Spin-resolved pair-correlation functions of quasi one dimensional electron gas at finite temperature

Materials Today: Proceedings xxx (xxxx) xxx

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Spin-resolved pair-correlation functions of quasi one dimensional electron gas at finite temperature Kulveer Kaur a, Akariti Sharma a, Vinayak Garg a,⇑, R.K. Moudgil b a b

Department of Physics, Punjabi University, Patiala 147 002, India Department of Physics, Kurukshetra University, Kurukshetra 136 119, India

a r t i c l e

i n f o

Article history: Received 16 August 2019 Received in revised form 4 November 2019 Accepted 15 November 2019 Available online xxxx Keywords: Electron quantum wire Pair-correlation function Static structure factor Spin-polarization Finite temperature

a b s t r a c t In a recent paper [Phys. Status Solidi B 255, 1800174 (2018)], we found that the total contact paircorrelation function g(r = 0, T) of an unpolarized quasi one dimensional electron gas (Q1DEG) exhibits non-monotonous dependence on temperature T, initially decreasing with rise in T and then increasing beyond a critical T for a fixed electron number density. Here, we calculate the spin-resolved paircorrelation functions i.e., g""(r, T) [= g;;(r, T)] and g";(r, T) of the unpolarized Q1DEG for different values of T at a fixed wire width and electron number density. To this endeavor, we are using the celebrated theory given by Singwi, Tosi, Land and Sjölander developed for an interacting three-dimensional manyelectron system. Our present study reveals that with rise in T, g""(r, T) at small electron separation becomes stronger in complete contrast to g";(r, T) and the non-monotonic T-dependence of g(r = 0, T) is governed by the cumulative behaviour of g""(r, T) and g";(r, T). Ó 2019 Elsevier Ltd. All rights reserved. Selection and peer-review under responsibility of the scientific committee of the International Conference on functional materials and simulation techniques.

1. Introduction The study of electron spin-polarization effects in quantum wire has attracted enormous [1–8] interest over the past few years due to its direct applications in the emerging field of spintronics. In quantum wires, motion of the electrons in transverse directions is somehow confined allowing them to move freely only in one spatial direction constituting a quasi one dimensional electron gas (Q1DEG). The electrons in Q1DEG may have any possible spin-polarization and the electronic properties of such a system can be calculated by averaging the corresponding spin-resolved components. At absolute zero, Shulenburger et al. [5,6] have used the quantum Monte Carlo (QMC) simulations to explore the spinresolved correlations in Q1DEG followed by some theoretical attempts [7,8] to elucidate the simulation results using different approximate methods. Importantly, the paramagneticferromagnetic phase transition has been predicted [7] in Q1DEG due to the pronounced spin correlations. Another excellent and direct manifestation of these correlation effects lie in the paircorrelation function grr’(r, T) which simply tells as how the elec-

⇑ Corresponding author.

trons with spin components r and r0 are distributed in space. Recently [9–12], we have explored the exchange–correlation effects in an unpolarized Q1DEG at finite-T by applying different dielectric formulation methods. Apart from other physical properties, the total contact pair-correlation function g(r = 0, T) was calculated which was found to display a non-monotonous behaviour with increasing T [12]. In this paper, we attempt to calculate the spin-resolved components of g(r, T) i.e., g""(r, T) [= g;;(r, T)] and g";(r, T), and hence g(r = 0, T) of the unpolarized Q1DEG using the Singwi-Tosi-Land-Sjölander (STLS) approach to seek out the possible explanation for the non-monotonous behaviour of g(r = 0, T) with rise in T. 2. Theoretical formalism In the present work, we assume the same Q1DEG model as described in our earlier work [12]. Here, the motion of electrons constituting Q1DEG is supposed to be restricted by a harmonic potential of the type Vc(r\) = r2\/(8b4) (b; the diameter of the wire), so that they are permitted to move only in one direction. The Fourier transform of the bare Coulomb interaction potential among the electrons is given by

E-mail address: [email protected] (V. Garg). https://doi.org/10.1016/j.matpr.2019.11.209 2214-7853/Ó 2019 Elsevier Ltd. All rights reserved. Selection and peer-review under responsibility of the scientific committee of the International Conference on functional materials and simulation techniques.

Please cite this article as: K. Kaur, A. Sharma, V. Garg et al., Spin-resolved pair-correlation functions of quasi one dimensional electron gas at finite temperature, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.11.209

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K. Kaur et al. / Materials Today: Proceedings xxx (xxxx) xxx

Fig. 1. The STLS grr’(r, s) is presented for b = a0*, f = 0 and rs = 2 at indicated s-values by solid lines. For comparison, the RPA results are also displayed at s = 0 and 1 by dashed lines.

wrr0 ðq ; T Þ ¼ VðqÞ½1  Grr0 ðq ; TÞ is the effective Coulomb interaction potential and Grr0 (q, T) is the spin-dependent local-field correction (LFC) factor which takes care of the exchange–correlation effects among electrons of spin r and r0 . In the STLS approach, Grr0 (q, T) is defined as

1 Grr0 ðq; TÞ ¼  pffiffiffiffiffiffiffiffiffiffiffiffi nr nr0

Z

1

1

dq0 q0 Vðq0 Þ ½Srr0 ðjq  q0 j; T Þ  drr0  : 2p qVðqÞ

ð4Þ

In Eq. (2), v0r(q,x,m,T) represents the density–density response function for free electrons with spin r confined in one dimension at temperature T. It is related to v0r(q,x,m,T = 0) as

8 R1 dx > v0 ðq; x ; l þkB Tlnjxj;T ¼ 0Þ ðxþ1Þ 2þ > eðl=kB TÞ r > > < R1 0 dx 0 v ðq; x ; l kB Tlnjxj;T ¼ 0Þ ðxþ1Þ2 ; for l > 0 ; vr ðq; x ; l;TÞ ¼ 0 r > > > R eðl=kB TÞ > dx : v0r ðq; x ; l kB Tlnjxj;T ¼ 0Þ ðxþ1Þ 2 ; for l < 0 0

Fig. 2. g(r = 0, s) displayed as a function of s with its "" (up-up) and "; (up-down) components for b = a0*, f = 0 and rs = 2. The STLS and the RPA results are shown by solid lines and dashed lines, respectively.

V ðqÞ ¼

e2

e0

2

2

E1 ðq2 b Þexpðq2 b Þ;

ð1Þ

where e0 is the dielectric constant of the substrate (background) and E1(z) the exponential-integral function. The Q1DEG having arbitrary spin-polarization f [=(n" - n;)/n] may be considered as a twocomponent electron system with n" and n; being the linear number density of the up-spin and down-spin electrons, respectively, and the correlational properties of electrons at any f depend upon their corresponding spin-resolved components. The finite-T STLS formalism given in Ref. [12] is suitably generalized here to the two-component electron system and the spinresolved linear density–density response function vrr0 (q,x,m,T) of the coupled system are obtained as

vrr0 ðq; x ; l;TÞ ¼

h i v0r ðq; x ; l;TÞ drr0 þ ð1Þdrr0 wr r0 ðq;T Þv0r ðq; x; l;TÞ Dðq; x; l;T Þ

;

ð2Þ 

here r, r0 are the spin indices and r denotes the spin orientation opposite to r. In Eq. (2), D(q,x,m,T) is given by

h ih i Dðq; x ; l;TÞ ¼ 1w"" ðq;T Þv0" ðq; x ; l;TÞ 1w## ðq;T Þv0# ðq; x; l;TÞ w2"# ðq;T Þv0" ðq; x ; l;TÞv0# ðq; x ; l;TÞ:

ð3Þ

ð5Þ where m represents the Q1DEG chemical potential. Grr0 (q, T) in Eq. (4) involves the spin-resolved static structure factors Srr0 (q, T) which in turn can be computed from vrr0 (q,x,m,T) as

X kB T l¼þ1 Srr0 ðq; TÞ ¼  pffiffiffiffiffiffiffiffiffiffiffiffi v 0 ðq; ixl ; l; TÞ nr nr0 l¼1 rr

ð6Þ

where ixl = 2plkBT/— h with l = 0, ±1, ±2, ±3. . . The spin-resolved paircorrelation functions grr0 (r, T) of the Q1DEG can be calculated by taking the inverse Fourier transform of Srr0 (q, T) as

g rr0 ðr; TÞ ¼ 1 þ

1 pffiffiffiffiffiffiffiffiffiffiffiffi 2p nr nr0

Z

1

1

dqeðiqrÞ ½Srr0 ðq; T Þ  drr0  :

ð7Þ

3. Results and discussion For computational purpose, we prefer to use a dimensionless system of units. We express wave vector q in the units of kF" (Fermi wave vector for up-spin electrons), the distance r in (kF")1, b in  and effective Bohr radius a0*, m and x in Fermi energy EF", h e0 = 1. Our Q1DEG model is completely characterized by three parameters, the dimensionless electron density parameter rs (=1/(2na0*)) with n = n" + n;, f, and the reduced temperature s (=kBT/ EF"). The computation of the spin-resolved pair-correlation functions grr0 (r, s) of the unpolarized Q1DEG requires [see Eq. (7)] the static structure factors Srr0 (q, s) which can be calculated from the itera-

Please cite this article as: K. Kaur, A. Sharma, V. Garg et al., Spin-resolved pair-correlation functions of quasi one dimensional electron gas at finite temperature, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.11.209

K. Kaur et al. / Materials Today: Proceedings xxx (xxxx) xxx

tive solution of set of Eqs. (2), (4) and (6). In Fig. 1, our results of grr0 (r, s) are shown at b = a0*, f = 0, rs = 2 for s = 0, 0.1, 0.5, 1, 3, 5 and 10 in the STLS approach. We note that in the limit r ? 0, g""(r, s) grows stronger (its deviation from unity increases) with increase in s in complete contrast to g";(r, s). This suggests that at r = 0, the probability of finding two electrons with parallel spin decreases and for anti-parallel spin increases with increasing s. The results obtained using the random phase approximation (RPA) are also presented at s = 0 and 1 for comparison. Interestingly, at small inter-electron separation the RPA g""(r, s) becomes negative for all s-values while the STLS g""(r, s) remains positive for s  0.5. This is attributed to the fact that the RPA does not include the electron correlations at small r and the STLS theory captures them only in a static way. On the other hand, g";(r, s) remains positive for all s in both the approximate schemes i.e. the STLS and RPA. In order to further explore the influence of s on electron correlations, the total (spin-averaged) contact pair-correlation function g(r = 0, s) is plotted in Fig. 2 along with its parallel ("") and antiparallel (";) spin components at rs = 2. The total pair-correlation function g(r, s) is a weighted mean of its parallel and antiparallel parts and it is given as

  gðr; sÞ ¼ g "" ðr; sÞ þ g "# ðr; sÞ =2:

ð8Þ

It should be noted that the results of the total (spin-averaged) g (r = 0, s) calculated from its spin-components and as obtained for the unpolarized Q1DEG (also reported in Ref. [12] but for b = 2a0*) at rs = 2 are exactly same. Further, an interesting interplay is observed between the spin-correlations and the s-effects in the behavior of g""(r = 0, s) and g";(r = 0, s). As can be seen from Fig. 2, g";(r = 0, s) consistently increases and g""(r = 0, s) decreases with rising s. In fact, it is this connection between g""(r = 0, s) and g";(r = 0, s) which is responsible for the non-monotonous sdependence of g(r = 0, s). For comparison, the RPA results are also reported and both the g""(r = 0, s) and g";(r = 0, s) curves show the non-monotonic s-dependence.

3

To summarize, we have explored the influence of s on spinresolved pair-correlation functions grr’(r, s) of an unpolarized Q1DEG using the self-consistent STLS approximation. An interesting interplay is observed between the electronic spin-correlations and thermal effects. It has been found that at contact, g""(r, s) becomes stronger with increasing s in complete contrast to g";(r, s) and the non-monotonic dependence of g(r = 0, s) on s is governed by the collective behaviour of g""(r, s) and g";(r, s). Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References [1] D.J. Reilly, T.M. Buehler, J.L. O’Brien, A.R. Hamilton, A.S. Dzurak, R.G. Clark, B.E. Kane, L.N. Pfeiffer, K.W. West 246801 Phys. Rev. Lett. 89 (246801) (2002) 1–4. [2] L.W. Smith, A.R. Hamilton, K.J. Thomas, M. Pepper, I. Farrer, J.P. Griffiths, G.A.C. Jones, D.A. Ritchie, Phys. Rev. Lett. 107 (126801) (2011) 1–5. [3] C. Yan, S. Kumar, K. Thomas, M. Pepper, P. See, I. Farrer, D. Ritchie, J. Griffiths, G. Jones, App. Phys. Lett. 11 (042107) (2017) 1–4. [4] A.A. Vasilchenko, Phys. Lett. A 379 (2015) 3013–3015. [5] L. Shulenburger, M. Casula, G. Senatore, Richard M. Martin, Phys. Rev. B 78 (165303) (2008) 1–12. [6] L. Shulenburger, M. Casula, G. Senatore, Richard M. Martin, J. Phys. A 42 (214021) (2009) 1–10. [7] V. Garg, R.K. Moudgil, Phys. E 47 (2013) 217–223. [8] R. Bala, R.K. Moudgil, S. Srivastava, K.N. Pathak, Eur. Phys. J. B 87 (5) (2014) 1– 12. [9] K. Kaur, A. Sharma, V. Garg, R.K. Moudgil, AIP Conf. Proc. 2093 (020008) (2019) 1–4. [10] A. Sharma, K. Kaur, V. Garg, R.K. Moudgil, AIP Conf Proc. 1953 (060012) (2018) 1–4. [11] V. Garg, A. Sharma, R.K. Moudgil, Mod. Phys. Lett. B 32 (1850060) (2018) 1–11. [12] A. Sharma, K. Kaur, V. Garg, R.K. Moudgil, Phys. Status Solidi B 255 (1800174) (2018) 1–9.

Please cite this article as: K. Kaur, A. Sharma, V. Garg et al., Spin-resolved pair-correlation functions of quasi one dimensional electron gas at finite temperature, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.11.209