42.0 42.0 [meV]
Γ M K Γ M K Γ M K Γ M K Γ t′ 15.0 23.0 20.0 20.0 [meV]
−′′
t 0.5 5.0 10.0 10.0 [meV]
FIG. 1. (Color online) Bandstructures of the α- √3 √3
phases for Si(111):X with X=Sn, Si, C, Pb [52]. The×color U0 1.4 1.1 1.0 0.9 [eV]
of the bands denotes their respective orbital character. Red U1 0.5 0.5 0.5 0.5 [eV]
color indicates a pz-like “apical” character, while the blue Un U1/ra
colordenotespx,y-like(i.e. planar)character. Theblackdots V0 6.0 4.7 4.4 4.3 [eV]
represent the tight binding fit given by Eq. (1) and hopping
V1 2.8 2.8 2.7 2.8 [eV]
parameters from Tab. I
. V1/εsStiasut.rf. 0.47 0.47 0.45 0.47 [eV]
(ω=0) 1.3 0.94 0.84 0.67(ins.) [eV]
U
the Fermi energy. In red (gray) we plot the contribu- 0.54(met.) [eV]
tions stemming from the p -orbital of the adatom while
z
we plot the adatom p -character in blue (dark gray). of1eV-abouttwicethesizeofthebandwidth-strongly
x,y
Even though the actual molecular orbital composition points towards Mott physics. This is, however,a prema-
might be complicated, the half-filled surface band has a ture conclusion due to the effect of non-local interaction
clear-cut “apical” (i.e. carrot-like) character. For our terms. Thefirstnon-localcontribution(nearest-neighbor
calculations presented below we directly use the ab ini- interaction) U1 [bare V1] is 0.5 eV [2.8 eV]. Remarkably,
tio deriveddispersionrelation. However,for the purpose thevalueis-opposedtoU0 [V0]-almostthe sameforall
of analysis we note that the tight-binding dispersion of materials. The reasonis that the intersite overlapof the
the half-filled surfaceband canbe wellfitted using up to orbitalsissosmallthattheCoulombenergycorresponds
third-nearest-neighbor hopping (t, t′, and t′′) by: totheelectrostaticenergyoftwopointcharges. Withthe
virial theorem Etot. = 1/2 V , we quantify this argu-
h i h i
εk =2t·(cid:16)cos(kx)+2cos(kx/2)cos(√3/2ky)(cid:17) mraedniutsbyofa6r˚Aes(caldeidsthayndcreoogfenadpartoobmlemsitwesit)h: effectiveBohr
≈
+2t′·(cid:16)cos(√3ky)+2cos(3kx/2)cos(√3/2ky)(cid:17) e2 = 1 VHatom = 1 2EHatom =2.3eV,(2)
+2t′′ cos(2k )+2cos(k )cos(√3k ) (1) DrrelE 12| pot | 12 | groundstate|
(cid:16) x x y (cid:17)
·
which roughly matches the value of our bare intersite
The values for the hopping integrals can be found in interaction parameters. The second, likewise remark-
Tab. I and we plot the analytically calculated bands in able, observation is that the screened values U are ex-
1
Fig. 1 as the black dashed line. The quality of the fit tremely close to the value we get by assuming a static
supports the picture of Wannier-like orbitals with a fast continuum approximation on the surface of a dielectric
decaying real space overlapon neighboring sites. medium: V /εsurf, where εsurf = 1(ε +1) is the static
1 Si Si 2 Si
In order to determine the interaction parameters as dielectric constant of silicon on the surface. The reason
partially screened matrix elements of the Coulomb in- is straightforward: The adatom distance (6˚A) is already
teraction within the cRPA one has to choose a suitable largeenoughcomparedtotheatomicstructureofthesil-
energy-window around the Fermi energy encompassing iconsubstrate( 2˚A) sothat local field effects (included
≈
the surface band. The bare interaction parameters are in cRPA)are negligible. Followingthis reasoningwe can
calculated by means of explicit evaluation of the ra- calculate longer range interaction terms by simply scal-
dial (Slater-) integrals of the Wannier functions. Sub- ing U with a/r, i.e., with the distance in units of the
1
sequently, the dielectric tensor is obtained within cRPA nearest-neighbordistance a, i.e., U =U /√3 andso on.
2 1
for local and non-local interaction parameters[34]. The In this respect U is not only the nearest-neighborinter-
1
results are summarized in Tab.I. action,but the parameterthat quantifies the strengthof
The bare onsite interaction parameters (V ) vary be- non-local interaction.
0
tween 6.0 eV for Si(111):C and 4.3 eV for Si(111):Pb To solve the effective low-energy Hamiltonians result-
decreasing monotonously within the series. The onsite ing from our parameter-free downfolding procedure we
U is reduced roughly by a factor of 4 5 due to cRPA implement the combined GW+DMFT scheme [35, 42]
0
−
screening. Atfirstglancetheonsite/staticU oftheorder and calculate spectral properties and charge-charge re-
0
3
1.5 (ω) 0) Pb 0.8
C = met. Sn
ω Si 0.6
1.0 k, C
(
χ
Si m 0.4
I
ins.
0.5 Sn
Pb ins. 0.2
met.
ω [eV]
0.0 0.0
0.0 0.4 0.8 1.2 Γ M K Γ
FIG. 3. (Color online) Left hand side: Frequency-dependent
(ω) (calculated from GW+DMFT) including both, insulat-
U
ing and metallic cases for the Pb system. Right hand side:
Imaginary part of the charge-charge susceptibility along the
usual path in theBrillouin zone.
0.8eV(Si),0.7eV(Sn),and0.5eV(Pb). However,specifi-
callyfortheSi(111): Sn,Pb wefindsubstantialspectral
{ }
weightalready at 0.2eV.Given this small gap, a siz-
≥−
FIG. 2. (Color online) Momentum-resolved spectral function able temperature dependence can be expected. We have
at T =116K of Si(111):X with X=Sn,Si, C, Pb obtained by extracted the value of the local (i.e., k-integrated) spec-
analytical continuationofGW+DMFTimaginary-time data. tralfunctionattheFermi-level[50](seeFig.2bottomleft
TheFermienergyissettoεF=0andindicatedbythewhite panel). While for Si(111):C the spectral weight transfer
dashedline. Onthebottomrightweshowthespectralweight
tothe Fermienergywithtemperatureisnegligibleasex-
at theFermi energy as a function of temperature.
pected from the spectral function, specifically Si(111):Si
and most of all Si(111):Sn display significant transfer of
spectralweightonatemperaturescalefrom50Ktoroom
sponse functions. Fully self-consistent GW+DMFT was temperature 300K.
applied to the extended Hubbard model in seminal work
Photoemission experiments for Si(111):Sn [10, 18]
by Sun et al. [37, 49], but only recently have numer-
(and, possibly[51], for Si(111):Pb[11]) observe, indeed,
ical techniques for the solution of dynamical impurity
such a temperature dependence and agree well with our
models[44–46]beensufficientlyadvancedtoextractreal-
results, both, concerning the gap size and temperature
frequency information from such calculations [38, 39].
scale. Our results - obtained without any adjustable pa-
We employ the techniques of the latter two works (in
rameters -alsostandasatheoreticalpredictionformore
particular a continuous-time quantum Monte Carlo im-
extensive studies on Si(111):Pb and the (experimentally
purity solver within the hybridization expansion [44]),
so far not studied) Si(111):C compound. Next, we an-
but implement them for the realistic Hamiltonian de-
alyze the spectral functions in view of the interaction
rived above. Moreover, we go beyond the ’standard’ ex-
strengthscalculatedby cRPA(see Tab.I). The gapsizes
tended Hubbard model and do not restrict the range of
no longer reflect the energy-scale of the onsite interac-
the non-local interaction terms. Rather, we include the
tion U but are reduced due to non-local interactions
0
entire 1/r tail by means of an Ewald-type lattice sum.
which screen the local interaction by non-local charge
InFig.2weshowmomentum-resolvedspectralfunctions
fluctuations. This physics is naturally present in the
fromGW+DMFTforallcompoundsinourseries: Asex-
GW+DMFT scheme, where non-local effects are incor-
pectedfromthelargeonsiteinteractionscomparedtothe
poratedintoaneffectiveretardedonsiteinteraction (ω)
bandwidth we obtaininsulating spectra for allfour com- U
(plotted in the left panel of Fig. 3). The shape of this
pounds. Interestingly,however,for the Pbcompound,in
quantity is reminiscent of screenedinteractions as calcu-
contrast to the other three systems, we find two stable
lated, e.g., within the cRPA[32], where retardation ef-
solutions at the temperature of our study (T = 116K) -
fects result from downfolding of high-energy degrees of
one metallic and one insulating. This indicates that we
freedom. The GW+DMFT (ω) can be viewed as an
areinacoexistenceregionofafirstorderphasetransition U
effective interaction, where the dynamical character re-
similar to that seen in the extended Hubbard model[39].
sults from downfoldingnon-localdegreesoffreedominto
In allcompounds the upper andlowerHubbard bands alocalquantity. Atlargefrequencies,screeningisnotef-
showsubstantialdispersionfollowingthebarebandstruc- ficient and, hence, (ω = ) =U . On the other hand,
0
U ∞
ture,asexpectedongeneralgrounds. Theinsulatinggap the static value (ω = 0) can be significantly reduced
U
decreaseswithin the seriesandwecanestimate fromthe (up to nearly a factor of 2 for Si(111):Pb). The latter
centerofmassoftheHubbardbandsvaluesof: 1.3eV(C), sets the energyscalefor the gaps weobserveinthe spec-
4
0.6
tral function. The transition between unscreened high- ]
V CO
frequencybehaviorandthestaticvaluetakesplaceatan e Pb Si* C
energy scale ω0 (plasmonic frequency) characteristic of n [ 3 x 3 Sn
o
thenon-localchargefluctuations. Thestrikinglydifferent ti 0.4
c
behavior of the dynamical effective interactions (ω) re- a √3 x √3
U r
flectstheobservedmaterialstrend: Si(111):C[Si(111):Si] e
t
is [nearly] unaffected by non-local interaction terms and n 0.2 Metal Mott
there is barely any screening. The remaining two com- al i
c
pounds show, however, large effects. The static values o
(ω = 0) are reduced compared to the onsite interac- n-l 0.0
U o 0.4 0.6 0.8 1.0 1.2 1.4
tion to 0.84eV for Si(111):Sn and to 0.67eV (0.54eV) for n
the insulating (metallic) solution for Si(111):Pb which local interaction [eV]
leads to the reduced gap sizes. Moreover,plasmonic res-
onances at energies between 0.6eV and 0.8eV stress the FIG. 4. (Color online) Schematic local/nonlocal- interaction
phase-diagram: Theblack-borderedcirclesmarkthepositions
importance of non-local interactions/charge-fluctuations
of the adatom systems of our study. Straight lines are guide
for these systems.
to the eyes and blurry color indicates coexistence regions.
Besidesleadingtoaretarded,frequency-dependentin-
Within the localized (i.e. insulating) phases CO and Mott,
teraction, the non-local charge fluctuations signal ten- small sketchesindicate the shapeof thesurface unit-cell.
dencies towards a charge-ordered(CO) state. Analyzing
the momentum-dependence of the imaginary part of the
charge-charge response function Imχ(k,ω = 0) for the
high symmetry points of the Brillouin zone, shown in
Fig. 3, we find for the different materials very particu-
coexistence region)are much closer to a phase boundary
lar behavior. The local double occupancy, which corre-
to a metallic phase. Even more peculiar is the obvious
sponds to the integral of the plotted quantity over all
tendency of Si(111):Pb towards a charge-ordered phase
momenta, becomes larger towards the end of the series.
of 3 3 symmetry indicated by the white region in our
Most interesting is the case of metallic Si(111):Pb for ×
phase diagram.
which we find a distinct structure within the Brillouin
zone: The maximum of Imχ(k,ω = 0) at the K sym-
metry point indicates strong charge fluctuations of the In conclusion, we have set up a fully self-consistent
so-called 3 3 symmetry, sketched in Fig, 4. This order GW+DMFT scheme for the realistic treatment of corre-
might even×tually be frozen in to form a charge ordered latedsurfacesystemstoaddresstheelectronicproperties
ground state which is actually seen in scanning tunnel- of the α-phases of adatoms on the Si(111) surface. We
ing microscopyfor this material[8]. An insulating charge reported on the ab initio construction of the materials-
ordered ground state of 3 3 symmetry is, in fact also specific low-energyHamiltonians and, mostimportantly,
seen in Ge(111):Sn[47] whe×re a concomittant structural on the respective interaction parameters including the
distortion(verticaldisplacementofadatoms)ofthesame long-range Coulomb tail. From these it becomes clear
symmetry is seen - our results show, that the instability that for the adatom systems taking into account non-
in the correlated electronic response function is a good local interaction effects is mandatory. We have solved
candidate for the key player of this feature. the derived many body Hamiltonians and discussed our
finding for momentum-resolved spectral functions, to be
We cansummarizeour results by drawinga schematic
compared to ARPES spectra. Without adjustable pa-
phase-diagramas a function of the strength of local and
rameters we reproduced available experimental findings,
non-local interactions (represented by the value of U )
1
or,in (most) cases where experiments are missing, made
as we show in Fig. 4. For zero non-local interactions
predictions. Specifically, the ARPES spectra for the se-
ourphasediagramdescribesthe Mott-Hubbardmetalto
ries, as well as the charge order instabilities in the case
insulator transition. The adatom systems are placed at
of Si(111):Pb are key conclusions/predictions which can
about 0.5eV of non-local interaction strength. However,
duetothedifferenceintheonsitetermU theirrespective provideguidanceforfurtherexperimentalandtheoretical
0
studies of semiconductor adatom structures.
position in the phase diagram and, hence, their ground
statecharacterisdifferent: Si(111):CisdeepintheMott
phase with a charge localization defined by one electron We acknowledge useful discussions with the authors
per adatom-site. of Ref. 17 and Ref. 18, as well as with M. Capone,
The Si(111):Sicompound[52]is alsoofMotttype with M. Casula, H. Hafermann, H. Jiang, M. Katsnelson,
only small values for the double occupancy and little O. Parcollet, and E. Tosatti. This work was sup-
effect of plasmon excitations. However, Si(111):Sn and ported by the French ANR under project SURMOTT
mostdramaticSi(111):Pb(whichisactuallyalreadyina and IDRIS/GENCI under project 129313.
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×