Induced Wigner crystallization in a coupled electron-hole quantum wire system at finite-temperature

We explore the possibility of induced Wigner crystallization in a coupled electron-hole (e-h) quantum wire system at finite-temperature T by calculating the intra- and inter-wire static structure factors in the dynamic mean-field approximation of Singwi et al (the qSTLS theory). We find that owing to the dynamics of severely correlated particles, the e-h system may favour an induced phase transition from quantum liquid to the Wigner crystal state at a non-zero T and sufficiently low particle (electron/hole) number density in the close proximity of two wires. On the other hand, the static STLS approximation does not predict such phase transition due to complete neglect of the dynamics of particle correlations.


Introduction
A coupled quantum wire system consists of two parallel quantum nano wires placed at sufficiently small (∼ nm) inter-wire distance.Depending upon the nature of charge carriers in both the wires, they are regarded as coupled electron-electron (e-e) or electron-hole (e-h) or hole-hole (h-h) quantum wire systems.
The coupled quantum wires have generated a great deal of theoretical [1]- [6] and experimental [7,8] curiosity over last few decades.Authors in Ref. [1][2][3] have predicted the charge-density-wave (CDW) instability in coupled quantum wire systems by including both the intra-and inter-wire particle correlations through the Singwi, Tosi, Land and Sjolander (STLS) approximation [9] at absolute zero.Very recently, Sharma et al [4] observed the CDW instability in a coupled e-h quantum wire system at sufficiently low temperature and particle number density using the STLS theory.On the other hand, no such instability was found in the coupled e-e quantum wire system at any non-zero T.
Notably, the correlations among particles are assumed to be static (i.e., time-independent) in the STLS theory.However, the dynamics of particle correlations becomes quite significant to determine the many-body properties of the system in the strongly correlated regime (i.e., low temperature, particle number density, transverse wire width and inter-wire separation), especially the Wigner crystallization.Though, Saini et al [5] included the dynamics of particle correlations through the quantum version of the STLS approach (the qSTLS theory [10]) to predict the CDW state in the ground-state of coupled ee and e-h quantum wire systems, yet, assumed equal effective mass of holes and electrons ( ℎ *   * ⁄ =1).Nevertheless, this may not be true in a real physical situation such as GaAs based e-h quantum wire system wherein  ℎ *   * ⁄ ≈ 7. Keeping this point in mind, Moudgil et al [6] considered the massasymmetry effect and employed the qSTLS approximation to study the ground-state properties of a coupled e-h wire system.It was deduced that the e-h system may favour a phase transition from quantum liquid to the CDW state or the Wigner crystallization at absolute zero in the close vicinity of two wires, and at sufficiently low particle (electron and hole) number density.The localization of holes in a coupled e-h system was also predicted by Steinberg and co-workers [7] at a hole density of 2 × 10 7  −1 , electron density 6 × 10 7  −1 and T = 0.25K.
Motivated by the experimental work of Steinberg et al, here we explore the possibility of the formation of an induced Wigner crystal (WC) phase in a coupled e-h quantum wire system at a non-zero temperature by employing the qSTLS approximation.

Results and discussion
We use a dimensionless system of units in which the wave vector  is taken in units of the electron's Fermi wave vector   , wire width  and inter-wire spacing  in electron's effective Bohr radius  0 * [=  0 ℏ 2 /(  *  2 )], ω and µ in electron's Fermi energy   and  in electron's Fermi temperature   (i.e.   ⁄ = ).We choose effective mass ratio of hole and electron to be 7 i.e.,  ℎ *   * ⁄ = 7, appropriate to a GaAs based e-h wire system, and   =  ℎ .The number density   of electrons or holes in each of the wire is taken to be same and is described by a dimensionless parameter   = 1/ (2   0 * );   = / (total no. of electrons/holes per unit length of the wire).Throughout our calculations, we set ℏ,   and  0 equal to unity.
Further, we use the subscripts ′ee′ and ′hh′ for the presentation of intra-wire static structure factors of the electron and hole wire, respectively.Nonetheless, the subscript 'e-h' is used for inter-wire static structure factor.The set of equations ( 1), ( 3) and ( 6) is solved numerically in a self-consistent manner for the intraand inter-wire static structure factors   ′ (; ) of the coupled e-h quantum wire system over a range of particle (electron/hole) number density (  ) for a fixed value of wire width and inter-wire separation.We accepted the numerical solution when convergence in the results of   ′ (; ) was better than 0.0001%, at each q and  in the chosen grid of qand -points.
In figure 1(a) and 1(b), we plot the intra-wire static structure factor   (; ) of the coupled e-h system at b= 0 * , d=2 0 * ,  = 0.1 and indicated   values for the eand h-wire, respectively.It is found that particle (electron or hole) correlations grow directly with increasing coupling (  ) among particles.