Wigner crystal

A Wigner crystal is the solid (crystalline) phase of electrons first predicted by Eugene Wigner in 1934.[1][2] A gas of electrons moving in 2D or 3D in a uniform, inert, neutralizing background will crystallize and form a lattice if the electron density is less than a critical value. This is because the potential energy dominates the kinetic energy at low densities, so the detailed spatial arrangement of the electrons becomes important. To minimize the potential energy, the electrons form a bcc (body-centered cubic) lattice in 3D, a triangular lattice in 2D and an evenly spaced lattice in 1D. Most experimentally observed Wigner clusters exist due to the presence of the external confinement, i.e. external potential trap. As a consequence, deviations from the b.c.c or triangular lattice are observed.[3] A crystalline state of the 2D electron gas can also be realized by applying a sufficiently strong magnetic field. However, it is still not clear whether it is the Wigner-crystallization that has led to observation of insulating behaviour in magnetotransport measurements on 2D electron systems, since other candidates are present, such as Anderson localization.

Structure of a two-dimensional Wigner crystal in a parabolic potential trap with 600 electrons. Triangles and squares mark positions of the topological defects.

More generally, a Wigner crystal phase can also refer to a crystal phase occurring in non-electronic systems at low density. In contrast, most crystals melt as the density is lowered. Examples seen in the laboratory are charged colloids or charged plastic spheres.

Description

A uniform electron gas at zero temperature is characterised by a single dimensionless parameter, the so-called Wigner-Seitz radius r_s = a / a_b, where a is the average inter-particle spacing and a_b is the Bohr radius. The kinetic energy of an electron gas scales as 1/r_s², this can be seen for instance by considering a simple Fermi gas. The potential energy, on the other hand, is proportional to 1/r_s. When r_s becomes larger at low density, the latter becomes dominant and forces the electrons as far apart as possible. As a consequence, they condense into a close-packed lattice. The resulting electron crystal is called the Wigner crystal. [4]

Based on the Lindemann criterion one can find an estimate for the critical r_s. The criterion states that the crystal melts when the root-mean-square displacement of the electrons ${\displaystyle {\sqrt {\langle r^{2}\rangle }}}$ is about a quarter of the lattice spacing a. On the assumption that vibrations of the electrons are approximately harmonic, one can use that for a quantum harmonic oscillator the root mean square displacement in the ground state (in 3D) is given by

${\displaystyle {\sqrt {\langle r^{2}\rangle }}=3{\frac {\hbar }{2m_{e}\omega }}}$

with ${\displaystyle \hbar }$ the Planck constant, m_e the electron mass and ω the characteristic frequency of the oscillations. The latter can be estimated by considering the electrostatic potential energy for an electron displaced by r from its lattice point. Say that the Wigner–Seitz cell associated to the lattice point is approximately a sphere of radius a/2. The uniform, neutralizing background then gives rise to a smeared positive charge of density ${\displaystyle 6e/a^{3}\pi }$ with ${\displaystyle e}$ the electron charge. The electric potential felt by the displaced electron as a result of this is given by

${\displaystyle \varphi (r)={\frac {e}{4\pi \epsilon _{0}}}\left({\frac {3}{a}}-{\frac {4r^{2}}{a^{3}}}\right)}$

with ε₀ the vacuum permittivity. Comparing ${\displaystyle -e\varphi (r)}$ to the energy of a harmonic oscillator, one can read off

${\displaystyle {\frac {1}{2}}m_{e}\omega ^{2}={\frac {e^{2}}{\pi \epsilon _{0}a^{3}}}}$

or, combining this with the result from the quantum harmonic oscillator for the root-mean-square displacement

${\displaystyle {\frac {\sqrt {\langle r^{2}\rangle }}{a}}={\sqrt {\frac {3}{8}}}\left({\frac {1}{r_{s}}}\right)^{1/4}}$

The Lindemann criterion than gives us the estimate that r_s > 40 is required to give a stable Wigner crystal. Quantum Monte Carlo simulations indicate that the uniform electron gas actually crystallizes at r_s = 106 in 3D[5][6] and r_s = 31 in 2D.[7][8][9]

For classical systems at elevated temperatures one uses the average interparticle interaction in units of the temperature: G = e² / (k_B Ta). The Wigner transition occurs at G = 170 in 3D[10] and G = 125 in 2D.[11] It is believed that ions, such as those of iron, form a Wigner crystal in the interiors of white dwarf stars.

Experimental realisation

In practice, it is difficult to experimentally realize a Wigner crystal because quantum mechanical fluctuations overpower the Coulomb repulsion and quickly cause disorder. Low electron density is needed. One notable example occurs in quantum dots with low electron densities or high magnetic fields where electrons will spontaneously localize in some situations, forming a so-called rotating "Wigner molecule",[12] a crystalline-like state adapted to the finite size of the quantum dot.

Wigner crystallization in a two-dimensional electron gas under high magnetic fields was predicted (and was observed experimentally[13] ) to occur for small filling factors[14] (less than ν=1/5) of the lowest Landau level. For larger fractional fillings, the Wigner crystal was thought to be unstable relative to the fractional quantum Hall effect (FQHE) liquid states. The recent observation[15] of a Wigner crystal in the immediate neighborhood of the large fractional filling ν=1/3 was unexpected, and has led to a new understanding[16] (based on the pinning of a rotating Wigner molecule) for the interplay between quantum-liquid and pinned-solid phases in the lowest Landau level.

Another experimental realisation of the Wigner crystal occurs in single electron transistors with very low currents, where a 1D Wigner crystal will form. The current due to each electron can be directly detected experimentally.[17]

Additionally, experiments using quantum wires - short quantum wires are sometimes referred to as ‘quantum point contacts’, (QPCs) - have also led to suggestions of Wigner crystallization in 1D systems.[18] In the experiment performed by Hew et al., a 1D channel was formed by confining electrons in both directions transverse to the electron transport, by the bandstructure of the GaAs/AlGaAs heterojunction and the potential from the QPC. The novel device design allowed the electron density in the 1D channel to be varied relatively independently of the strength of transverse confining potential, thus allowing experiments to be performed in the regime in which Coulomb interactions between electrons dominate over the kinetic energy. It is a well-established result that the conductance through a QPC shows a series of plateaux quantized in units of the conductance quantum, 2e²/h However, this experiment reported a disappearance of the first plateau (resulting in a jump in conductance of 4e²/h), which was attributed to the formation of two parallel rows of electrons. In a strictly 1D system, electrons will occupy equidistant points along a line, i.e. a 1D Wigner crystal. As the electron density increases, the Coulomb repulsion becomes large enough to overcome the electrostatic potential confining the 1D Wigner crystal in the transverse direction, leading to a lateral rearrangement of the electrons into a double-row structure.[19][20] The evidence of a double row observed by Hew et al. may point towards the beginnings of a Wigner crystal in a 1D system.

New evidence

In 2018 a transverse magnetic focusing which combines charge and spin detection was used to directly probe the Wigner crystal and its spin properties in 1D quantum wires with tunable width. It provides not merely a direct evidence but also a better understanding of the nature of zigzag Wigner crystallization by unveiling both the structural and the spin phase diagrams.[21]

An article about "Imaging the electronic Wigner crystal in one dimension" was published in Science magazine by May 2019, providing a direct evidence for the formation of small Wigner crystals.[22]

References

1. Wigner, E. (1934). "On the Interaction of Electrons in Metals". Physical Review. 46 (11): 1002–1011. Bibcode:1934PhRv...46.1002W. doi:10.1103/PhysRev.46.1002.
2. Wigner, E. P. (1938). "Effects of the electron interaction on the energy levels of electrons in metals". Transactions of the Faraday Society. 34: 678. doi:10.1039/TF9383400678.
3. Radzvilavicius, A.; Anisimovas, E. (2011). "Topological defect motifs in two-dimensional Coulomb clusters". Journal of Physics: Condensed Matter. 23 (38): 385301. arXiv:1204.6028. Bibcode:2011JPCM...23L5301R. doi:10.1088/0953-8984/23/38/385301. PMID 21891854.
4. Jenö, S. (2010). Fundamentals of the Physics of Solids: Volume 3-Normal, Broken-Symmetry, and Correlated Systems. Vol. 3. Springer Science & Business Media.
5. Ceperley, D. M. (1980). "Ground State of the Electron Gas by a Stochastic Method". Physical Review Letters. 45 (7): 566–569. Bibcode:1980PhRvL..45..566C. doi:10.1103/PhysRevLett.45.566.
6. Drummond, N.; Radnai, Z.; Trail, J.; Towler, M.; Needs, R. (2004). "Diffusion quantum Monte Carlo study of three-dimensional Wigner crystals". Physical Review B. 69 (8): 085116. arXiv:0801.0377. Bibcode:2004PhRvB..69h5116D. doi:10.1103/PhysRevB.69.085116.
7. Tanatar, B.; Ceperley, D. (1989). "Ground state of the two-dimensional electron gas". Physical Review B. 39 (8): 5005–5016. Bibcode:1989PhRvB..39.5005T. doi:10.1103/PhysRevB.39.5005. PMID 9948889.
8. Rapisarda, F.; Senatore, G. (1996). "Diffusion Monte Carlo study of electrons in two-dimensional layers". Australian Journal of Physics. 49: 161. Bibcode:1996AuJPh..49..161R. doi:10.1071/PH960161.
9. Drummond, N.D.; Needs, R.J. (2009). "Phase diagram of the low-density two-dimensional homogeneous electron gas". Physical Review Letters. 102 (12): 126402. arXiv:1002.2101. Bibcode:2009PhRvL.102l6402D. doi:10.1103/PhysRevLett.102.126402. PMID 19392300.
10. Dubin, D. H. E.; O'neil, T. M. (1999). "Trapped nonneutral plasmas, liquids, and crystals (the thermal equilibrium states)". Reviews of Modern Physics. 71 (1): 87–172. Bibcode:1999RvMP...71...87D. doi:10.1103/RevModPhys.71.87.
11. Imai, Y.; Kawakami, N.; Tsunetsugu, H. (2003). "Low-energy excitations of the Hubbard model on the Kagomé lattice". Physical Review B. 68 (19): 195103. arXiv:cond-mat/0305144. Bibcode:2003PhRvB..68s5103I. doi:10.1103/PhysRevB.68.195103.
12. Yannouleas, C.; Landman, U. (2007). "Symmetry breaking and quantum correlations in finite systems: studies of quantum dots and ultracold Bose gases and related nuclear and chemical methods". Reports on Progress in Physics. 70 (12): 2067–2148. arXiv:0711.0637. Bibcode:2007RPPh...70.2067Y. doi:10.1088/0034-4885/70/12/R02.
13. Andrei, E. Y.; Deville, G.; Glattli, D. C.; Williams, F. I. B.; Paris, E.; Etienne, B. (1988). "Observation of a magnetically induced Wigner solid". Physical Review Letters. 60 (26): 2765–2768. Bibcode:1988PhRvL..60.2765A. doi:10.1103/PhysRevLett.60.2765. PMID 10038446.
14. Jain, J.K. (2007). Composite Fermions. Cambridge, England: Cambridge University Press.
15. Zhu, H.; Chen, Y.P.; Jiang, P.; Engel, L.W.; Tsui, D.C.; Pfeiffer, L.N.; West, K.W. (2010). "Observation of a pinning mode in a Wigner solid with ν=1/3 fractional quantum Hall excitations". Physical Review Letters. 105 (12): 126803. arXiv:1006.2335. Bibcode:2010PhRvL.105l6803Z. doi:10.1103/PhysRevLett.105.126803. PMID 20867666.
16. Yannouleas, C.; Landman, U. (2011). "Unified microscopic approach to the interplay of pinned-Wigner-solid and liquid behavior of the lowest-Landau-level states in the neighborhood of ν=1/3". Physical Review B. 84 (16): 165327. arXiv:1111.0019. Bibcode:2011PhRvB..84p5327Y. doi:10.1103/PhysRevB.84.165327.
17. Bylander, Jonas; Duty, Tim; Delsing, Per (2005). "Current measurement by real-time counting of single electrons". Nature. 434 (7031): 361–364. arXiv:cond-mat/0411420. Bibcode:2005Natur.434..361B. doi:10.1038/nature03375. PMID 15772655. (see also the Nature review article here
18. Hew, W.K.; Thomas, K.J.; Pepper, M.; Farrer, I.; Anderson, D.; Jones, G.A.C.; Ritchie, D.A. (2009). "Incipient Formation of an Electron Lattice in a Weakly Confined Quantum Wire". Physical Review Letters. 102 (5): 056804. arXiv:0907.1634. Bibcode:2009PhRvL.102e6804H. doi:10.1103/PhysRevLett.102.056804. PMID 19257536.
19. Meyer, J. S.; Matveev, K. A. (January 2009). "Wigner Crystal Physics in Quantum Wires". J. Phys.: Condens. Matter. 21 (2): 023203. arXiv:0808.2076. Bibcode:2009JPCM...21b3203M. doi:10.1088/0953-8984/21/2/023203. PMID 21813970.
20. Klironomos, A. D.; Meyer, J. S.; Matveev, K. A. (May 2006). "Spontaneous Spin Polarization in Quantum Wires". Europhysics Letters. 74 (4): 679–685. arXiv:cond-mat/0507387. Bibcode:2006EL.....74..679K. doi:10.1209/epl/i2006-10024-x.
21. Ho, Sheng-Chin; Chang, Heng-Jian; Chang, Chia-Hua; Lo, Shun-Tsung; Creeth, Graham; Kumar, Sanjeev; Farrer, Ian; Ritchie, David; Griffiths, Jonathan; Jones, Geraint; Pepper, Michael; Chen, Tse-Ming (6 September 2018). "Imaging the Zigzag Wigner Crystal in Confinement-Tunable Quantum Wires". Physical Review Letters. 121 (10): 106801. doi:10.1103/PhysRevLett.121.106801.
22. Imaging the electronic Wigner crystal in one dimension, Science magazine