Signatures of Wigner Localization in Epitaxially Grown Nanowires

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by  L. H. Kristinsdóttir

Signatures of Wigner Localization in Epitaxially Grown Nanowires
L.H. Kristinsdóttir,1 J.C. Cremon,1 H.A. Nilsson,2 H.Q. Xu,2
L. Samuelson,2 H. Linke,2 A. Wacker,1 and S.M. Reimann1,∗
(Nanometer Structure Consortium, nmC@LU)
1Division of Mathematical Physics, Lund University, Box 118, 22100 Lund, Sweden
2Division of Solid State Physics, Lund University, Box 118, 22100 Lund, Sweden
(Dated: December 1, 2010)
It was predicted by Wigner in 1934 that the electron gas will undergo a transition to a crystal-
lized state when its density is very low. Whereas significant progress has been made towards the
1 detection of electronic Wigner states, their clear and direct experimental verification still remains
1 a challenge. Here we address signatures of Wigner molecule formation in the transport properties
0 ofInSbnanowirequantumdotsystems, whereafewelectronsmayformlocalizedstatesdepending
2 onthesizeofthedot(i.e. theelectrondensity). Byaconfigurationinteractionapproachcombined
with an appropriate transport formalism, we are able to predict the transport properties of these
n
systems, in excellent agreement with experimental data. We identify specific signatures of Wigner
a
J state formation, such as the strong suppression of the antiferromagnetic coupling, and are able to
detecttheonsetofWignerlocalization,bothexperimentallyandtheoretically,bystudyingdifferent
1
dot sizes.
3

PACSnumbers: 73.21.Hb,73.22.Gk,73.22.Lp,73.23.Hk,73.63.Nm
]
l
l
a
h The transition to a Wigner crystal1 can be viewed as
- a contest between the electronic Coulomb repulsion and
s
the quantum mechanical kinetic energy. If the Coulomb
e
m repulsiondominates,themany-particlegroundstateand
its excitations resemble a distribution of classical par-
.
t ticles located in a lattice minimizing the Coulomb en-
a
m ergy. In the bulk, the transition to a Wigner crystal
is only expected for extremely dilute systems2,3, while
-
d in lower dimensions, or for broken translational invari-
n ance, it becomes accessible at higher densities4–6. A lot
o ofworkhasfocusedonfinite-sizedtwo-dimensionalquan-
c tumdots7–11,wherethecrossoverfromliquidtolocalized
[
states in the transport properties of the nanostructure

2 has been addressed12,13. For one-dimensional systems,
v localizationhasbeenreportedincleavededgeovergrowth
7 structures14 and for holes in carbon nanotubes15. These
4
highly correlated one-dimensional systems exhibit a va-
1
riety of fascinating features as reviewed recently16. Here
5
0. wseemiinctornodduuccteorantahnirodwsiryesst,ewmh,ibchasaeldloownseapsittraaxiigahlltyfogrrwoawrnd Figure 1: (a) SEM-image of the InSb nanowire on a SiO2
1 capped Si substrate, where the quantum dot is defined by
application of tunneling spectroscopy compared to the
0 Schottky barriers of the gold contacts (‘source’ and ‘drain’).
1 rather involved cleaved edge overgrowth structures and Calculated electron density in nanowires of lengths 70 nm,
: avoids further complications due to the isospin degree of 160 nm, and 300 nm is displayed in panels (b,c,d), respec-
v
freedom in carbon nanotubes. tively, for the lowest two-electron states (excitation energies
i
X InSb nanowires17, as used here, allow for the realiza- are given; ‘S’ stands for singlet and ‘T’ for triplet). For the
r tion of quantum dots, where the electronic confinement two-particlegroundstatethepair-correlateddensityisshown
a alongthenanowireisestablishedbySchottkybarriersto with the position of one electron marked by a black arrow.
gold contact stripes, see Fig. 1(a). Varying the distance
between the stripes (here: 70 nm and 160 nm) allows for
the systematic realization of wires with specific length evaluated with the configuration interaction method.
and thereby controlled electron densities. For our cal- Theresultscanbeunderstoodintermsoftwolimiting
culations we model the nanowire as a hard-wall cylinder cases: a short wire with no electron localization and a
with the experimental radius 35 nm. The Schottky bar- long wire with Wigner localization25.
rier at the semiconductor-metal interface creates a stan- The first limiting case, where interaction is dominated
dard quantum well with a width equal to the contact by kinetic energy, can be described by the independent-
spacing. The Coulomb interaction between the electrons particleshellmodel. Therethetwo-particlegroundstate
is approximated as that in a cylinder embedded in ho- is obtained by populating the lowest single-particle level
mogeneous matter, taking into account the different di- withaspin-upandaspin-downelectron. Thusthespatial
electric constants of the wire and the surrounding ma- electron density follows that of the lowest single-particle
terial18,19. Exact many-particle states in the wire are level and exhibits a peak in the center of the quantum
2
dot. The lowest excited two-particle state is obtained
bymovingoneelectrontothefirstexcitedsingle-particle
levelatthecostofthelevelspacingenergy∆ε. Thusone
expects the two-particle excitation energy ∆E ≈ ∆ε.
2
Furthermore the spin degrees allow for four realizations
of such an excited two-particle state, which are typically
split into a triplet and a singlet due to exchange interac-
tion.
In the second limiting case, Wigner localization, the
electrons are localized at different positions along the
wire, minimizing the Coulomb repulsion. Thus the two-
particle ground state density exhibits two peaks and a
minimum in the center of the nanowire segment. As the
electrons can have arbitrary spin on each site, one has
four realizations of this configuration, with a minor en-
ergy split between a singlet and a triplet. Hence, we
expectaverysmall∆E (cid:28)∆ε,whilefurtherexcitations
2
are significantly higher in energy and exhibit a different
spatial distribution of charge.
At the onset of localization, the electron density is ex-
pected to resemble two weakly separated peaks in the
two-particle ground state. The interaction of the elec-
trons is substantial, without yet dominating the kinetic
part. Hence the two-particle excitation energy is consid-
erably lower than the single-particle excitation energy,
∆E < ∆ε. However, as the two electrons are not yet
2
fully crystallized, ∆E is expected to be well above zero.
2
Tunneling spectroscopy is a convenient way to study
groundandexcitedstatesinquantumdotsystems. Here
we can use the gold contacts (Fig. 1(a)) as source and
drain by applying a bias Vsd between both stripes. The Figure 2: Results for an InSb nanowire of length L=70 nm.
nanowire is located on a highly doped Si substrate cov- (a) Simulated differential conductance as a function of bias
ered by an insulating SiO layer, which allows for appli- (V ) and gate energy E . The number of particles in the
2 sd g
cation of a back-gate voltage V providing an approxi- dot, N, is shown in each diamond. (b) A closer look at the
bg
mately homogeneous shift in energy of all levels in the area marked by a dashed box in panel (a). The conduction
dot. Varying V and V provides the characteristic lines, where tunneling into the N = 1 ground state and first
chargingdiagramssd(seee.gb.1g1)displayedinFigs.2(c)and excited state sets in, are marked by the symbols (cid:192) and (cid:193),
respectively. The corresponding lines for the entering of the
3(c) at a temperature of 300 mK. Here high differential
second electron, where the dot reaches the N = 2 ground
conductance indicates that the electron addition energy state and the N = 2 excited state, are marked by (cid:194) and
(affinity) coincides with the chemical potential in either (cid:195) symbols, respectively. The separation between these lines
of the gates. The diamonds of vanishing conductance provides the excitation energies from the N = 1 and N = 2
centered around zero Vsd are the regions of Coulomb ground states, ∆ε and ∆E2, respectively, which are depicted
blockade,wherethechemicalpotentialsofbothreservoirs by arrows. (c) Experimental differential conductance as a
areabovetheenergydifferencebetweenthe(N−1)-and functionofbias(V )andgatevoltage(V ). (d)Experimen-
sd bg
N-electron ground state and below the energy difference taldifferentialconductanceasafunctionofmagneticfield(B)
betweentheN-and(N+1)-electrongroundstate. Asno and gate voltage (Vbg).
furtherlinesofhighconductancearefoundforlowergate
bias, we assume that the lowest diamond corresponds to
N =1. Halfthewidthofthisdiamonddefinesthecharg- ical calculations allow for a verification of the Wigner
ing energy U. localization scenario described above.
Based on the calculated many-particle states, elec- Fora70nmwire,the2-electrondensityalongthewire
tron transport is treated within the master equation is a single peak, see Fig. 1(b). This corresponds to the
model20–22 with tunneling matrix elements calculated as independent-particle shell model as described above. In
in Ref.12. The results are displayed in Figs. 2(a) and Fig. 2(b) we have marked the lines, where the first elec-
3(a) for the respective experimental samples displayed tron enters the one-electron ground state and the one-
in panel (c). We find that all Coulomb diamonds agree electron excited state, by the symbols (cid:192) and (cid:193), respec-
rather well, which indicates that the radial excitations, tively. This reflects the level spacing ∆ε = 12 meV as
which are disregarded in our effectively one-dimensional shown by the horizontal arrow. Similarly, starting from
model,onlybecomeofrelevanceforhigherparticlenum- the one-electron ground state, the second electron enters
bers in the dot. the dot reaching the two-electron ground state and the
Nowwefocusontheexcitedstatesandshow, thatthe two-electron excited state at lines marked by the (cid:194) and
experimental conductance data along with our theoret- (cid:195) symbols. The separation between these two lines rep-
3
particleshellmodelbeingvalidwhenkineticenergydom-
inates interaction.
Forthe160nmwire,the2-electrondensityinFig.1(c)
resemblestwosemi-separatedpeaks,indicatingtheonset
ofWignerlocalization(asalsoseeninthepair-correlated
density). In Fig. 3, the lines (cid:192)-(cid:195) can be identified both
in the simulation and the experiment. The theoretical
resultsgive∆E =1.0meVand∆ε=2.8meV,asinthe
2
experiments we observe ∆Eexp = 1.0 meV < ∆εexp =
2
3.2meV. Again, this is in agreement with the scenario of
onset of Wigner localization discussed above.
Note that if we would neglect the different dielectric
consant outside the wire, the onset of Wigner localiza-
tion would first appear at double the actual wire length.
Hence the screening due to the different dielectric con-
stants of the wire and the surrounding material is an
important effect and must be included in the modelling.
The energy separation between the singlet and the
triplettwo-electronstate,theantiferromagneticcoupling,
can also be manifested by the magnetic field dependence
of the differential conductance. The S = 1 part of the
z
triplet is lowered in energy by a magnetic field with re-
specttothesingletstatebygµ B,whereµ istheBohr
B B
magneton. Fig. 3(d) shows that there is a level cross-
ing at B ≈ 0.4 T (marked by an arrow). According
cross
to Ref.17 the electronic g-factors are around 40 for two
electrons in the dot. This provides an energy splitting
∆Emag =gµ B ≈1 meV in full agreement with the
2 B cross
calculated value for the 160 nm wire. Note that for the
70 nm wire, the level splitting is no longer linear in the
high magnetic field, B ≈ 4 T, at which the cross-
cross
ing appears (marked by an arrow in Fig. 2(d)). Hence
we cannot apply the same method to find ∆Emag for
2
the 70 nm wire, although its result ∆Emag ≈ 10 meV is
2
of the correct order of magnitude. The strong suppres-
Figure3: ResultsforanInSbnanowireoflengthL=160nm, sion of this antiferromagnetic coupling between the two
panels as in Fig. 2 electrons (by an order of magnitude, while changing the
length by about a factor of two) is one of the hallmarks
of the Wigner crystal state16.
resents the excitation energy ∆E = 11 meV. The four Finally, our theoretical results indicate complete
2
lines, (cid:192)-(cid:195), can be observed in the experimental data in Wigner localization for a 300 nm long wire. Fig. 1(d)
Fig. 2(c) (this is clearer for negative bias, as the mea- showsthatinthetwo-particlegroundstate,theelectrons
surement results in the positive bias region most likely are strongly localized, i.e. they form a Wigner molecule.
suffer from charging of impurity states). From this fig- From Fig. 4(b) we observe that the conductance line of
ure, we read ∆Eexp = 15 meV ≈ ∆εexp = 16 meV, and the N = 2 triplet first excited state ((cid:194)) has merged
2
hence for the sample of length 70 nm, the experimental intothelineofthesingletgroundstate((cid:195)),asexpected:
dataareingoodagreementwiththeindependent-particle There is no difference in the energy of these two states,
shell model discussed above. as there should be no difference between the singlet and
Note that there is some discrepancy between theory triplet states of two strongly localized particles. More
andexperimentregardingthevalueof∆εand∆E . This precisely we find ∆E = 9.3 µeV and ∆ε = 0.84 meV,
2 2
could be due to bending of energy levels at the interface i.e. ∆E (cid:28) ∆ε. Furthermore we find U = 5.7 meV,
2
of the wire and the gold contacts (Schottky barriers), that is ∆ε (cid:28) U. This conforms to Wigner localization
whichmakesthewireeffectivelyshorterthanthespacing being present when kinetic energy is strongly dominated
of the contacts. Indeed, simulations of a 60 nm wire give by interaction.
∆ε=16 meV and ∆E =15 meV. Even for the N = 3 ground state the theoretical cal-
2
We can quantify the electron-electron interaction culations suggest the onset of Wigner localization in a
strength by the energy difference between the two- 300 nm wire, as seen in Fig. 4(c). The small energy dif-
particle ground state and twice the energy of the lowest ference between the three lowest N = 3 states results
single-particle level (half-width of the N = 1 Coulomb in a broad conduction line, marked by the symbol (cid:196)
diamond). This provides the charging energy Uexp = in Fig. 4(b). Unfortunately, we could not obtain exper-
6.5 meV for the 70 nm sample, as read from Fig. 2(c). imental data for this length, since for such a long sam-
That is U < ∆ε, in accordance with the independent- ple and low charge densities the effect of disorder is too
4
strong, creating an effective double quantum dot. This
can be identified in a charge stability diagram as addi-
tional kinks in the conductance lines that comprise the
N = 1 Coulomb diamond23. Such kinks are not present
in the stability diagram for the 160 nm wire shown in
Fig. 3(c), implying that disorder has no significant effect
in that case. Also, Coulomb interaction has been shown
to decrease the effect of Anderson localization24. How-
ever the theoretical results demonstrate the prospects of
our approach, if more efficient gating schemes are devel-
oped.
We have demonstrated the transition from the
independent-particle shell model to Wigner localiza-
tion with increasing length of a semiconductor nanowire
sample. While the excitation spectrum follows the
independent-particle shell model for the 70 nm wire
(∆E ≈ ∆ε), the onset of Wigner localization is ob-
2
served for the 160 nm wire (∆E < ∆ε) and finally our
2
simulations show complete Wigner localization in a wire
of length 300 nm. There the excitation energy of the
two-particle state is almost negligible and much lower
than the level spacing, ∆E (cid:28) ∆ε, and the calculated
Figure 4: Simulation of a 300 nm long wire. (a) Charge sta- 2
bility diagram. (b) A closer look at the area in the dashed electron density exhibits two peaks. This shows that
box in panel (a). Symbols (cid:192)-(cid:195) as in Fig. 2. The two lowest InSb nanowires form a convenient system to investigate
N =2stateshaveapproximatelythesameenergy,andhence stronglycorrelatedsystemsbywellestablishedtransport
the double conduction line of the 160 nm wire ((cid:194) and (cid:195) in measurement techniques.
Fig. 3b) has merged into a single line leading to the N = 2
Coulomb diamond. The broad conduction line consisting of
three lines for the three lowest N = 3 states, is marked by This work was supported by the Swedish Research
the symbol (cid:196). (c) Electron density of the four lowest N =3 Council (VR) as well as the Swedish Foundation for
states. Strategic Research (SSF).
∗ Corresponding author, [email protected] Pfeiffer, and K. W. West, Science (New York, N.Y.) 308,
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