The von Neumann-Wigner theorem in quantum dot molecules

0.31 MB·English

by  L. -X. Zhang

The von Neumann-Wigner theorem in quantum dot molecules
L.-X. Zhang, D. V. Melnikov, and J.-P. Leburton
Beckman Institute for Advanced Science & Technology and Department of Electrical and Computer Engineering,
University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
(Dated: February 2, 2008)
8
We show that electrons in coupled quantum dots characterized by high aspect ratios undergo
0
abrupt density rotations when the dots are biased into an asymmetric confinement configuration.
0
Density rotations occur with electron transfer to a single dot, and give rise to sharp variations of
2
theexchangecouplingbetweenelectronsasafunction ofinter-dotdetuning. Ouranalysisbasedon

n exact diagonalization technique indicates that this unusual behavior is in agreement with the von
a Neumann-Wignertheoremthatdictatesthevariationsoftheenergyspectrumfromthesymmetries
J of the molecular states during the detuning process. It is also shown that the overall effect is

6 quenched by the presence of magnetic fields, which by adding angular momentum to the electron
1 motion affects thespatial symmetry of themolecular states.

] PACSnumbers: 73.21.La,73.21.-b
l
l
a
h
The experimental observation of atomic-like proper- interaction.
-
s ties in manmade semiconductor quantum dots (QD) In this Letter, we show that two electrons in coupled
e
with feature sizes of a few hundreds to thousands of QDswithnon-zeroellipticityexperiencedensityrotation
m
Angstroms has stimulated intense research in the last and abrupt change of the ordering of their energy lev-
.
t decade [1, 2, 3, 4, 5, 6]. The existence of shell structures els with inter-dot detuning. The charge density rotation
a
andthe demonstrationofHund’s firstrule withshellfill- occurs at specific detuning values, andleads to a discon-
m
ingcontrolledbyelectricgatesinQDs haveprovidedthe tinuity in the derivative of the exchange energy J with
-
d conditionsforinvestigatingtunablemany-bodyeffectsin respect to the detuning bias, which is in agreement with
n solid state systems [7, 8]. As natural extensions of the
o concept of artificial atoms, coupled QDs resemble natu-
c ( r=1 ) ( r=3 )
ral molecules and offer enhanced many-body tunability
[ 60
byelectricallyvaryingthecouplingbarrierordistancebe-
2 tween the dots [9, 10], and detuning the energy spectra nm] 0
v y [
of the individual QDs by appropriate gate biasing [11],
6 -60
4 whicharefundamentaltechnicalingredientsfor realizing -60 x [n0m] 60
0 a quantum logic gate [12, 13] (a) (b)
-20 -30
1.0 funInctitohnissicsoonftcerxutc,iatlhiemspyomrtmanectreyfoorfththeeeveolelucttiroonnowfathvee V]-30 U V V]-35 U' PyS Px
me me
:071 eeWnnecigrengoyefrsptpheerectoturrreubmmat[io1of4n]sm,.eonAleeccrcugolyardrleinsvygeslstteocmrtoshsesunvfoodrnersNttaehtueemsinathflnaunt-- Energy [---456000 28.[5meV2]9.5--3397.5 [meV] Energy [---445050 24.[5meV2]5.5--3357.[meV]5 V'
v beardifferentsymmetries,whiletheyanti-crossforstates 21 26 31 36 41 21 23 25 27 29 31
Xi withthe samesymmetry. Thisfundamentaltheoremhas VL [meV] VL [meV]
found successful applications in the spectroscopy of al-
r FIG. 1: (color online) Top panels: potential contour plots of
a kaline salts [15], the interpretation of the Zeeman spec-
coupled QD with r=1 (left two) and r=3 (right two). For
trumofhydrogenmolecules[16],andmorerecentlyinthe
eachr,thepanelontheleft(right)isforVL =VR =21(VL=
explanation of the magnetic anisotropy in thin metallic 31, VR =21) meV. (a) r=1. Main panel: two lowest singlet
films [17]. (red,solid lines)andtriplet(blue,dashedlines) energylevels
Spatial symmetry effects can be mostly evidenced in asafunctionofdetuning. U (V)indicatesanti-crossingpoints
forthetwolowestsinglet(triplet)levels. Lowerinset: zoom-in
double QDs with large aspect ratios, i.e., for which the
regionneartheanti-crossingpointofthesingletenergylevels.
confinement strength in the coupling direction is com-
(b) r = 3. Main panel and lowest inset same as in (a) but
parable to or larger than that in the perpendicular di- forr=3. U′ (V′)indicatestheanti-crossing(crossing) point
rection. Hence inter-dot detuning is expected to induce of the two lowest singlet (triplet) energy levels [19]. Upper
significant conformational changes in the electron wave inset: threelowestsingle-particleenergylevelsinasymmetric
functions, which rearranges the electronic spectra of the confinement. In (a) and (b), the singlet energy levels are
individual QDs with drastic effects on the many-body lowered by 3 meV for clarity.
2
comparing the results using 8 and 9 harmonic oscillator
basis states in each Cartesian direction (we use 9 states
in each direction for this work). We ignore the Zeeman
effect which simply lowers the triplet energy and J by
∼25 µeV/T at a given magnetic field.
InFig. 1,bottompanels,weplotthetwolowestsinglet
and the two lowest triplet energy levels for the coupled
QDs with r = 1 [Fig. 1(a)] and r = 3 [Fig. 1(b)] as
a function of detuning ǫ (from here on, if not specified,
“singlet” and “triplet” refer to the singlet and triplet
states of lowest energy, respectively). All energy levels
decreaseasV increasesbecausethe single-particleener-
L
gies that give the main contribution to the two-electron
energy are lowered with the electric potential. For both
aspect ratios, r =1 and 3, and at small ǫ [21 ≤ V ≤ 29
L
meV in Fig. 1(a) and 21 ≤ V ≤ 25 meV Fig. 1(b)], we
L
observe singlet and triplet energies decrease with nearly
the same slope as ǫ increases. Then, the singlet energy
slope becomes steeper as both electrons localize into one
FIG. 2: (color online) (a) Two-dimensional density plots for
′
r = 1 at zero magnetic field. Columns (I), (II) and (III) QD, which are indicated by points U and U in Fig. 1.
correspond to VL = 21, 29 and 42 meV, respectively. First For r = 1 and VL > 34 meV, the slope of the triplet
(second) row, labeled S (T), is for thelowest singlet (triplet) energy increases in value (yet smaller than that of sin-
states. (b) Same as (a) but for r =3. Columns (I), (II) and glet), while for r = 3 the triplet energy experiences a
(III) correspond to VL = 25, 25.45 and 25.47 meV, respec- sudden change of slope at V = 25.46 meV [point V′
L
tively. Coordinates are shown in lower left panel and are the
in Fig. 1(b)], and with increasing V values afterwards,
same for all panels. For r = 3, the peak separation along L
the x (y) direction in the density of the triplet state is ∼ 60 the spacing between singlet and triplet energies first de-
(∼40) nm for column (II) [(III)]. creases,and then gradually increases for VL ≥27.2 meV
(cf. J curve in Fig. 3).
The detuning effects on the singlet and triplet states
the von Neumann-Wigner theorem [14]. We also show canbe further illustratedbyinspecting the electronden-
thatthepresenceofmagneticfieldsenhanceselectronlo- sity variations. In Fig. 2(a) we see that for coupled cir-
calization in the QDs and breaks the spatial symmetry cular QDs (r = 1) with increasing ǫ electrons gradually
in the two-electron states, which quenches the density moveintothelower(left)QDforbothsingletandtriplet
rotation effect. states, albeit more quickly for the singlet state. When
We model the coupled QDs with the ad hoc potential bothelectronsareintheleftdot[Fig. 2(a),column(III)],
V(r)=−V exp[−(x+d/2)2/R2−y2/R2]−V exp[−(x− the singlet density has a single maximum corresponding
L x y R
d/2)2/R2 −y2/R2], where d defines the inter-dot sepa- to the state with nearly zero angular momentum [20],
x y
ration, R and R are the extensions of the Gaussian whilethetripletstateexhibitstwomaximaalongtheQD
x y
potential for each dot in the coupling direction x and couplingdirection(x)astheconstituentQDsareslightly
perpendicular direction y. In this work, we take d = 60 moreextendedinthex-directionthaninthey-direction.
nm, R = 30 nm, and define the QD aspect ratio as For r = 3, as ǫ increases, the singlet density gradually
x
r =R /R . The difference betweenV and V givesthe localizesintothe left dotwith the electrondensityshow-
y x L R
inter-dot detuning ǫ = V −V . We fix V at 21 meV ing two maxima in the y-direction once V > 25 meV
L R R L
and increase V from 21 meV to higher values, which is because of the relaxed confinement in that direction [cf.
L
equivalent to increasingǫ. Figure 1, top panels show the Fig. 2(b)columns(I)and(II)].Forthetripletstate,how-
potentialcontourplotsforcircular(r=1,lefttwopanels) ever,itisseenthattheelectrondensityabruptlychanges
andelliptical(r=3,righttwopanels)QDsundersymmet- frombeingspreadoverthetwoQDswithahigherpeakin
ricandasymmetricbiases. AsǫincreasesthecoupledQD the left dot [V = 25.45 meV, column (II) in Fig. 2(b)]
L
system becomes effectively a single QD centered at the to occupying only the left dot with two peaks of equal
lowest point of the confinement potential (the left dot height in the y-direction [V = 25.47 meV, column (III)
L
center for both cases). in Fig. 2(b)]. This abrupt transition (or “rotation”) of
′
We use numerical exact diagonalization of the two- the electron density occurs at VL =25.46 meV [point V
electron Hamiltonian to solve for the system energies in Fig. 2(b)].
within the effective mass approximation. Details for the In order to understand the drastic difference in the
method and the material parameters are published else- dependences of the energy levels on ǫ between the
where [18]. The convergence of the energies is tested by QD configurations with r = 1 and r = 3, we com-
3
the level crossing, it becomes energetically favorable for
4.5
(x1) thetripletelectronstobeinthesameQDasansp -pair,
4 y
(x1.5) eventhoughthe expectationvalue ofthe Coulombinter-
action in the triplet state increases from 2.0 meV before
3
V] (x5) the transition spx →spy to 2.9 meV afterwards.
me 2 Figure 3 displays the exchange coupling J as a func-
J [ tion of VL, or effectively as a function of ǫ = VL −VR.
(x10) EachcurveinFig. 3depictsthevariationofJ asthecou-
1
pled QDs undergo the transition from the (1,1) to (2,0)
charge states (the right end of each curve is chosen at a
0
bias point where ǫ is large enough to localize both elec-
21 26 31 36 41 trons into the left dot). For small departure from equal
VL [meV] biases (ǫ & 0), as V increases from 21 meV, J main-
L
tains a small and approximately constantvalue before it
FIG. 3: (color online) Exchange coupling J as a function of increasessuperlinearly,aswasrecentlyobservedbothex-
VL (VR fixed at 21 meV) for QD aspect ratios r = 1 (black, perimentally [11] and reproduced theoretically [22]. For
solid), r = 1.5 (blue, dashed), r = 3 (green, dashed-dotted) r =1,J continuestoincreasemonotonicallywithǫasthe
and r = 5 (red, dotted) at zero magnetic field. The scaling
twoelectronsstronglylocalizeinoneQD.Inthiscase,the
factor for each curveis shown in parenthesis.
localization of the first excited single-particle state that
hasp -characterinthe leftdotoccursatlargerdetuning
x
valuethanthegroundstate[pointsU andV inFig. 1(a)],
pute the expectation values of the parity operator and J can only increase as the inter-level separation in
DPˆE = hΨ(x1,y1,x2,y2)|Ψ(−x1,−y1,−x2,−y2)i, and of the individual QD is larger than that in the coupled QD
system. However,withincreasingaspectratios(r ≥1.5),
the parity operator with respect to the y-axis DPˆyE = dramatic changes in the J behavior appear: a kink in J
hΨ(x1,y1,x2,y2)|Ψ(x1,−y1,x2,−y2)i. Here, Ψ is the occursatVL =30.70meVwithasuddenslopechangefor
two-electronwavefunction, and (x1, y1), (x2, y2) are the r =1.5,whilesharpmaximainJ ariseat25.46and23.30
coordinatesofthefirstandsecondelectrons,respectively. meVforr =3andr =5,respectively. Inthelattercases,
In the investigated detuning range, we find that for the there is also a localJ minimum at VL =27.20and 26.20
singlet state, 0 < P < 1 and P = 1 for all r; while for meV, respectively. For increasingr, electronlocalization
y
the tripletstate atr =1,P <0 andP =1,asexpected in the left dot occurs at progressively smaller detuning
y
from the symmetry of the Hamiltonian. However,in the (Fig. 3), as it becomes easier for the system to lower
tripletstateforr =3atV =25.46meV,ourcalculation its energy in a configurationwhere the two electrons are
L
showsthatthevalueof Pˆ sharplyincreasesfrom−0.98 located at the opposite “ends” of the elliptical QD [Fig.
D E
2(b)]tominimizetheirCoulombinteraction. Asaresult,
to −0.02, while DPˆyE changes sign from 1 to −1. This both singlet and triplet electrons move into the left dot
indicates thatthe parityofthe singletstate with respect atcloseVL values[e.g., points U′ andV′ inFig. 1(b) for
to the y-axis remains even, while the triplet wavefunc- r =3]. Dependinguponthedifferencebetweenthebiases
tion parity changes from even to odd, thereby changing requiredtolocalizesingletandtripletstatesintoasingle
its symmetry along the y-direction. In fact, we find that dot, the crossing between the two lowest triplet states
astheinter-dotdetuningisincreased,thetwolowestsin- associated with the electron density rotation is observed
glet states at any r have the same symmetry, and their as a kink (r & 1) or a sharp maximum (r ≫ 1) in the
energylevelsanticross;thesameistrueforthetwolowest exchange coupling at intermediate values of detuning.
triplet states at r = 1 (see Fig. 1), whereas the energy Figure 4, main panel shows the J-dependence on V
L
levels of the two lowest triplet states at r >1 [Fig. 1(b)] for r = 3 in the detuning range corresponding to the
crossduetotheirdifferentsymmetriesinagreementwith (1,1) ⇒ (2,0) transition (24 ≤ V ≤ 26 meV; V = 21
L R
the von Neumann-Wigner theorem [14]. Near the cross- meV)fordifferentmagneticfieldsnormaltothexy-plane.
′
ing pointofthe twolowesttripletenergylevels[pointV We observethatthe sharpdiscontinuities indJ/dV due
L
inFig. 1(b)],theanalysisofthespectralfunction[21]re- to abrupt electron density rotation with detuning dis-
veals that the first excited single-particle state localized appears for 0 < B ≤ 2 T, and evolves into a smooth
intheleftQDhasthep -character[seeschematicenergy J-increase with V . By adding angular momentum to
y L
diagraminthe upperinsetofFig. 1(b)]. Beforethe level the electronmotion, the magnetic field mixes the p and
x
crossing, this p -like orbital is unoccupied so that the p state parities, which affect the symmetry of the two-
y y
electronic states form a sp -pair consisting of the s-like electron wavefunction, and quenches the abruptness of
x
statemostlylocalizedintheleftQD,andthelowestanti- the density rotation. This symmetry breaking (from p -
y
symmetric p -like state spreading over both QDs. After symmetrytop -p symmetrymixing)removesthecross-
x x y
4
-36.4 In conclusion,we show thatinter-dotdetuning in cou-
V] pledellipticalquantumdotsinducesabruptdensityrota-
0.6 --3366..86nergy [me 0 T tiniocnresa.sWese, efinnedrgthyaltevaetlzcerroossminaggnoectciucrfiseflodratshtehtewdoetluowniensgt
0.45 E tripletstatesbearingdifferent(even/odd)symmetry. De-
-37 0.5 T
25.3 25.4 25.5 25.6 pendingonthe dotaspectratior,suchacrossingresults
] 0.3 VL [meV] 1 T in a kink (r & 1) or sharp local maximum (r ≫ 1) in
V 0.1 T
e 0.15 the dependence of the exchange coupling on the detun-
m
[ 2 T ing. For small nonzero magnetic field, the triplet energy
J 0
level crossing in coupled dots with large aspect ratios
3 T evolves into anti-crossing due to the mixing of single-
- 0.15
6 T particle states bearing different spatial symmetry. For
- 0.3 even larger magnetic fields, the localization of electrons
24 24.5 25 25.5 26
by detuning is more abrupt due to the strong compres-
VL [meV] sion of electronic orbitals.
This work is supported by the DARPA QUIST pro-
FIG. 4: (color online) Main panel: J as a function of VL for gramandNSFthroughtheMaterialComputationalCen-
r = 3 at different magnetic fields. The magnetic field value
ter at the University of Illinois. LXZ thanks the Beck-
is given on top of each curve. The curve for B = 0.1 T is
indicatedbyanarrow. Forclarity,curvesatdifferentB fields man Institute and Computer Science and Engineering
are shifted vertically in multiples of 0.03 meV. The dashed programat the University of Illinois.
(dotted) gray line is a guidefor theeyestracing thelocaliza-
tion point of the singlet (triplet) state at different magnetic
fields. Inset: twolowesttripletenergylevelsatB=0T(red,
dashed) and B=0.1 T (blue,solid).
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[6] W.G.vanderWieletal.,Rev.Mod.Phys.75,1(2003).
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6 T), the localization of both singlet and triplet states
(1998).
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[17] Sˇ, Pick and H. Dreyss´e, Phys.Rev.B 48, 13588 (1993).
same detuning [shown by the dashed line on Fig. 4(b)]
[18] L.-X. Zhang et al.,Phys. Rev.B 74, 205306 (2006).
for all investigated magnetic fields [23].
[19] At VL =25.46 meV, the computed spacing between the
Experimentally, the predicted variations in the two- two lowest triplet energy levels is less than ∼1 µeV. By
electron density and exchange coupling can be obtained the inspection of the energy level slopes and wavefunc-
from the conductance responses of quantum point con- tionsymmetry,weconcludethatacrossingoccursatthis
tacts adjacent to the dots, which are sensitive enough to point.
registerveryslightchangesintheCoulombpotentialdis- [20] M. Wagner et al.,Phys. Rev.B 45, R1951 (1992).
[21] D. V. Melnikov and J.-P. Leburton, Phys. Rev. B 73,
tributions in the coupled QD system [10, 11]. One can
155301 (2006).
alsodetecttheexistenceofthemaximumintheexchange
[22] M. Stopaand C. M. Marcus, cond-mat/0604008 (2006).
coupling from the coherent Rabi oscillations in the two- [23] The localization of the singlet and triplet states is de-
electron system by properly biasing the QD system [11]. termined at the point where the two lowest two-particle
Finally, electron imaging spectroscopy can be utilized to energy levels cross or anti-cross as detuningincreases.
directly probe the electron density in QDs [24]. [24] P. Fallahi et al.,Nano Lett. 5, 223 (2005).