Electron crystallization in two dimensions

Physica B 169 (1991) North-Holland

Electron

328-335

crystallization

in two dimensions

D.C. Glattli”, E.Y. Andreih, R.G. Clark’, G. Deville”, C. Dorind, B. Etienned, C.T. Foxone, J.J. Harris’. E. Parisd, 0. Probst”, F.I.B. Williams” and P.A. Wright“ “Lahoratoire de Physique du solide et de resonance Magnetique, C. E. N. Saclay, 91191 Gif-sur-Yvette hDepartment of Physics, Rutgers University. P&caraway, NJ 088RSS. USA ‘Clarendon Laboratory. Parks Road. Oxford. OX1 3PlJ, UK ‘Lahoratoire de microstructures et microelectronique. 196 Av. H. Ravera, 92220 Bagneux. France ‘Phillips Rewarch Laboratory. Redhill, Surrey RHl SHA. UK The invited

talk was given

by D.C.

Cedex,

France

Glattli.

Electrons confined at the interface of a GaAsiGaAlAs heterojunction form a 2D quantum electron liquid. Under a strong magnetic field a phase transition to an electron (Wigner) solid takes place in the low filling factor regime of the Fractional Quantum Hall Effect (FQHE). We describe experimental evidence for such electron solid formation obtained both by radiofrequency (RF) study of the low-frequency collective excitations and by conductivity measurements. A finite-threshold electric field for DC conduction reflecting the electron crystallite pinning in the sample random potential is found associated to a small gap in the solid phase low-lying collective excitation branch. The v = l/S FQHE liquid reenters the solid domain at low temperature.

1. Introduction A 2D electron system confined by a smooth disitribution of positive charges undergoes a transition to a solid phase when the Coulomb interaction energy V,. = e’lca strongly exceeds the kinetic energy K per particle (a = (7~~1~) lf2, n, is the density). At low density, K = k,T, the

thermodynamic properties are a function of the r = Vc/ K = e’(rn,)’ ‘*/ correlation parameter

lk,T. A classical electron solid is formed when r > r, = 127. Such a classical solid has been observed at low T for dilute electron or ion systems trapped at the 4He liquid-vapor interface [l], a smooth deformable substrate (T < 700mK, II, < 10’ cm-‘). At higher density or low effective mass m* the zero-point motion (K = h2/m*a2) becomes relevant and even at T = 0 can melt the solid. The thermodynamic properties are now functions of rs = V,i K = phase transiala Bohr - n, Ii’ and T. A quantum tion from a (Wigner) solid to a liquid is expected two-dimenfor r, = 33 [2]. Present high-density sional electron system (2DES) realizations in semiconductors are far from the Quantum solid

domain (r, = 1) but well in the quantum liquid regime. However, a perpendicular magnetic field B, can squeeze the zero-point motion into cyclotron orbits of size I, = (hc/eB)“’ to restore the correlation. The Landau state spatial degeneracy in 2D gives rise first to the quantum Hall effect [3] followed, at low Landau-level filling factors I, = hn,/eB =S 1, by the Coulomb correlated liquids of the Fractional Quantum Hall Effect (FQHE) [4] at v = p/q before a Magnetically Induced Wigner Solidification (MIWS) takes place [5,6] for v < V, @ I. The thermodynamic quantities are a function of v and T for low r,. An infinite magnetic field classicizes the 2DES and the melting temperature T,( u + 0) + T,,, = e’(rn,)‘!‘/rcEk,, the equivalent classical melting temperature. The solid phase is then expected in the lower left corner of the (T, v) plane. 2DES confined at the interface of Molecular Beam Epitaxy (MBE) grown GaAsiGa, ox. (Al).rAs semiconductor heterojunctions, for which r, = l2 (m*= 0.068, E = 13), are able to reach the MIWS domain at low temperature, <400mK. and high magnetic field. = 15-30 T, provided the disorder potential produced by the quasi-

D. C. Glattli et al. I Electron crystallization in 20

random donor and acceptor impurity distribution is low enough compared with the Coulomb energy e”lEa. The aim of this paper is to describe recent experiments on the 2DES low-lying collective excitation branch [6,7] and conductivity [8,9] for high-quality heterojunctions at low temperature and high magnetic field showing definitive evidence of MIWS formation. Samples are high-quality MBE grown heterojunctions with large doping setback distance d, from the interface and low residual acceptor density to reduce the random potential (C804: double delta doped, di = lOOnm, II, = 710 X 10’” cmP2, p = 1.2-1.9 cm2 V’ s-l [lo]; G641: bulk doped, di = 150 nm, dark and illuminated: 12, = 4-9.2 x 10” cm-*, p = 4-9 cm2 V’S~’ [ll]). They are mounted in a (35mK) dilution refrigerator mixing chamber and characterized by their resistivity p,, (contacted sample) or finite frequency conductivity a,,(o) (if contactless) Subnikov-de Haas (SdH) variations.

2. Low-frequency

collective excitation study

Like any liquid, an uncorrelated charged system has only one (longitudinal) phonon branch tip(k). The restoring force to compression is long-range and oy = wP = (2Tn,e2klm*e)“2, the 2D plasmon frequency. The magnetic field opens a gap w, = eBlm*c in the so-called magnetoplasmon branch w, = (wz + w:)“~. For high B,

329

the quantum liquid shows Coulomb correlations (FQHE) and presents a finite gap magnetoroton branch A - e2/elc for partially filled Landau levels (v = p/q, not integer) but smaller than fiw, (Y integer) reflecting an incompressible state. The transition to a compressible MIWS gives rise to a second but gapless magnetophonon branch o_(k) associated with a shear modulus K (as in an ordinary solid) induced by the onset of longrange positional order

w,o,/w,-

w_ =

l/2

k‘

,

(1)

where w, = (Klm*n,) “*k, and K is the shear modulus. The existence of a low-lying gapless collective excitation branch is then a decisive electron crystallization signature. The first observation of this signature was reported in ref. [6] where a broadband RF spectrometer is used to detect the solid phase gapless magnetophonon branch. From a 0.01-4 GHz swept frequency source, a PO= 0.1 nW wave is sent through a 501R matched strip-line which meanders with spatial period 2w parallel and close (h ~0.3-0.5 pm) to the electrons (w = 16 km, see fig. 1). The RF electric field on the charge plane contains up to p = 12 spatial harmonics: E, = E,J,(

ipn)

exp(-k,h)

exp(ik,x)

,

J, is a Bessel function and k, = pqo+ line disW/C,,ne = pqo = p2Trl w is the meander

where

W

Electrons

W

Dielectric Ground Plane Fig. 1. (a) Meander line dispersion relation. (b) Schematic section of the meander line with GaAsiGaAlAs face-down.

sample mounted

D.C.

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Glattli et al.

/ Electron crvstallization

persion relation. A diminution of the transmitted RF power signals the electron absorption Pahs = C,] Efj~x,(w. k,?), where v~,,(w, k) is the longitudinal dynamical conductivity. A low-frequency backgate voltate modulation is applied such that the synchronously demodulated signal is proportional to the electronic absorption derivative. If a low-lying electronic branch w_(k) is present (solid case) a pole in (T,,(o, k) will be detected for each crossing between the meander line and electron dispersion relation, otherwise (liquid case) a fat response is expected (for p = 10, o = 1 GHz at T= 0, ~2, = 10” cm-? and B = 30 T). At low-field the absorption derivative indeed shows no resonance but only a broad signal with conductivity dependent amplitude. At fixed w and swept B, the SdH. oscillations of the (T,,(W) derivative are clearly seen at integer and fractional u (see fig. 2). At low T, I, = 115 is the last high-field FQHE state shown by this technique. At higher field, at a precisely defined filling factor V~= 0.192 -+ 0.003. a qualitative spectrum discontinuity occurs in the form of a well-defined resonance whose frequency rapidly increases

0

in 20

from zero, sharpens and reaches a plateau (see fig. 3). Heating the sample decreases the frequency until the signal broadens and vanishes at a (less well-defined) filling factor dependent temperature T,(v). The T,(v) points define in the (T, V) plane a boundary separating the domain with resonance but no SdH. oscillations from the FQHE liquid. For different samples over a wide density range (=3.5), the boundaries find a common representation when plotted in Coulomb energy reduced temperature t = T,(u) IT,,, and Y units, see fig. 4 open circles. The constancy of V, for T+ 0 is striking. These features indicate that the basic phenomenon observed is not related to disorder but to magnetic degeneracy plus Coulomb interaction. In addition. the very steep frequency rise at V, is suggestive of a phase transition. The lowfrequency collective mode not compatible with a liquid and the proximity of the high-field limit of T,(u) + (0.85 k 0.15) T,,, to the classical limit strongly suggests that the phase is a MIWS. The resonance contains much information: in a pure electron solid we expect a multiplet spectrum at frequency w_( pq,,) - ~1~“. p = l-=12.

15

5 MAGNETIC

Fig. 2. Shubnikov-de Haas oscillation of the fixed RF (4OOMHz) modulation; n< = 6.9 ? 0.15 x 10”’ cm I, 7‘= SO mK. sample C804.

TESLA

absorption

derivative

under

low-frequency

backgatc

D. C. Glattli et al.

0

0.5

1

FREQUENCY

1.5

I Electron

2

(GHz 1

crystallization

in 20

331

Fig. 4. Phase diagram in reduced units t = TIT,,, v = n,hcl eB. Open circles: deduced from the appearance of the magnetophonon mode for different densities and sample, n, = 3.4-12 x 10”’ cm-‘. Crosses: from the appearance of threshold voltage for conduction (crosses, G641 dark, n, = 3.8 x 101ncm~‘; black squares, G641 after illumination by a brief red light LED pulse at low T, n, = 9 X 10’” cm-‘). In (t, v) units the diagram is universal and identical for both types of measurement.

Fig. 3. RF absorption derivative spectrum around v = I/S and in the solid phase, showing the sharp onset of the magnetophonon mode; n, = 10 t 0.1 x 10”’ cm- ‘, T = 42 mK sample C804. The fine structure on the first trace is a result of non monotonic spectrometer response.

3. Threshold

Early observations showed part of such multiplet [6]. A plausible p”” fit suggested a =prn cutoff length to explain the lack of low harmonics. However, other samples show only a single broader resonance reaching higher (1.5-2 GHz) frequency. For the same (v, t) the frequency is higher for low density or less good samples [7]. Short illumination increases the sample density but mostly reduces the impurity disorder. Figure 5 shows how illumination transforms the singlet high-frequency resonance of a sample in the dark into a low-frequency multiplet. It is likely that impurity disorder plays an important role, broadening and opening a small gap (Ghz) in the MIWS magnetophonon branch. New transport measurements better adapted to the physical situation give independent information on disorder and a coherent explanation of the small magnetophonon gap [8,9].

In contrast with the RF spectrum discontinuiry, four-point AC current source resistivity measurements show a puzzling continuous (but apparent, see below) p,, divergence [12] upon crossing the phase boundary, see fig. 6. However, more significant is the non-linear voltage or ‘R,y,’ in response to the source current amplitude i. A recently noticed (81 voltage phase shift 6’- 1 li, due to competing lead capacitance, suggested a threshold voltage for conduction, V, (0 reflects the time to charge the lead capacitance to V,). As demonstrated in ref. [9], the appropriate tool is a two-point voltage source measurement of the current-voltage characteristic (IVC) i(U). Here the Hall voltage reduces the potential drop V= I/ - Ri for longitudinal conduction (R = R,,,, + Rlrad) and the observed i(U) is the solution of i = Z( U - Ri) where I(V) is the basic longitudinal IVC. At high field and low T, the IVC does indeed show threshold conduction with zero or

voltage for MIWS conduction

332

D. C. Glattli

et al.

I Electron

crystallization

in 20

INVERSE FILLINGv-' r,

3

5

-‘-7-_-7---y

1

1

12

1 0

16

20 24 MAGNETIC FIELD [TESLAI

28

Fig. 6. Diagonal magnetoresistivity measured by four-point7 Hz current-source technique for G641 illuminated sample. I

I

I

2

1 FREQUENCY

(GHz)

Fig. 5. Frequency spectrum showing the effect of reduction in disorder on the magnetophonon mode. (a) and (b) Sample G641 as cooled (in the dark) n, = 3.6 x 10’” cm -* at -40 mK. (c) After illumination n, = 7.4 x 10” cm-‘, same T. At low disorder a multiplet spectrum like the one seen in ref. [6] is recovered (for technical reasons the distribution of the wavevectors q of the exciting RF electric field is slightly different from the meander line comb and illumination of the sample has not been completely homogeneous).

extremely small conductivity before a voltage V, is reached after which the current suddenly flows through the sample, see fig. 7. Hysteretic current jumps with sometime chaotic time oscillations are observed. For lower B or larger T, V, is smaller and the IVC is smoothed. Figure 8 shows the low-T V, with B variations for the G641 sample in the dark and illuminated. For both systems (and other samples) V, vanishes within small uncertainties at v,, at low T, suggesting the same basic phenomenon as the one observed for RF measurements. This is confirmed by comparing some points (t, v) at which V, disappears with the RF deduced solid FQHE liquid phase boundary in fig. 5 (black squares and crosses). In addition, the highest mobility illuminated sample

shows a (finite V,) region between 219 and 115 suggesting a reentrant but fragile solid phase. The small characteristic single-electron energy e&a - mK associated with a threshold field E, which is nonetheless still observed at T 3 300 mK gives evidence for a collective phenomenon of N 3 100 electrons. A natural explanation is pinned electron crystallites requiring a finite electric field for conduction. The IVC have similarities with pinned charge density wave CDW systems [14] and conduction properties of a 2D classical electron solid on Helium films [15]. In a random potential (here the ionized donor and (mostly) acceptor impurities) a 2D CDW is unstable to domain formation [16] (the elastic strain energy cost balances the energy gained in a potential fluctuation). The Nd = n,Lfj electrons in a domain (size Ld) move to screen an external driving field E until the total force N,eE reaches NdeE, = KU , and the E, and

(2)

the domain slides: conduction occurs. Using v = 0 and t = 0 values of K and the observed 30-300 mV cm- ‘, eq. (2) gives Nd - 1000 L, - pm. This value is close to the cut-off

D. C. Glattli et al. I Electron crystallization in 2D

333

nS = 8.80

I

-50

0 APPLIED

11

9 G 6L1

I

,

POTENTIAL

ImVl

III

lLLU?llNATEO

\

0

-50

APPLlE3

Fig. 7. i(U) characteristics for G641 illuminated. schematic plot (lower left corner).

The filling factor

length deduced independently in [6] for the magnetophonon mode. A longitudinal or transverse force uniform on the scale L, induces an elastic strain due to the constraint imposed by domain size and pinning potential. An associated pinning frequency upin - (~ln,m*)“*nlL, = (eE,l 4nam*)” affects the basic Lorentz free trans-

and reduced

POTENTIAL

temperature

50 U ImV]

for each curve

are identified

on a

verse and longitudinal branches as O, - upin and -(W;,” + W;)1’2 for kl, < 1. From eq. (1) the WI magnetophonon branch acquires a small gap and the observed frequency in the regime where mpin < wP is w- = wpin wP/w, - Eit2/B

.

(3)

334

D.C.

Glattli

I

et al.

Electron

crvstallization

in 2D

MAGNETIC FIELD [TEsLAI 10

-

20 l_

I G6Ll

dark

n.: 3.7'4? 006 1.~37

lO*tm-'

? L mK

1,,=3LS mK

. . . v,.

i

t-o.11? 0.01 vc=0.187! 0.006

l

.

.’

_. d--

_I

Ll

__L~

15

10

5

INVERSE FILLINGl/v

MAGNETIC FIELD ITESLAI 0

20

10

r---7-

.

G6Ll "lllumulatcd"

i

n,=Ic.O 10" cm-' .

T =36!LrnK

t

zoo7

. .

I

. l

v-0216 -, 1;

. ‘\ I

. . . I,

.

i 5 INVERSE FILLING

INVERSE FILLING Fig. 8. Threshold potential values deduced finite V, values are observed also between

from IVC for sample 219 and 117.

Magnetophonon frequency and threshold voltage measured simultaneously on the same sample are described well by eq. (3), see fig. 9. As up,” is now a simple function of measurable parameters, a calculation of o_ from eq. (3) is possible and it agrees with the observed value to within 35%. The IVC study gives additional evidence for MIWS formation. It explains quantitatively the small gap in the magnetophonon branch and the

(3641 as cooled

(dark)

and after illumination.

In the latter

case

continuous variation at the phase boundary in current source p,, measurements (as V, rises smoothly from zero at solidification the apparent Rxx = V,li + dUldi is continuous). Very recently, a study of the luminescence of this system has shown that a shoulder appears on the luminescence spectrum for v < 115 at 80 mK with simultaneous dephasing of current source AC transport. Probably this feature must also be

D. C. Glattli et al. I Electron crystallization in 2 D

335

MAGNETIC FIELD ITESLAI

4

0 6 INVERSE FILLING FACTOR

O/ 4

0 6 INVERSE FILLING FACTOR

Fig. 9. Simultaneous measured magnetophonon resonance (open circles) and threshold voltage (triangles) on sample B/E:’ showing that eq. 3 is well-verified (arbitrary n, = 7.0 X 10’” cm-? at 50 mK. The black circles give the values of w

associated P91. Finally, reveals a 219 < v < Tat 2111 challenge

with the formation

of the solid phase

the low-disorder illuminated sample fascinating reentrant solid phase for 1 I.5 and a conductivity anomaly at low and l/7 (see also refs. [S, 181). A new for theorists.

References [I] CC. Grimes and G. Adams, Phys. Rev. Lett. 42 (1979) 795; C. Deville, J. Low Temp. Phys. 72 (1988) 135, and references therein. [2] E.P. Wigner, Phys. Rev. 46 (1934) 1002; D. Ceperley Phys. Rev. B 18 (1978) 3126; M. Imada and M. Takahashi, J. Phys. Sot. Jpn. 53 (1984) 3770. [3] K. von Klitzing, G. Dorda and M. Pepper, Phys. Rev. Lett. 45 (1980) 494. [4] D.C. Tsui, H.L. Stormer and A.C. Gossard, Phys. Rev. Lett. 48 (1982) 1559; R.B. Laughlin, Phys. Rev. Lett. 50 (1983) 1395. [5] H. Fukuyama and D. Yoshioka, J. Phys. Sot. Jpn. 48 (1980) 1853; see also references in T. Chakraborty and P. Pietilanen, The Fractional Quantum Hall Effect, Springer Series in Solid States Science 8.5 (Springer, New York 1988). [6] E.Y. Andrei, G. Deville, D.C. Glattli, F.I.B. Williams, E. Paris and B. Etienne, Phys. Rev. Lett. 60 (1988) 2765. [7] D.C. Glattli, G. Deville, V. Duburcq, F.I.B. Williams, E. Paris, B. Etienne and E.Y. Andrei, Surf. Sci. 229 (1990) 344. (81 F.I.B. Williams, E.Y. Andrei, R. G. Clark, G. Deville, B. Etienne, C.T. Foxon, D.C. Glattli, J.J. Harris, E.

C804, units).

Paris and P.A. Wright, in: Localisation and confinement of electrons in semiconductor, Proc. Austr. Phys. Sec. Winter School, Mauterndorf, Feb. 1990; a preprint from V.J. Goldman also indicates a threshold electric field for conduction. (91 F.I.B. Williams, P.A. Wright, R.G. Clark, E.Y. Andrei, G. Deville, D.C. Glattli, 0. Probst, B. Etienne, C. Dorin, C.T. Foxon and J.J. Harris, submitted to Phys. Rev. Lett. WI B. Etienne and E. Paris, J. Phys. (Paris) 48 (1987) 2049. Pll C.T. Foxon. J.J. Harris. D. Hilton, J. Hewett and C. Roberts, J. Semicon. Sci. Techn. 4 (1989) 5. [I21 S.M. Girvin, A.H. Mac Donald and P.M. Platzman. Phys. Rev. Lett. 54 (1985) 581. [I31 R.L. Willett, H.L. Stormer, S.T. Tsui, L.N. Pfeiffer. K.W. West and K.W. Baldwin, Phys. Rev. Lett. 61 (1988) 881. 1141 See review by G. Grunen and A. Zettl, Physics Reports 119 (1985) 117. 1151 H.-W. Jiang and A.J. Dahm, Phys. Rev. Lett. 62 (1989) 13x9. [1’51Y. Imry and S. Ma, Phys. Rev. Lett. 35 (1975) 399. Phys. Rev. B 15 (171 L. Bonsall and A.A. Maradudin. (1977) 15; G. Meissner. H. Namaizawa and M. Voss. Phys. Rev. B 13 (1976) 1370. [I81 V.J. Goldman, M. Shayegan and D.C. Tsui. Phys. Rev. Lett. 61 (1988) 881. I191 R.G. Clark, R.A. Ford, S.R. Haynes, J.F. Ryan, A.J. Turberfield, P.A. Wright, CT. Foxon and J.J. Harris, in: Proc. of Application of High Magnetic Fields in Semiconductor Physics III, Wurzburg, August 1990, to be published; R.G. Clark, R.A. Ford, S.R. Haynes, J.F. Ryan, A.J. Turberfield, P.A. Wright, F.I.B. Williams. G. Deville, D.C. Glattli, J.R. Mallett, M. van der Burgt, P.M.W. Oswald. F. Herlach, C.T. Foxton and J.J. Harris, in: Proceedings of the 19th International Conference on Low Temperature Physics. D.S. Betts, ed., Physica B 169 (North-Holland, Amsterdam, 1991).