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Thermally enhanced Wigner oscillations in two-electron 1D quantum dots
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 505301 (http://iopscience.iop.org/0953-8984/26/50/505301) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 147.8.31.43 This content was downloaded on 15/06/2017 at 16:44 Please note that terms and conditions apply.
You may also be interested in: Correlation functions for the detection of Wigner molecules in a one-channel Luttinger liquid quantum dot F. M. Gambetta, N. Traverso Ziani, F. Cavaliere et al. Current noise as a probe for Wigner molecules F Cavaliere, F M Gambetta, N Traverso Ziani et al. Probing Wigner correlations in a suspended carbon nanotube N Traverso Ziani, F Cavaliere and M Sassetti Temperature-induced emergence of Wigner correlations in a STM-probed one-dimensional quantum dot N Traverso Ziani, F Cavaliere and M Sassetti Spin exchange energy for a pair of valence band holes in artificial molecules W J Pasek, B Szafran and M P Nowak Theory of the STM detection of Wigner molecules in spin-incoherent CNTs N. Traverso Ziani, F. Cavaliere and M. Sassetti Transport properties of quantum dots in the Wigner molecule regime F Cavaliere, U De Giovannini, M Sassetti et al. Non-linear Coulomb blockade microscopy of a correlated one-dimensional quantum dot D Mantelli, F Cavaliere and M Sassetti Symmetry breaking and quantum correlations Constantine Yannouleas and Uzi Landman
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 505301 (8pp)
doi:10.1088/0953-8984/26/50/505301
Thermally enhanced Wigner oscillations in two-electron 1D quantum dots F Cavaliere1,2 , N Traverso Ziani3 , F Negro4 and M Sassetti1,2 1
Dipartimento di Fisica, Universit`a di Genova, Via Dodecaneso 33, 16146 Genova, Italy CNR-SPIN, Via Dodecaneso 33, 16146 Genova, Italy 3 Institute for Theoretical Physics and Astrophysics, University of Würzburg, 97074 W¨urzburg, Germany 4 Dipartimento di Fisica dell’Universi`a di Pisa and INFN—Sezione di Pisa, Largo Pontecorvo 3, I-56127 Pisa, Italy 2
E-mail: [email protected] Received 21 July 2014, revised 27 October 2014 Accepted for publication 28 October 2014 Published 24 November 2014 Abstract
Motivated by a recent experiment (Pecker et al 2013 Nat. Phys. 9 576), we study the stability, with respect to thermal effects, of Friedel and Wigner density fluctuations for two electrons trapped in a one-dimensional quantum dot. Diagonalizing the system exactly, the finite-temperature average electron density is computed. While the weak and strong interaction regimes display a Friedel oscillation or a Wigner molecule state at zero temperature, which as expected smear and melt as the temperature increases, a peculiar thermal enhancement of Wigner correlations in the intermediate interaction regime is found. We demonstrate that this effect is due to the presence of two different characteristic temperature scales: TF , dictating the smearing of Friedel oscillations, and TW , smoothing Wigner oscillations. In the early Wigner molecule regime, for intermediate interactions, TF < TW leading to the enhancement of the visibility of Wigner oscillations. These results complement those obtained within the Luttinger liquid picture, valid for larger numbers of particles. Keywords: quantum dot, Wigner molecule, correlated electrons (Some figures may appear in colour only in the online journal)
WM [2, 3]. Few works have considered the effects of finitetemperature, essentially demonstrating how spherical [27] and circular [28, 29] Wigner molecules eventually melt resulting in an uncorrelated, liquid-like state. In 1D quantum dots, several numerical techniques have also been adopted, such as density matrix renormalization group analysis (DMRG) [30], density functional theory [31–36], quantum Montecarlo [37], HartreeFock [38] and exact diagonalization [39–44]. In addition, the restricted phase space allows also for analytical methods such as the Bethe ansatz solution [31–34, 45] or spin-coherent [46–51] and spin-incoherent [52–57] Luttinger liquids [58]. Due to the broken translational symmetry, signatures of the WM can already be found in the electron density. This is particularly interesting since the possibility of imaging the electron density in local transport experiments has been demonstrated [59, 60]. The peaks of the linear conductance are in fact shifted, with respect to the gate potential, when the dot is coupled to a local capacitive coupling [59–61].
1. Introduction
Eighty years since their seminal proposal, Wigner crystals [1] and their finite-size analogue Wigner molecules (WM) [2, 3] still represent intriguing systems, only partially understood so far. In particular, Wigner molecules have recently triggered a lot of interest with several experiments reporting their observation in two-dimensional (2D) semiconducting heterostructures [4, 5], in semiconducting circular [6] quantum dots [7] and in one-dimensional (1D) quantum wires [8–11] and carbon-nanotubes (CNTs) [12, 13]. Due to strong electronic correlations, WMs have often been studied numerically [2, 3], with a particular focus on circular quantum dots [14–25]. In such 2D systems, the electron density only shows weak signatures of a radial ordering in the molecular state, while no angular oscillations are found due to the rotational symmetry. Density density correlation functions [26] are hence required to investigate the 0953-8984/14/505301+08$33.00
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The shift depends on the position of the local coupling, and its characterization allows for the extraction of the electron density profile inside the dot. It is worth pointing out that, on the other side, a local tunnel coupling reveals the local density of states instead of the electron density. The two quantities can be significantly different [41, 49]. For 1D finite systems, most of the theoretical efforts have concentrated on the zero-temperature regime. Here, a competition in the density between finite size effects, inducing Friedel oscillations with wavevector 2kF (kF the Fermi wavevector) and electron correlations inducing Wigner oscillations with 4kF momentum occurs. Friedel oscillations produce N/2 (resp. (N + 1)/2) peaks for even (resp. odd) N , while Wigner oscillations are always characterized by N distinct peaks. At zero temperature, Friedel oscillations dominate for weak interactions, while Wigner oscillations mark the formation of a WM in the strong interaction regime. This behaviour is confirmed by both analytical and numerical results [31, 39, 43, 46–49]. It has been pointed out that, in the case of short-range delta-like electron interactions in 1D, even in the presence of N peaks in the density, the densitydensity correlation functions may reveal a liquid-like state of the dot [32]. However, for interactions whose range exceeds the size of the 1D quantum dot such as those studied in the present paper, N distinct peaks in the density are associated with the presence of a well defined molecular state [32]. More intriguing is the finite temperature behaviour, which has been analysed by means of the Luttinger liquid theory in the regime of large numbers of particles [49, 57]. There, it is shown that Friedel oscillations are stable at energy scales smaller than those of spin collective excitations εσ = kB Tσ , while Wigner oscillations are stable at energies smaller than those of the plasmonic charge modes [50, 51] ερ = kB Tρ . Here, Tσ < Tρ for an interacting system, with Tσ , related to the exchange interaction among electrons, exponentially suppressed in the strong interaction limit [62, 63]. Three regimes exist. For T < Tσ , both Friedel and Wigner are stable and the relative importance is essentially set by the interaction strength. For Tσ < T < Tρ , Friedel oscillations smear out allowing the Wigner oscillations to be enhanced in the total density. In this regime, if Tσ and Tρ are sufficiently well separated, a rather counter-intuitive effect occurs, namely an enhancement of the Wigner oscillations as temperature increases, even if at T = 0 the density is dominated by Friedel oscillations. Note that for weak interactions, Tσ ∼ Tρ , while increasing interactions Tσ → 0 while Tρ sharply increases over the weakly interacting case. Thus, to observe the enhancement of the Wigner oscillations, an intermediate-to-strong interaction regime is best suited. Finally, for T > Tρ the Wigner molecule melts and the density no longer exhibits peculiar oscillations. The above regime describes a dot with a large number of particles. On the other hand, few-electron quantum dots also represent a natural playground where one can investigate the effects of strong correlations—requiring numerical techniques for their theoretical investigation. Stringent evidence of the formation of a 1D Wigner molecule composed by two electrons in a carbon nanotube was provided very recently [13]. The proof of the existence
of such a molecule relies on the observation—via transport measurement—of a sharp collapse of the energy gap , between the ground state and the first excited state. Indeed, one of the strongest spectroscopical signatures of the formation of a Wigner molecule is a tendency towards the degeneracy of all lowest-lying spin multiplets (which implies → 0 for two electrons) as the interaction strength increases [2, 13, 39]. Note that in CNTs the degeneracies of the ground state and the first excited state are 6 and 10 respectively [13] when neglecting spin–orbit interaction. In the presence of spin–orbit coupling, the sixteen states further reorganize into three multiplets with degeneracies 4,8,4 [41]). In the case of semiconducting quantum wires with negligible spin orbit coupling, the ground state is non degenerate, and the first excited states are characterized by a threefold degeneracy. Stimulated by this experiment, we analyse the stability of Friedel and Wigner oscillations in a two-electron quantum dot as a function of temperature and interaction strength. Employing an exact diagonalization technique, we evaluate the dot electron density as a function of temperature. We will demonstrate that the results obtained for large particle numbers have a wide range of validity and that a similar phenomenon of enhancement of the Wigner oscillations occurs also in the opposite regime of two electrons. Although our model applies directly to the case of quantum dots built in semiconducting quantum wires, we do not expect qualitative differences for the case of CNTs, the different degeneracy of the ground and excited states leading to a renormalization of the characteristic temperatures discussed later in this work. Our main conclusions are the following. For zero temperature we recover the standard behaviour of a continuous transition from a singly-peaked density, in which the Friedel contribution is dominating at weak interaction, to a density pattern with two peaks characteristic of the formation of a Wigner molecule for strong interactions. For finite temperature, however, an intermediate regime of the Coulomb interaction strength and an intermediate regime of temperatures exists, in which the visibility of the Wigner oscillations is actually enhanced with respect to the zero temperature case. Outside this regime of interaction strengths, namely for weak or strong interactions, increasing the system temperature leads to a more conventional melting of either the Friedel or the Wigner oscillations. This behaviour is schematically represented in the scheme shown in figure 7. The threshold values for the interaction strength and the temperature are defined and discussed in detail in section 3. The outline of the paper is as follows: in section 2.1 the model and general features of the numerical method are discussed. The results are presented and discussed in section 3. In section 4 conclusions are drawn. 2. Model and method 2.1. Model
The system consists of two electrons harmonically confined in a 1D quantum dot, interacting via the potential V (y) [41]. The 2
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Hamiltonian is Hˆ =
−
i=1,2
absence of a magnetic field, all triplet states are degenerate. Furthermore, ϕn (R) are the eigenfunctions of a free harmonic oscillator with dimensionless energy ε˜ n = n + 1/2. To find m (r) and the corresponding energy spectrum ˜m we employ a numerical exact diagonalization technique, projecting the hamiltonian sector containing r on the space spanned by M bare harmonic oscillator states. Up to M = 50 states have been employed in our calculation, and convergence with a relative error below 5 · 10−3 has been achieved. Our task is to investigate the behaviour of the dimensionless electron density ρ(x) ˆ [26, 66]
h ¯ 2 ∂2 mω2 2 + yˆ + Vˆ (yˆ2 − yˆ1 ) , 2m ∂ yˆi2 2 i
(1)
where yˆi is the position operator of the i-th electron and the effective 1D soft Coulomb interaction is [41] c Vˆ (y) ˆ = . 2 λ + yˆ 2
(2)
Here, λ is a short-distance cutoff due to the effective width of the channel [46, 64, 65] while c is the interaction strength. To obtain the spectrum and wavefunctions of the system, we employ an exact diagonalization method [39–44]. This method provides results with an excellent precision in all ranges of parameters considered in this paper, while retaining a bearable level of computational complexity. Introducing the ˆ and relative (ˆr ) coordinates dimensionless centre of mass (R) yˆ1 + yˆ2 yˆ1 − yˆ2 Rˆ = ; rˆ =
(3)
1 1 ∂2 1 c˜ 1 ∂2 + Rˆ 2 − + rˆ 2 + hˆ = − 2 ∂ Rˆ 2 2 2 ∂ rˆ 2 2 ˜λ2 + rˆ 2
(4)
ρ(x) ˆ =
1 δ x − xˆi , 2 i=1,2
(5)
at finite temperature. Here, we have introduced xˆi = yˆi /. The quantum average of the density is ˜ˆ e −β h ρ(x) ˆ (6) ρ(x) = Tr Z
˜ˆ where Z = Tr e−β h with β˜ = E0 /kB T and T is the system temperature. One finally obtains 2 1 −β˜ ˜n,m e Dm dz n,m (x + z, x − z) (7) ρ(x) = Z n,m
√ with = 2¯h/mω the typical length scale, one obtains
¯ ω and λ˜ = λ/, c˜ = c/(E0 ) where hˆ = Hˆ /E0 with E0 = h are dimensionless parameters governing the interaction. Note that in the case of a one dimensional quantum dot, the condition λ˜ 1 must hold. The coordinates R and r decouple, which allows searching for solutions of the form n,m (R, r; σ1 , σ2 ) = ϕn (R) m (r)|χm (σ1 , σ2 ) with energy spectrum ˜n,m = ε˜ n + ˜m . Here, σi are the spin directions for i-th electron and χm (σ1 , σ2 ) is the total spin wavefunction of the system. Since parity is a good quantum number and ϕn (R) is symmetric under particle exchange, the spin wavefunction is uniquely determined by the quantum number m, with χm (σ1 , σ2 ) being a singlet or a triplet according to m (r) being even or odd. Note that in the absence of a magnetic field, all triplet states are degenerate. Furthermore, ϕn (R) are the eigenfunctions of a free harmonic oscillator with dimensionless energy ε˜ n = n + 1/2. To find m (r) and the corresponding energy spectrum ˜m we employ a numerical exact diagonalization technique, projecting the hamiltonian sector containing r on the space spanned by M bare harmonic oscillator states. Up to M = 50 states have been employed in our calculation, and convergence with a relative error below 5 · 10−3 has been achieved. Where hˆ = Hˆ /E0 with E0 = h ¯ ω and λ˜ = λ/, c˜ = c/(E0 ) are dimensionless parameters governing the interaction. The coordinates R and r decouple, which allows searching for solutions of the form n,m (R, r; σ1 , σ2 ) = ϕn (R) m (r)|χm (σ1 , σ2 ) with energy spectrum ˜n,m = ε˜ n + ˜m . Here, σi are the spin directions for i-th electron and χm (σ1 , σ2 ) is the total spin wavefunction of the system. Since parity is a good quantum number and ϕn (R) is symmetric under particle exchange, the spin wavefunction is uniquely determined by the quantum number m, with χm (σ1 , σ2 ) being a singlet or a triplet according to m (r) being even or odd. Note that in the
Z=
1
˜ 2 sinh (β/2)
m
˜
e−β ˜m Dm ,
(8)
where Dm is the degeneracy of (n,m) (R, r), with Dm = 1 if the total spin is a singlet, Dm = 3 if the total spin is a triplet. The summations are performed numerically, truncating the number of involved states when convergence with a relative precision of less than 10−2 is achieved. 2.2. Spectrum and zero-temperature density
Figure 1 shows the energy spectrum (referred to the ground state) ˜n,m − ˜0,0 as a function of c˜ in a wide range of interaction strength spanning, as will be shown, from noninteracting electrons to a strongly interacting regime. In the non-interacting limit, the spectrum ˜n,m = n + m + 1 exhibits a shell structure with degenerate levels at energy k + 1 satisfying the condition n + m = k (n, m, k 0). The degeneracy of the k-th shell (including spin degeneracy) is 2k + 1 if k is even and 2k + 2 if k is odd. As such, the singlet ground state is non-degenerate. In the presence of Coulomb interactions, several level crossings occur. However, the dot ground state always remains a spin singlet and the lowest-lying excited state is always a triplet. Ultimately, for c˜ 1 the spectrum rearranges in a sequence of fourfold degenerate spin multiplets composed of a singlet and a triplet. This reordered spectrum is described by √ corr − ˜0,0 = p + 3q , (9)
˜p,q with p, q 0. Let us now summarize the behaviour of the system in the zero temperature limit. Figure 2 shows the zero temperature 3
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Figure 1. Energy spectrum ˜n,m − ˜0,0 as a function of the interaction parameter c. ˜ Solid (blue) lines represent a singlet state, dashed (red) lines a triplet state. Here, λ˜ = 0.02.
Figure 3. Density profile ρ(x) (units −1 ) as a function of x for c˜ = 0.05 (solid, black); c˜ = 0.3 (dashed, red); c˜ = 3 (dotted, blue); c˜ = 10 (dash–dotted, green). Other parameters: T˜ = 0 and λ˜ = 0.02.
the fourfold degeneracy of the spectrum in this regime— corresponding to the four distinct manners to arrange spins on the two sites of the Wigner molecule. 3. Results 3.1. Thermal–Wigner enhancement
Let us now turn to the question of the stability of Friedel and Wigner oscillations against thermal effects, by studying the temperature dependence of the electron density. Figure 4(a) shows the weakly interacting case c˜ = 0.1, in the Friedel regime, for five different temperatures. Increasing T˜ the density smears and broadens. Melting is a continuous process, as is typical in finite-size 1D systems, and is due to the thermal activation of excited states. In particular, the lowest lying triplet excitation which lies at energy ε˜ 0,1 − ε˜ 0,0 < 1 is the first to come into play, see figure 1. We can thus customarily identify a temperature scale TF (c) ˜ for melting in the Friedel ˜ ∼ (c)/E ˜ , where (c) ˜ is the lowest-lying regime as TF (c) 0 singlet–triplet energy gap. Increasing c˜ (while still retaining c˜ < c∗ ) one observes a qualitatively similar behaviour, with a melting process which however occurs at a lower temperature, in agreement with the fact that the singlet–triplet gap shrinks to zero as c˜ increases (not shown). In the opposite case of very strong interactions, the system is dominated by Wigner correlations. For example, consider the case c˜ = 3 shown in figure 4(b). Here, the two sharp peaks become weaker as T˜ is increased and, for large enough T˜ , the density exhibits a single peak becoming similar to the thermally smeared state discussed above. This smearing is a continuous process, analogous with the melting of Friedel oscillations. Also, here we can define a temperature scale TW such that for T˜ > TW the two peaks disappear and thus the Wigner molecule is melted. √ We observe that in this strong interaction regime for T˜ ≈ 3 the√density flattens out, allowing us to identify TW (c˜ = 3) = 3. This energy scale corresponds to the radial breathing mode of the Wigner molecule and thus suggests that the leading mechanism responsible for the melting are vibrations of the two electrons about their equilibrium positions.
Figure 2. Density profile ρ(x) (units −1 ) as a function of x and c˜
for T˜ = 0 and λ˜ = 0.02.
electron density of the system as a function of x, in the same range of c˜ spanned in figure 1. It is clear that the density exhibits a single peak for weak interactions, while for strong interactions, two peaks are present. To be more quantitative, figure 3 shows ρ(x) for selected values of the interaction strength c. ˜ For weak interactions, the single peak in the electron density corresponds to the physical picture of two electrons packed near the centre of the dot with antiparallel spins. This is the hallmark of the Friedel oscillation for a 1D dot with two electrons. As interactions increase, the density broadens while the peak gets lower. At a critical value c = c∗ ≈ 0.3 the density flattens out and for c > c∗ two small bumps emerge signalling the incipience of a Wigner molecule. Indeed, for c c∗ two fully developed peaks appear, distinctive of the Wigner molecule. The strongly correlated molecular state allows us to understand the spectrum reordering occurring for c c∗ . The quantum number p in equation (9) describes the centre of √ mass excitations of the molecule, while the mode at energy 3 with quantum number q is connected to the radial breathing mode involving the relative coordinate only. In this strongly correlated regime, the overlapping between the electrons’ wavefunctions near the centre of the dot is very small and, as a result, exchange interaction is strongly suppressed. The energy gap between different spin states collapses, which motivates 4
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Figure 5. Density profile ρ(x) (units −1 ) as a function of x for
(a) T˜ = 0.0 (black); T˜ = 0.1 (red); T˜ = 0.3 (green); T˜ = 0.6 (blue); T˜ = 1.2 (orange) and (b) T˜ = 0.0 (black); T˜ = 0.1 (red); T˜ = 0.4 (green); T˜ = 0.7 (blue); T˜ = 1.6 (orange). The arrows mark the evolution of the density as temperature increases. In panel (a) c˜ = 0.4, in panel (b) c˜ = 0.6. In all panels λ˜ = 0.02.
Figure 4. Density profile ρ(x) (units −1 ) as a function of x for (a) T˜ = 0.0 (black); T˜ = 0.4 (red); T˜ = 0.8 (green); T˜ = 1.2 (blue); T˜ = 1.6 (orange) and (b) T˜ = 0.0 (black); T˜ = 0.6 (red); T˜ = 1.2 (green); T˜ = 1.73 (blue); T˜ = 2.0 (orange). The arrows mark the evolution of the density as temperature increases. In panel (a) c˜ = 0.1, in panel (b) c˜ = 3. In all panels λ˜ = 0.02.
regime of Wigner enhancement, such values are forbidden by the condition λ˜ 1, needed to ensure the 1D character of the dot. To understand in more detail how the visibility of the Wigner oscillations is enhanced by temperature, the wavefunction of the ground state and of the first three excited states of the dot are shown in figure 6. As is clear, the lowest lying triplet excited state has a much more pronounced oscillating pattern than the singlet ground state. Indeed we have checked that as soon as c > c∗ the ground state wavefunction begins to show a double bump structure, whereas the first excited state already has a more developed peak ˜ structure. Since the singlet–triplet energy gap, i.e. TF (c), is already of the order of 0.1 in the range c∗ < c˜ < cW , even a slight thermal excitation promotes the population of the triplet state, which produces a more marked oscillating pattern of the density. When T˜ > ε˜ 1,0 − ε˜ 0,0 = 1 the next excited states become populated, promoting the occupation of the centre of the dot even further as is clear by inspecting their ˜ is found to lie between wavefunctions (see figure 6), and TW (c) the energy gap of the second and the third excited states. As c˜ increases towards cW the Wigner character of the ground state gets more pronounced and the wavefunctions of singlet and triplet states tend to exhibit peaks of similar height and position (not shown). As a consequence, the temperatureinduced enhancement becomes less visible and the melting
We have checked that the simple melting of Friedel and Wigner oscillations described above occurs only in specific ranges of interaction strength. More specifically, the simple smearing of the Friedel peak as T˜ > TF occurs for c < c∗ , while the monotonous disappearance of the Wigner molecule oscillations in the density for which T˜ > TW leaving place to a single peak only occurs for large enough interactions c > cW ∼ 1. We turn now to the more interesting regime of intermediate interactions c∗ < c˜ < cW . Consider figure 5(a), for c∗ < c˜ = 0.4 < cW : here, a double-peak structure can be detected already for T = 0 as discussed above. However, in sharp contrast to what was already discussed, increasing the temperature leads to an enhancement of the visibility of the Wigner molecule, with the double-peak structure becoming more pronounced as T˜ is raised. Only for large enough T˜ TW (c˜ = 0.4) ≈ 1.1 will the Wigner molecule eventually melt and the two peaks disappear. Figure 5(b) shows the case c∗ < c˜ = 0.6 < cW which exhibits the same qualitative behaviour, with TW (c) ˜ ≈ 1.4. The same qualitative behaviour is observed throughout the whole c∗ < c˜ < cW range. Moreover, in the Wigner enhancement regime, the effect of varying λ˜ is weak. In fact the properties of the system strongly depend on λ˜ only starting from λ˜ ∼ c˜1/3 , as can be seen by studying the minima of the potential energy. However, in the 5
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Figure 6. Wavefunction modulus squared | n,m (x)|2 (units −1 ) as a function of x for (a) n = m = 0, ground state (singlet); (b) n = 0, m = 1, first excited state (triplet); (c) n = 1, m = 0, second excited state (singlet); (d) n = m = 1, third excited state (triplet). In all panels c˜ = 0.4, λ˜ = 0.02.
Figure 7. Qualitative scheme of the different regimes of the two electrons quantum dots as a function of the dimensionless interaction strength c˜ and temperature T˜ . The region labelled F represents the predominance of the Friedel contribution, F/W denotes a regime where Friedel and Wigner coexist, W marks the predominance of the Wigner molecule, Enhanced Wigner the regime where Wigner correlations are enhanced by temperature. For large temperature, the two-electron states melt regardless of their character at small temperatures. The curves TF,W (c) ˜ represent the qualitative behaviour of these quantities, as inferred by our numerical observations.
temperature TW (c) ˜ increases, since the contribution of the third excited state becomes crucial to cancel the double-peak structure of the density. Finally, when c˜ = cW the singlet and ˜ ≈ 0 and no triplet wavefunctions essentially coincide, TF (c) enhancement occurs. ˜ < T˜ < We dub the range c∗ < c˜ < cW and TF (c) ˜ (see figure 7) the Wigner enhancement regime in which TW (c) the Wigner features of the electron density become more prominent as temperature is increased. The existence of this regime is clearly due to the strong separation of the ˜ and TW (c) ˜ induced by interactions. This behaviour TF (c) is in perfect agreement with the findings of the Luttinger theory [49, 52, 56, 57]. This provides strong evidence that for N 3 particles also, one would observe a qualitatively similar behaviour with a universal trend of enhancement of the Wigner correlations. This will be the subject of future numerical investigations. A summary of our conclusions is schematically represented in the picture shown in figure 7. In order to identify different regimes, the sharp lines T˜ = TF (c), T˜ = TW (c), c = c∗ and c = cW have been drawn even though it is clear that all the transitions discussed here are continuous processes, occurring in proximity of the threshold values denoted in the figure.
One of the possible techniques could involve a linear transport measurement in the presence of a charged AFM tip [48, 59, 60] which has also been suggested as a possible way to obtain direct information about the spatial fluctuations of the electron density in the Wigner molecule regime. When this kind of experiment is performed with a tip weakly coupled to the dot, the chemical potential of the latter (and thus the position of the linear conductance peak) is predicted to shift by an amount proportional to the electron density. One of the limiting factors of this approach would be the thermal broadening of conductance traces. By converting the peakto-valley swing between the central minimum and the lateral maxima of the density in the Wigner enhancement regime, see figure 5, to energy scales for a typical setup [59, 60], we obtain a chemical potential shift slightly larger than 1 K, which should allow us to observe the predicted effect at least up to the temperature when Wigner oscillations in the density begin to surmise.
3.2. Observability of the enhancement of Wigner oscillations
4. Conclusion
We conclude by briefly commenting on the observability of the enhancement of Wigner phenomena predicted here. A typical ˜ in the range c∗ < c˜ < cW is T˜F ∼ 0.1. estimate for T˜F (c) Assuming E0 ≈ 1 meV, the maximum enhancement would occur at a temperature of about 1 K, while the complete melting of the Wigner molecule would occur at a temperature that is one order of magnitude higher.
We have studied the influence of temperature on the states of two correlated electrons harmonically confined in a one-dimensional quantum dot. By exactly diagonalizing the system, the finite-temperature density profile has been calculated. 6
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In the weak interaction limit, the density exhibits a centred maximum attributed to finite-size Friedel effects, which is smeared by temperature. For strong interactions, the lowtemperature density profile confirms the formation of a Wigner molecule, with two marked peaks symmetrically placed with respect to the potential well. Increasing the temperature melts the Wigner molecule resulting in the disappearance of the double-peak structure. The case of intermediate interactions is peculiar: at low temperature, the system exhibits very weak Wigner molecule features. However, the latter are enhanced as the temperature is increased. The above behaviour has been interpreted introducing two temperature scales TF and TW , which resemble those introduced in the Luttinger liquid theory of the system [49, 56, 57], valid for a large number of electrons. Indeed, all the results presented in this paper confirm those obtained in the Luttinger theory, suggesting that the Wigner enhancement mechanism could be a universal phenomenon. In order to prove this universality, it would be interesting to extend this analysis to numbers of particles N 3. More general-purpose techniques such as path-integral Montecarlo [67, 68] may be employed to tackle this problem.
[17] Nishi Y, Maksym P A, Austing D G, Hatano T, Kouwenhoven L P, Aoki H and Tarucha S 2006 Phys. Rev. B 74 033306 [18] Rontani M, Cavazzoni C, Bellucci D and Goldoni G 2006 J. Chem. Phys. 124 124102 [19] Harju A, Saarikoski H and R¨as¨anen E 2006 Phys. Rev. Lett. 96 126805 [20] Yannouleas C and Landman U 2000 Phys. Rev. Lett. 85 1726 [21] De Giovannini U, Cavaliere F, Cenni R, Sassetti M and Kramer B 2007 New J. Phys. 9 93 [22] De Giovannini U, Cavaliere F, Cenni R, Sassetti M and Kramer B 2008 Phys. Rev. B 77 035325 [23] Cavaliere F, De Giovannini U, Sassetti M and Kramer B 2009 New J. Phys. 11 123004 [24] Puente A, Serra L and Nazmitdinov R G 2004 Phys. Rev. B 69 125315 [25] Rontani M, Molinari E, Maruccio G, Janson M, Schramm A, Meyer C, Matsui T, Heyn C, Hansen W and Wiesendanger R 2007 J. Appl. Phys. 101 081714 [26] Giuliani G F and Vignale G 2005 Quantum Theory of the Electron Liquid (Cambridge: Cambridge University Press) [27] Lozovik Yu E, Volkov S Yu and Willander M 2003 Solid State Commun. 125 127 [28] Filinov A V, Bonitz M and Lozovik Yu E 2001 Phys. Rev. Lett. 86 3851 [29] B¨oning J, Filinov A, Ludwig P, Baumgartner H, Bonitz M and Lozovik Yu E 2008 Phys. Rev. Lett. 100 113401 [30] S¨offing S A, Bortz M, Schneider I, Struck A, Fleischhauer M and Eggert S 2009 Phys. Rev. B 79 195114 [31] Xianlong G 2012 Phys. Rev. A 86 023616 [32] Wang J-J, Li W, Chen S, Xianlong G, Rontani M and Polini M 2012 Phys. Rev. B 86 075110 [33] Xianlong G and Asgari R 2008 Phys. Rev. A 77 033604 [34] Magyar R J and Burkev K 2004 Phys. Rev. A 70 032508 [35] Abedinpour S H, Polini M, Xianlong G and Tosi M P 2007 Eur. Phys. J. B 56 127 [36] Abedinpour S H, Polini M, Xianlong G and Tosi M P 2007 Phys. Rev. A 75 015602 [37] Casula M, Sorella S and Senatore G 2006 Phys. Rev. B 74 245427 [38] Mueller E J 2005 Phys. Rev. B 72 075322 [39] H¨ausler W and Kramer B 1993 Phys. Rev. B 47 16353 [40] Jauregui K, Häusler W and Kramer B 1993 Europhys. Lett. 24 581 [41] Secchi A and Rontani M 2009 Phys. Rev. B 80 041404 [42] Secchi A and Rontani M 2010 Phys. Rev. B 82 035417 [43] Secchi A and Rontani M 2012 Phys. Rev. B 85 121410 [44] Secchi A and Rontani M 2013 Phys. Rev. B 88 125403 [45] Bethe H A 1931 Z. Phys. 71 205 [46] Schulz H J 1993 Phys. Rev. Lett. 71 1864 [47] Safi I and Shulz H J 1999 Phys. Rev. B 59 3040 [48] Mantelli D, Cavaliere F and Sassetti M 2012 J. Phys.: Condens. Matter 24 432202 Gambetta F M, Traverso Ziani N, Cavaliere F and Sassetti M 2014 Europhys. Lett. 107 47010 [49] Traverso Ziani N, Cavaliere F and Sassetti M 2013 New J. Phys. 15 063002 [50] Sassetti M and Kramer B 1998 Phys. Rev. Lett. 80 1485 [51] Cavaliere F, Braggio A, Stockburger J T, Sassetti M and Kramer B 2004 Phys. Rev. Lett. 93 036803 Sassetti M and Kramer B 1996 Phys. Rev. B 54 R5203 [52] Fiete G A 2007 Rev. Mod. Phys. 79 801 [53] Fiete G A, Le Hur K and Balents L 2006 Phys. Rev. B 73 165104 [54] Fiete G A, Le Hur K and Balents L 2005 Phys. Rev. B 72 125416 [55] Matveev K A 2004 Phys. Rev. Lett. 92 106801 [56] Matveev K A, Furusaki A and Glazman L I 2007 Phys. Rev. Lett. 98 096403
Acknowledgments
We gratefully acknowledged the support of EU-FP7 via Grant No. ITN-2008-234970 NANOCTM and of MIUR via MIUR-FIRB2012—Project HybridNanoDev, Grant No.RBFR1236VV. FN acknowledges the financial support from the EU under project Hadron Physics 3 (Grant Agreement no. 283286). References [1] [2] [3] [4] [5] [6] [7]
[8] [9] [10] [11] [12] [13] [14] [15] [16]
Wigner E 1934 Phys. Rev. 46 1002 Reimann S M and Manninen M 2002 Rev. Mod. Phys. 74 1283 Yannouleas C and Landman U 2007 Rep. Prog. Phys. 70 2067 Andrei E Y, Deville G, Glattli D C, Williams F I B, Paris E and Etienne B 1998 Phys. Rev. Lett. 60 2765 Piot B A, Jiang Z, Dean C R, Engel L W, Gervais G, Pfeiffer L N and West K W 2008 Nat. Phys. 4 936 Kalliakos S, Rontani M, Pellegrini V, Garcia C P, Pinczuk A, Goldoni G, Molinari E, Pfeiffer L N and West K W 2008 Nat. Phys. 4 467 Kouwenhoven L P, Marcus C M, McEuen P L, Tarucha S, Westervelt R M and Wingreen N S 2007 Electron Transport in Quantum Dots (NATO Advanced Studies Institute, Series E: Applied Science) ed L L Sohn et al (Dordrecht: Kluwer) p 105 Ellenberger C et al 2006 Phys. Rev. Lett. 96 126806 Singha A et al 2010 Phys. Rev. Lett. 104 246802 Kristinsd´ottir L et al 2011 Phys. Rev. B 83 041101 Meyer J S and Matveev K A 2009 J. Phys.: Condens. Matter 21 023203 Deshpande V V and Bockrath M 2008 Nat. Phys. 4 314 Pecker S, Kuemmeth F, Secchi A, Rontani M, Ralph D C, McEuen P L and Ilani S 2013 Nat. Phys. 9 576 Dineykhan M and Nazmitdinov R G 1997 Phys. Rev. B 55 13707 Hawrylak P and Pfannkuche D 1993 Phys. Rev. Lett. 70 485 Tavernier M B, Anisimovas E, Peeters F M, Szafran B, Adamowski J and Bednarek S 2003 Phys. Rev. B 68 205305 7
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F Cavaliere et al
[57] Traverso Ziani N, Cavaliere F and Sassetti M 2013 Europhys. Lett. 102 47006 [58] Haldane F D 1981 Phys. Rev. Lett. 47 1840 [59] Boyd E E and Westervelt R M 2011 Phys. Rev. B 84 205308 [60] Traverso Ziani N, Cavaliere F and Sassetti M 2012 Phys. Rev. B 86 125451 [61] Wach E, Zebrowski D P and Szafran B 2013 J. Phys.: Condens. Matter 25 335801
[62] Hausler W 1996 Z. Phys. B 99 551 [63] Matveev K A 2004 Phys. Rev. Lett. 92 106801 [64] Giamarchi T 2004 Quantum Physics in One Dimension (Oxford: Oxford Science) [65] Gold A and Ghazali A 1990 Phys. Rev. B 41 7626 [66] L¨owdin P O 1955 Phys. Rev. 97 1474 [67] Ceperley D M 1995 Rev. Mod. Phys. 67 279 [68] Lyubartsev A P and Vorontsov-Velyaminov P N 1993 Phys. Rev. A 48 4075
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Thermally enhanced Wigner oscillations in two-electron 1D quantum dots
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 505301 (http://iopscience.iop.org/0953-8984/26/50/505301) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 147.8.31.43 This content was downloaded on 15/06/2017 at 16:44 Please note that terms and conditions apply.
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 505301 (8pp)
doi:10.1088/0953-8984/26/50/505301
Thermally enhanced Wigner oscillations in two-electron 1D quantum dots F Cavaliere1,2 , N Traverso Ziani3 , F Negro4 and M Sassetti1,2 1
Dipartimento di Fisica, Universit`a di Genova, Via Dodecaneso 33, 16146 Genova, Italy CNR-SPIN, Via Dodecaneso 33, 16146 Genova, Italy 3 Institute for Theoretical Physics and Astrophysics, University of Würzburg, 97074 W¨urzburg, Germany 4 Dipartimento di Fisica dell’Universi`a di Pisa and INFN—Sezione di Pisa, Largo Pontecorvo 3, I-56127 Pisa, Italy 2
E-mail: [email protected] Received 21 July 2014, revised 27 October 2014 Accepted for publication 28 October 2014 Published 24 November 2014 Abstract
Motivated by a recent experiment (Pecker et al 2013 Nat. Phys. 9 576), we study the stability, with respect to thermal effects, of Friedel and Wigner density fluctuations for two electrons trapped in a one-dimensional quantum dot. Diagonalizing the system exactly, the finite-temperature average electron density is computed. While the weak and strong interaction regimes display a Friedel oscillation or a Wigner molecule state at zero temperature, which as expected smear and melt as the temperature increases, a peculiar thermal enhancement of Wigner correlations in the intermediate interaction regime is found. We demonstrate that this effect is due to the presence of two different characteristic temperature scales: TF , dictating the smearing of Friedel oscillations, and TW , smoothing Wigner oscillations. In the early Wigner molecule regime, for intermediate interactions, TF < TW leading to the enhancement of the visibility of Wigner oscillations. These results complement those obtained within the Luttinger liquid picture, valid for larger numbers of particles. Keywords: quantum dot, Wigner molecule, correlated electrons (Some figures may appear in colour only in the online journal)
WM [2, 3]. Few works have considered the effects of finitetemperature, essentially demonstrating how spherical [27] and circular [28, 29] Wigner molecules eventually melt resulting in an uncorrelated, liquid-like state. In 1D quantum dots, several numerical techniques have also been adopted, such as density matrix renormalization group analysis (DMRG) [30], density functional theory [31–36], quantum Montecarlo [37], HartreeFock [38] and exact diagonalization [39–44]. In addition, the restricted phase space allows also for analytical methods such as the Bethe ansatz solution [31–34, 45] or spin-coherent [46–51] and spin-incoherent [52–57] Luttinger liquids [58]. Due to the broken translational symmetry, signatures of the WM can already be found in the electron density. This is particularly interesting since the possibility of imaging the electron density in local transport experiments has been demonstrated [59, 60]. The peaks of the linear conductance are in fact shifted, with respect to the gate potential, when the dot is coupled to a local capacitive coupling [59–61].
1. Introduction
Eighty years since their seminal proposal, Wigner crystals [1] and their finite-size analogue Wigner molecules (WM) [2, 3] still represent intriguing systems, only partially understood so far. In particular, Wigner molecules have recently triggered a lot of interest with several experiments reporting their observation in two-dimensional (2D) semiconducting heterostructures [4, 5], in semiconducting circular [6] quantum dots [7] and in one-dimensional (1D) quantum wires [8–11] and carbon-nanotubes (CNTs) [12, 13]. Due to strong electronic correlations, WMs have often been studied numerically [2, 3], with a particular focus on circular quantum dots [14–25]. In such 2D systems, the electron density only shows weak signatures of a radial ordering in the molecular state, while no angular oscillations are found due to the rotational symmetry. Density density correlation functions [26] are hence required to investigate the 0953-8984/14/505301+08$33.00
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The shift depends on the position of the local coupling, and its characterization allows for the extraction of the electron density profile inside the dot. It is worth pointing out that, on the other side, a local tunnel coupling reveals the local density of states instead of the electron density. The two quantities can be significantly different [41, 49]. For 1D finite systems, most of the theoretical efforts have concentrated on the zero-temperature regime. Here, a competition in the density between finite size effects, inducing Friedel oscillations with wavevector 2kF (kF the Fermi wavevector) and electron correlations inducing Wigner oscillations with 4kF momentum occurs. Friedel oscillations produce N/2 (resp. (N + 1)/2) peaks for even (resp. odd) N , while Wigner oscillations are always characterized by N distinct peaks. At zero temperature, Friedel oscillations dominate for weak interactions, while Wigner oscillations mark the formation of a WM in the strong interaction regime. This behaviour is confirmed by both analytical and numerical results [31, 39, 43, 46–49]. It has been pointed out that, in the case of short-range delta-like electron interactions in 1D, even in the presence of N peaks in the density, the densitydensity correlation functions may reveal a liquid-like state of the dot [32]. However, for interactions whose range exceeds the size of the 1D quantum dot such as those studied in the present paper, N distinct peaks in the density are associated with the presence of a well defined molecular state [32]. More intriguing is the finite temperature behaviour, which has been analysed by means of the Luttinger liquid theory in the regime of large numbers of particles [49, 57]. There, it is shown that Friedel oscillations are stable at energy scales smaller than those of spin collective excitations εσ = kB Tσ , while Wigner oscillations are stable at energies smaller than those of the plasmonic charge modes [50, 51] ερ = kB Tρ . Here, Tσ < Tρ for an interacting system, with Tσ , related to the exchange interaction among electrons, exponentially suppressed in the strong interaction limit [62, 63]. Three regimes exist. For T < Tσ , both Friedel and Wigner are stable and the relative importance is essentially set by the interaction strength. For Tσ < T < Tρ , Friedel oscillations smear out allowing the Wigner oscillations to be enhanced in the total density. In this regime, if Tσ and Tρ are sufficiently well separated, a rather counter-intuitive effect occurs, namely an enhancement of the Wigner oscillations as temperature increases, even if at T = 0 the density is dominated by Friedel oscillations. Note that for weak interactions, Tσ ∼ Tρ , while increasing interactions Tσ → 0 while Tρ sharply increases over the weakly interacting case. Thus, to observe the enhancement of the Wigner oscillations, an intermediate-to-strong interaction regime is best suited. Finally, for T > Tρ the Wigner molecule melts and the density no longer exhibits peculiar oscillations. The above regime describes a dot with a large number of particles. On the other hand, few-electron quantum dots also represent a natural playground where one can investigate the effects of strong correlations—requiring numerical techniques for their theoretical investigation. Stringent evidence of the formation of a 1D Wigner molecule composed by two electrons in a carbon nanotube was provided very recently [13]. The proof of the existence
of such a molecule relies on the observation—via transport measurement—of a sharp collapse of the energy gap , between the ground state and the first excited state. Indeed, one of the strongest spectroscopical signatures of the formation of a Wigner molecule is a tendency towards the degeneracy of all lowest-lying spin multiplets (which implies → 0 for two electrons) as the interaction strength increases [2, 13, 39]. Note that in CNTs the degeneracies of the ground state and the first excited state are 6 and 10 respectively [13] when neglecting spin–orbit interaction. In the presence of spin–orbit coupling, the sixteen states further reorganize into three multiplets with degeneracies 4,8,4 [41]). In the case of semiconducting quantum wires with negligible spin orbit coupling, the ground state is non degenerate, and the first excited states are characterized by a threefold degeneracy. Stimulated by this experiment, we analyse the stability of Friedel and Wigner oscillations in a two-electron quantum dot as a function of temperature and interaction strength. Employing an exact diagonalization technique, we evaluate the dot electron density as a function of temperature. We will demonstrate that the results obtained for large particle numbers have a wide range of validity and that a similar phenomenon of enhancement of the Wigner oscillations occurs also in the opposite regime of two electrons. Although our model applies directly to the case of quantum dots built in semiconducting quantum wires, we do not expect qualitative differences for the case of CNTs, the different degeneracy of the ground and excited states leading to a renormalization of the characteristic temperatures discussed later in this work. Our main conclusions are the following. For zero temperature we recover the standard behaviour of a continuous transition from a singly-peaked density, in which the Friedel contribution is dominating at weak interaction, to a density pattern with two peaks characteristic of the formation of a Wigner molecule for strong interactions. For finite temperature, however, an intermediate regime of the Coulomb interaction strength and an intermediate regime of temperatures exists, in which the visibility of the Wigner oscillations is actually enhanced with respect to the zero temperature case. Outside this regime of interaction strengths, namely for weak or strong interactions, increasing the system temperature leads to a more conventional melting of either the Friedel or the Wigner oscillations. This behaviour is schematically represented in the scheme shown in figure 7. The threshold values for the interaction strength and the temperature are defined and discussed in detail in section 3. The outline of the paper is as follows: in section 2.1 the model and general features of the numerical method are discussed. The results are presented and discussed in section 3. In section 4 conclusions are drawn. 2. Model and method 2.1. Model
The system consists of two electrons harmonically confined in a 1D quantum dot, interacting via the potential V (y) [41]. The 2
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Hamiltonian is Hˆ =
−
i=1,2
absence of a magnetic field, all triplet states are degenerate. Furthermore, ϕn (R) are the eigenfunctions of a free harmonic oscillator with dimensionless energy ε˜ n = n + 1/2. To find m (r) and the corresponding energy spectrum ˜m we employ a numerical exact diagonalization technique, projecting the hamiltonian sector containing r on the space spanned by M bare harmonic oscillator states. Up to M = 50 states have been employed in our calculation, and convergence with a relative error below 5 · 10−3 has been achieved. Our task is to investigate the behaviour of the dimensionless electron density ρ(x) ˆ [26, 66]
h ¯ 2 ∂2 mω2 2 + yˆ + Vˆ (yˆ2 − yˆ1 ) , 2m ∂ yˆi2 2 i
(1)
where yˆi is the position operator of the i-th electron and the effective 1D soft Coulomb interaction is [41] c Vˆ (y) ˆ = . 2 λ + yˆ 2
(2)
Here, λ is a short-distance cutoff due to the effective width of the channel [46, 64, 65] while c is the interaction strength. To obtain the spectrum and wavefunctions of the system, we employ an exact diagonalization method [39–44]. This method provides results with an excellent precision in all ranges of parameters considered in this paper, while retaining a bearable level of computational complexity. Introducing the ˆ and relative (ˆr ) coordinates dimensionless centre of mass (R) yˆ1 + yˆ2 yˆ1 − yˆ2 Rˆ = ; rˆ =
(3)
1 1 ∂2 1 c˜ 1 ∂2 + Rˆ 2 − + rˆ 2 + hˆ = − 2 ∂ Rˆ 2 2 2 ∂ rˆ 2 2 ˜λ2 + rˆ 2
(4)
ρ(x) ˆ =
1 δ x − xˆi , 2 i=1,2
(5)
at finite temperature. Here, we have introduced xˆi = yˆi /. The quantum average of the density is ˜ˆ e −β h ρ(x) ˆ (6) ρ(x) = Tr Z
˜ˆ where Z = Tr e−β h with β˜ = E0 /kB T and T is the system temperature. One finally obtains 2 1 −β˜ ˜n,m e Dm dz n,m (x + z, x − z) (7) ρ(x) = Z n,m
√ with = 2¯h/mω the typical length scale, one obtains
¯ ω and λ˜ = λ/, c˜ = c/(E0 ) where hˆ = Hˆ /E0 with E0 = h are dimensionless parameters governing the interaction. Note that in the case of a one dimensional quantum dot, the condition λ˜ 1 must hold. The coordinates R and r decouple, which allows searching for solutions of the form n,m (R, r; σ1 , σ2 ) = ϕn (R) m (r)|χm (σ1 , σ2 ) with energy spectrum ˜n,m = ε˜ n + ˜m . Here, σi are the spin directions for i-th electron and χm (σ1 , σ2 ) is the total spin wavefunction of the system. Since parity is a good quantum number and ϕn (R) is symmetric under particle exchange, the spin wavefunction is uniquely determined by the quantum number m, with χm (σ1 , σ2 ) being a singlet or a triplet according to m (r) being even or odd. Note that in the absence of a magnetic field, all triplet states are degenerate. Furthermore, ϕn (R) are the eigenfunctions of a free harmonic oscillator with dimensionless energy ε˜ n = n + 1/2. To find m (r) and the corresponding energy spectrum ˜m we employ a numerical exact diagonalization technique, projecting the hamiltonian sector containing r on the space spanned by M bare harmonic oscillator states. Up to M = 50 states have been employed in our calculation, and convergence with a relative error below 5 · 10−3 has been achieved. Where hˆ = Hˆ /E0 with E0 = h ¯ ω and λ˜ = λ/, c˜ = c/(E0 ) are dimensionless parameters governing the interaction. The coordinates R and r decouple, which allows searching for solutions of the form n,m (R, r; σ1 , σ2 ) = ϕn (R) m (r)|χm (σ1 , σ2 ) with energy spectrum ˜n,m = ε˜ n + ˜m . Here, σi are the spin directions for i-th electron and χm (σ1 , σ2 ) is the total spin wavefunction of the system. Since parity is a good quantum number and ϕn (R) is symmetric under particle exchange, the spin wavefunction is uniquely determined by the quantum number m, with χm (σ1 , σ2 ) being a singlet or a triplet according to m (r) being even or odd. Note that in the
Z=
1
˜ 2 sinh (β/2)
m
˜
e−β ˜m Dm ,
(8)
where Dm is the degeneracy of (n,m) (R, r), with Dm = 1 if the total spin is a singlet, Dm = 3 if the total spin is a triplet. The summations are performed numerically, truncating the number of involved states when convergence with a relative precision of less than 10−2 is achieved. 2.2. Spectrum and zero-temperature density
Figure 1 shows the energy spectrum (referred to the ground state) ˜n,m − ˜0,0 as a function of c˜ in a wide range of interaction strength spanning, as will be shown, from noninteracting electrons to a strongly interacting regime. In the non-interacting limit, the spectrum ˜n,m = n + m + 1 exhibits a shell structure with degenerate levels at energy k + 1 satisfying the condition n + m = k (n, m, k 0). The degeneracy of the k-th shell (including spin degeneracy) is 2k + 1 if k is even and 2k + 2 if k is odd. As such, the singlet ground state is non-degenerate. In the presence of Coulomb interactions, several level crossings occur. However, the dot ground state always remains a spin singlet and the lowest-lying excited state is always a triplet. Ultimately, for c˜ 1 the spectrum rearranges in a sequence of fourfold degenerate spin multiplets composed of a singlet and a triplet. This reordered spectrum is described by √ corr − ˜0,0 = p + 3q , (9)
˜p,q with p, q 0. Let us now summarize the behaviour of the system in the zero temperature limit. Figure 2 shows the zero temperature 3
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Figure 1. Energy spectrum ˜n,m − ˜0,0 as a function of the interaction parameter c. ˜ Solid (blue) lines represent a singlet state, dashed (red) lines a triplet state. Here, λ˜ = 0.02.
Figure 3. Density profile ρ(x) (units −1 ) as a function of x for c˜ = 0.05 (solid, black); c˜ = 0.3 (dashed, red); c˜ = 3 (dotted, blue); c˜ = 10 (dash–dotted, green). Other parameters: T˜ = 0 and λ˜ = 0.02.
the fourfold degeneracy of the spectrum in this regime— corresponding to the four distinct manners to arrange spins on the two sites of the Wigner molecule. 3. Results 3.1. Thermal–Wigner enhancement
Let us now turn to the question of the stability of Friedel and Wigner oscillations against thermal effects, by studying the temperature dependence of the electron density. Figure 4(a) shows the weakly interacting case c˜ = 0.1, in the Friedel regime, for five different temperatures. Increasing T˜ the density smears and broadens. Melting is a continuous process, as is typical in finite-size 1D systems, and is due to the thermal activation of excited states. In particular, the lowest lying triplet excitation which lies at energy ε˜ 0,1 − ε˜ 0,0 < 1 is the first to come into play, see figure 1. We can thus customarily identify a temperature scale TF (c) ˜ for melting in the Friedel ˜ ∼ (c)/E ˜ , where (c) ˜ is the lowest-lying regime as TF (c) 0 singlet–triplet energy gap. Increasing c˜ (while still retaining c˜ < c∗ ) one observes a qualitatively similar behaviour, with a melting process which however occurs at a lower temperature, in agreement with the fact that the singlet–triplet gap shrinks to zero as c˜ increases (not shown). In the opposite case of very strong interactions, the system is dominated by Wigner correlations. For example, consider the case c˜ = 3 shown in figure 4(b). Here, the two sharp peaks become weaker as T˜ is increased and, for large enough T˜ , the density exhibits a single peak becoming similar to the thermally smeared state discussed above. This smearing is a continuous process, analogous with the melting of Friedel oscillations. Also, here we can define a temperature scale TW such that for T˜ > TW the two peaks disappear and thus the Wigner molecule is melted. √ We observe that in this strong interaction regime for T˜ ≈ 3 the√density flattens out, allowing us to identify TW (c˜ = 3) = 3. This energy scale corresponds to the radial breathing mode of the Wigner molecule and thus suggests that the leading mechanism responsible for the melting are vibrations of the two electrons about their equilibrium positions.
Figure 2. Density profile ρ(x) (units −1 ) as a function of x and c˜
for T˜ = 0 and λ˜ = 0.02.
electron density of the system as a function of x, in the same range of c˜ spanned in figure 1. It is clear that the density exhibits a single peak for weak interactions, while for strong interactions, two peaks are present. To be more quantitative, figure 3 shows ρ(x) for selected values of the interaction strength c. ˜ For weak interactions, the single peak in the electron density corresponds to the physical picture of two electrons packed near the centre of the dot with antiparallel spins. This is the hallmark of the Friedel oscillation for a 1D dot with two electrons. As interactions increase, the density broadens while the peak gets lower. At a critical value c = c∗ ≈ 0.3 the density flattens out and for c > c∗ two small bumps emerge signalling the incipience of a Wigner molecule. Indeed, for c c∗ two fully developed peaks appear, distinctive of the Wigner molecule. The strongly correlated molecular state allows us to understand the spectrum reordering occurring for c c∗ . The quantum number p in equation (9) describes the centre of √ mass excitations of the molecule, while the mode at energy 3 with quantum number q is connected to the radial breathing mode involving the relative coordinate only. In this strongly correlated regime, the overlapping between the electrons’ wavefunctions near the centre of the dot is very small and, as a result, exchange interaction is strongly suppressed. The energy gap between different spin states collapses, which motivates 4
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Figure 5. Density profile ρ(x) (units −1 ) as a function of x for
(a) T˜ = 0.0 (black); T˜ = 0.1 (red); T˜ = 0.3 (green); T˜ = 0.6 (blue); T˜ = 1.2 (orange) and (b) T˜ = 0.0 (black); T˜ = 0.1 (red); T˜ = 0.4 (green); T˜ = 0.7 (blue); T˜ = 1.6 (orange). The arrows mark the evolution of the density as temperature increases. In panel (a) c˜ = 0.4, in panel (b) c˜ = 0.6. In all panels λ˜ = 0.02.
Figure 4. Density profile ρ(x) (units −1 ) as a function of x for (a) T˜ = 0.0 (black); T˜ = 0.4 (red); T˜ = 0.8 (green); T˜ = 1.2 (blue); T˜ = 1.6 (orange) and (b) T˜ = 0.0 (black); T˜ = 0.6 (red); T˜ = 1.2 (green); T˜ = 1.73 (blue); T˜ = 2.0 (orange). The arrows mark the evolution of the density as temperature increases. In panel (a) c˜ = 0.1, in panel (b) c˜ = 3. In all panels λ˜ = 0.02.
regime of Wigner enhancement, such values are forbidden by the condition λ˜ 1, needed to ensure the 1D character of the dot. To understand in more detail how the visibility of the Wigner oscillations is enhanced by temperature, the wavefunction of the ground state and of the first three excited states of the dot are shown in figure 6. As is clear, the lowest lying triplet excited state has a much more pronounced oscillating pattern than the singlet ground state. Indeed we have checked that as soon as c > c∗ the ground state wavefunction begins to show a double bump structure, whereas the first excited state already has a more developed peak ˜ structure. Since the singlet–triplet energy gap, i.e. TF (c), is already of the order of 0.1 in the range c∗ < c˜ < cW , even a slight thermal excitation promotes the population of the triplet state, which produces a more marked oscillating pattern of the density. When T˜ > ε˜ 1,0 − ε˜ 0,0 = 1 the next excited states become populated, promoting the occupation of the centre of the dot even further as is clear by inspecting their ˜ is found to lie between wavefunctions (see figure 6), and TW (c) the energy gap of the second and the third excited states. As c˜ increases towards cW the Wigner character of the ground state gets more pronounced and the wavefunctions of singlet and triplet states tend to exhibit peaks of similar height and position (not shown). As a consequence, the temperatureinduced enhancement becomes less visible and the melting
We have checked that the simple melting of Friedel and Wigner oscillations described above occurs only in specific ranges of interaction strength. More specifically, the simple smearing of the Friedel peak as T˜ > TF occurs for c < c∗ , while the monotonous disappearance of the Wigner molecule oscillations in the density for which T˜ > TW leaving place to a single peak only occurs for large enough interactions c > cW ∼ 1. We turn now to the more interesting regime of intermediate interactions c∗ < c˜ < cW . Consider figure 5(a), for c∗ < c˜ = 0.4 < cW : here, a double-peak structure can be detected already for T = 0 as discussed above. However, in sharp contrast to what was already discussed, increasing the temperature leads to an enhancement of the visibility of the Wigner molecule, with the double-peak structure becoming more pronounced as T˜ is raised. Only for large enough T˜ TW (c˜ = 0.4) ≈ 1.1 will the Wigner molecule eventually melt and the two peaks disappear. Figure 5(b) shows the case c∗ < c˜ = 0.6 < cW which exhibits the same qualitative behaviour, with TW (c) ˜ ≈ 1.4. The same qualitative behaviour is observed throughout the whole c∗ < c˜ < cW range. Moreover, in the Wigner enhancement regime, the effect of varying λ˜ is weak. In fact the properties of the system strongly depend on λ˜ only starting from λ˜ ∼ c˜1/3 , as can be seen by studying the minima of the potential energy. However, in the 5
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Figure 6. Wavefunction modulus squared | n,m (x)|2 (units −1 ) as a function of x for (a) n = m = 0, ground state (singlet); (b) n = 0, m = 1, first excited state (triplet); (c) n = 1, m = 0, second excited state (singlet); (d) n = m = 1, third excited state (triplet). In all panels c˜ = 0.4, λ˜ = 0.02.
Figure 7. Qualitative scheme of the different regimes of the two electrons quantum dots as a function of the dimensionless interaction strength c˜ and temperature T˜ . The region labelled F represents the predominance of the Friedel contribution, F/W denotes a regime where Friedel and Wigner coexist, W marks the predominance of the Wigner molecule, Enhanced Wigner the regime where Wigner correlations are enhanced by temperature. For large temperature, the two-electron states melt regardless of their character at small temperatures. The curves TF,W (c) ˜ represent the qualitative behaviour of these quantities, as inferred by our numerical observations.
temperature TW (c) ˜ increases, since the contribution of the third excited state becomes crucial to cancel the double-peak structure of the density. Finally, when c˜ = cW the singlet and ˜ ≈ 0 and no triplet wavefunctions essentially coincide, TF (c) enhancement occurs. ˜ < T˜ < We dub the range c∗ < c˜ < cW and TF (c) ˜ (see figure 7) the Wigner enhancement regime in which TW (c) the Wigner features of the electron density become more prominent as temperature is increased. The existence of this regime is clearly due to the strong separation of the ˜ and TW (c) ˜ induced by interactions. This behaviour TF (c) is in perfect agreement with the findings of the Luttinger theory [49, 52, 56, 57]. This provides strong evidence that for N 3 particles also, one would observe a qualitatively similar behaviour with a universal trend of enhancement of the Wigner correlations. This will be the subject of future numerical investigations. A summary of our conclusions is schematically represented in the picture shown in figure 7. In order to identify different regimes, the sharp lines T˜ = TF (c), T˜ = TW (c), c = c∗ and c = cW have been drawn even though it is clear that all the transitions discussed here are continuous processes, occurring in proximity of the threshold values denoted in the figure.
One of the possible techniques could involve a linear transport measurement in the presence of a charged AFM tip [48, 59, 60] which has also been suggested as a possible way to obtain direct information about the spatial fluctuations of the electron density in the Wigner molecule regime. When this kind of experiment is performed with a tip weakly coupled to the dot, the chemical potential of the latter (and thus the position of the linear conductance peak) is predicted to shift by an amount proportional to the electron density. One of the limiting factors of this approach would be the thermal broadening of conductance traces. By converting the peakto-valley swing between the central minimum and the lateral maxima of the density in the Wigner enhancement regime, see figure 5, to energy scales for a typical setup [59, 60], we obtain a chemical potential shift slightly larger than 1 K, which should allow us to observe the predicted effect at least up to the temperature when Wigner oscillations in the density begin to surmise.
3.2. Observability of the enhancement of Wigner oscillations
4. Conclusion
We conclude by briefly commenting on the observability of the enhancement of Wigner phenomena predicted here. A typical ˜ in the range c∗ < c˜ < cW is T˜F ∼ 0.1. estimate for T˜F (c) Assuming E0 ≈ 1 meV, the maximum enhancement would occur at a temperature of about 1 K, while the complete melting of the Wigner molecule would occur at a temperature that is one order of magnitude higher.
We have studied the influence of temperature on the states of two correlated electrons harmonically confined in a one-dimensional quantum dot. By exactly diagonalizing the system, the finite-temperature density profile has been calculated. 6
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In the weak interaction limit, the density exhibits a centred maximum attributed to finite-size Friedel effects, which is smeared by temperature. For strong interactions, the lowtemperature density profile confirms the formation of a Wigner molecule, with two marked peaks symmetrically placed with respect to the potential well. Increasing the temperature melts the Wigner molecule resulting in the disappearance of the double-peak structure. The case of intermediate interactions is peculiar: at low temperature, the system exhibits very weak Wigner molecule features. However, the latter are enhanced as the temperature is increased. The above behaviour has been interpreted introducing two temperature scales TF and TW , which resemble those introduced in the Luttinger liquid theory of the system [49, 56, 57], valid for a large number of electrons. Indeed, all the results presented in this paper confirm those obtained in the Luttinger theory, suggesting that the Wigner enhancement mechanism could be a universal phenomenon. In order to prove this universality, it would be interesting to extend this analysis to numbers of particles N 3. More general-purpose techniques such as path-integral Montecarlo [67, 68] may be employed to tackle this problem.
[17] Nishi Y, Maksym P A, Austing D G, Hatano T, Kouwenhoven L P, Aoki H and Tarucha S 2006 Phys. Rev. B 74 033306 [18] Rontani M, Cavazzoni C, Bellucci D and Goldoni G 2006 J. Chem. Phys. 124 124102 [19] Harju A, Saarikoski H and R¨as¨anen E 2006 Phys. Rev. Lett. 96 126805 [20] Yannouleas C and Landman U 2000 Phys. Rev. Lett. 85 1726 [21] De Giovannini U, Cavaliere F, Cenni R, Sassetti M and Kramer B 2007 New J. Phys. 9 93 [22] De Giovannini U, Cavaliere F, Cenni R, Sassetti M and Kramer B 2008 Phys. Rev. B 77 035325 [23] Cavaliere F, De Giovannini U, Sassetti M and Kramer B 2009 New J. Phys. 11 123004 [24] Puente A, Serra L and Nazmitdinov R G 2004 Phys. Rev. B 69 125315 [25] Rontani M, Molinari E, Maruccio G, Janson M, Schramm A, Meyer C, Matsui T, Heyn C, Hansen W and Wiesendanger R 2007 J. Appl. Phys. 101 081714 [26] Giuliani G F and Vignale G 2005 Quantum Theory of the Electron Liquid (Cambridge: Cambridge University Press) [27] Lozovik Yu E, Volkov S Yu and Willander M 2003 Solid State Commun. 125 127 [28] Filinov A V, Bonitz M and Lozovik Yu E 2001 Phys. Rev. Lett. 86 3851 [29] B¨oning J, Filinov A, Ludwig P, Baumgartner H, Bonitz M and Lozovik Yu E 2008 Phys. Rev. Lett. 100 113401 [30] S¨offing S A, Bortz M, Schneider I, Struck A, Fleischhauer M and Eggert S 2009 Phys. Rev. B 79 195114 [31] Xianlong G 2012 Phys. Rev. A 86 023616 [32] Wang J-J, Li W, Chen S, Xianlong G, Rontani M and Polini M 2012 Phys. Rev. B 86 075110 [33] Xianlong G and Asgari R 2008 Phys. Rev. A 77 033604 [34] Magyar R J and Burkev K 2004 Phys. Rev. A 70 032508 [35] Abedinpour S H, Polini M, Xianlong G and Tosi M P 2007 Eur. Phys. J. B 56 127 [36] Abedinpour S H, Polini M, Xianlong G and Tosi M P 2007 Phys. Rev. A 75 015602 [37] Casula M, Sorella S and Senatore G 2006 Phys. Rev. B 74 245427 [38] Mueller E J 2005 Phys. Rev. B 72 075322 [39] H¨ausler W and Kramer B 1993 Phys. Rev. B 47 16353 [40] Jauregui K, Häusler W and Kramer B 1993 Europhys. Lett. 24 581 [41] Secchi A and Rontani M 2009 Phys. Rev. B 80 041404 [42] Secchi A and Rontani M 2010 Phys. Rev. B 82 035417 [43] Secchi A and Rontani M 2012 Phys. Rev. B 85 121410 [44] Secchi A and Rontani M 2013 Phys. Rev. B 88 125403 [45] Bethe H A 1931 Z. Phys. 71 205 [46] Schulz H J 1993 Phys. Rev. Lett. 71 1864 [47] Safi I and Shulz H J 1999 Phys. Rev. B 59 3040 [48] Mantelli D, Cavaliere F and Sassetti M 2012 J. Phys.: Condens. Matter 24 432202 Gambetta F M, Traverso Ziani N, Cavaliere F and Sassetti M 2014 Europhys. Lett. 107 47010 [49] Traverso Ziani N, Cavaliere F and Sassetti M 2013 New J. Phys. 15 063002 [50] Sassetti M and Kramer B 1998 Phys. Rev. Lett. 80 1485 [51] Cavaliere F, Braggio A, Stockburger J T, Sassetti M and Kramer B 2004 Phys. Rev. Lett. 93 036803 Sassetti M and Kramer B 1996 Phys. Rev. B 54 R5203 [52] Fiete G A 2007 Rev. Mod. Phys. 79 801 [53] Fiete G A, Le Hur K and Balents L 2006 Phys. Rev. B 73 165104 [54] Fiete G A, Le Hur K and Balents L 2005 Phys. Rev. B 72 125416 [55] Matveev K A 2004 Phys. Rev. Lett. 92 106801 [56] Matveev K A, Furusaki A and Glazman L I 2007 Phys. Rev. Lett. 98 096403
Acknowledgments
We gratefully acknowledged the support of EU-FP7 via Grant No. ITN-2008-234970 NANOCTM and of MIUR via MIUR-FIRB2012—Project HybridNanoDev, Grant No.RBFR1236VV. FN acknowledges the financial support from the EU under project Hadron Physics 3 (Grant Agreement no. 283286). References [1] [2] [3] [4] [5] [6] [7]
[8] [9] [10] [11] [12] [13] [14] [15] [16]
Wigner E 1934 Phys. Rev. 46 1002 Reimann S M and Manninen M 2002 Rev. Mod. Phys. 74 1283 Yannouleas C and Landman U 2007 Rep. Prog. Phys. 70 2067 Andrei E Y, Deville G, Glattli D C, Williams F I B, Paris E and Etienne B 1998 Phys. Rev. Lett. 60 2765 Piot B A, Jiang Z, Dean C R, Engel L W, Gervais G, Pfeiffer L N and West K W 2008 Nat. Phys. 4 936 Kalliakos S, Rontani M, Pellegrini V, Garcia C P, Pinczuk A, Goldoni G, Molinari E, Pfeiffer L N and West K W 2008 Nat. Phys. 4 467 Kouwenhoven L P, Marcus C M, McEuen P L, Tarucha S, Westervelt R M and Wingreen N S 2007 Electron Transport in Quantum Dots (NATO Advanced Studies Institute, Series E: Applied Science) ed L L Sohn et al (Dordrecht: Kluwer) p 105 Ellenberger C et al 2006 Phys. Rev. Lett. 96 126806 Singha A et al 2010 Phys. Rev. Lett. 104 246802 Kristinsd´ottir L et al 2011 Phys. Rev. B 83 041101 Meyer J S and Matveev K A 2009 J. Phys.: Condens. Matter 21 023203 Deshpande V V and Bockrath M 2008 Nat. Phys. 4 314 Pecker S, Kuemmeth F, Secchi A, Rontani M, Ralph D C, McEuen P L and Ilani S 2013 Nat. Phys. 9 576 Dineykhan M and Nazmitdinov R G 1997 Phys. Rev. B 55 13707 Hawrylak P and Pfannkuche D 1993 Phys. Rev. Lett. 70 485 Tavernier M B, Anisimovas E, Peeters F M, Szafran B, Adamowski J and Bednarek S 2003 Phys. Rev. B 68 205305 7
J. Phys.: Condens. Matter 26 (2014) 505301
F Cavaliere et al
[57] Traverso Ziani N, Cavaliere F and Sassetti M 2013 Europhys. Lett. 102 47006 [58] Haldane F D 1981 Phys. Rev. Lett. 47 1840 [59] Boyd E E and Westervelt R M 2011 Phys. Rev. B 84 205308 [60] Traverso Ziani N, Cavaliere F and Sassetti M 2012 Phys. Rev. B 86 125451 [61] Wach E, Zebrowski D P and Szafran B 2013 J. Phys.: Condens. Matter 25 335801
[62] Hausler W 1996 Z. Phys. B 99 551 [63] Matveev K A 2004 Phys. Rev. Lett. 92 106801 [64] Giamarchi T 2004 Quantum Physics in One Dimension (Oxford: Oxford Science) [65] Gold A and Ghazali A 1990 Phys. Rev. B 41 7626 [66] L¨owdin P O 1955 Phys. Rev. 97 1474 [67] Ceperley D M 1995 Rev. Mod. Phys. 67 279 [68] Lyubartsev A P and Vorontsov-Velyaminov P N 1993 Phys. Rev. A 48 4075
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