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Effect of a tilted magnetic field on the orientation of Wigner crystals Shi-Jie Yang1 1Department of Physics, Beijing Normal University, Beijing 100875, China We study the effect of a tilted magnetic field on the orientation of Wigner crystals by taking 5 account of the width of a quantum well in the z-direction. It is found that the cohesive energy of 0 theelectroniccrystalisalwayslowerforthe[110]directionparalleltothein-planefield. Inarealistic 0 sample,adomainstructureformsintheelectronicsolidandeachdomainorientsrandomlywhenthe 2 magnetic field is normal to the quantum well. As the field is tilted an angle, the electronic crystal n favors to align along a preferred direction which is determined by the in-plane magnetic field. The a orientation stabilization is strengthened for wider quantum wells as well as for larger tilted angles. J Possibleconsequenceofthetiltedfieldonthetransportpropertyintheelectronicsolidisdiscussed. 0 1 PACSnumbers: 73.20.Qt,73.40.-c,73.21.Fg ] l I. INTRODUCTION electroniccrystalinaHartree-Fock(HF)approximation. l a We find that it becomes anisotropic in the tilted mag- h netic field. The [110] axis of the hexagonal electronic - It was initially predicted by Wigner that two- s crystalfavorsto align along the direction of the in-plane dimensional (2D) electrons crystallize into a triangular e magnetic field. This trend of orientation stabilization m latticeinthelowdensitylimitwheretheelectron-electron is strengthened for larger tilted angles. It also shows interactions dominate over the kinetic energy. In an ide- . thatthe energydifference betweentwo orthogonalorien- t ally clean 2D system, the critical r (r = U/ǫ , cor- a s s F tations of the electronic crystal increases with the width m responding to the ratio of the Coulomb energy scale of the quantum well. The in-plane field favors the do- U to the kinetic energy scale of the Fermi energy ǫ ) - F mainstoorienttothesamedirection. Thus,theeffective d was presented to be 37±5 from quantum Monte Carlo impurity density is reduced as the field is tilted. We will n simulations1. A strong magnetic field perpendicular to discuss the possible consequence of such effect on the o the 2D plane can effectively localize electron wave func- c tions while keeping the kinetic energy controlled2. Since transport properties in the electronic solids. [ thislessensthe otherwiseseverelow-densitycondition,it 1 is believed that the Wigner crystal (WC) can be stabi- II. ANISOTROPIC COHESIVE ENERGY OF v lizedina sufficiently strongmagneticfield3,4,5. Approxi- THE ELECTRONIC CRYSTAL 9 matecalculations6 haveshownthattheWCbecomesthe 8 lowestenergystatewhenthefillingfactorν <1/6forthe 1 Consider an electron moving on a x-y plane under the 1 GaAs/AlGaAs electron system and around ν = 1/3 for influence ofa strongmagneticfieldwhichistiltedanan- 0 the hole system. Since the impurities pin the electronic 5 crystal, a domain structure forms in a realistic sample7. gle θ to the normal, with B~ = (Btanθ,0,B). The elec- 0 While the electrons in a domain have an order as they tronisconfinedinaharmonicpotentialV(z)= 21mbΩ2z2 t/ are in the ideal crystal, the orientations of the domains inthez-direction,wheremb isthebandmassoftheelec- a tronandΩthecharacteristicfrequency. Suchaquantum are random. m well has been chosen to deal with many quantum Hall - Presently,themeasurementinatiltedfieldhasbecome systems11,12 to substitute the realistic potential which is d anestablishedtechniquetoexplorethevariouscorrelated either triangular or square. It was also used to discuss n properties in single layer as well as in double layers 2D the giant magneto-resistance induced by a parallel mag- o electron systems8. In a previous work9, we have com- neticfield13. Weworkinthe”Landaugauge”bychoosing v:c plianrelidquthide10grtoounthdestealetceternoenricgiesosloidftshteatgeenaetraaligzievdenLatiultgehd- the vector potential A~ = {0,xBz −zBx,0}. The single particle wave function for the lowest LL are: i angle. It was found that the critical filling factor ν at X c ar athneIgnlseot.lhidis-lwiqourikd,twreanwsiiltlioenxainmcirneeastehsewreitlahtiinoncroeafstihnegotriliteend- φX(~r) = 1Lye−iXy/l2BΦω0+ −(x−X)sli+nθ˜+zcosθ˜! tation of the hexagonal WC with the in-plane magnetic p (x−X)cosθ˜+zsinθ˜ ×Φω− , (1) field as well as the width of the 2D quantum well. In a 0 l ! wide quantum well, the electron wave function extends − in the z-direction, hence may reduce the coulomb inter- where l is the magnetic length and l2 = h¯/mω . X B actions. The in-plane field deforms the electron wave is an integer multiple of 2πl2/L . Φω±±is the harm±onic function, causing the interaction energy to vary accord- B y 0 oscillator wave function in the lowest energy level corre- ing to the different patterns or orientations of the electronic crystals. We calculate the cohesive energy of the sponding to the frequencies ω± and tanθ˜= ω+2ω−c2ωc2 tanθ, 2 withthecyclotronfrequencyω =eB/m c. Thefrequen- where∆(Q~)istheorderparameter. Thecohesiveenergy c b cies ω are given by10 canbe calculatedin the same wayas ithas been done in ± Refs.[2,14,15]: 1 ω2 1 ω2 ω±2 = 2(Ω2+cosc2θ)±r4(Ω2− cosc2θ)2+Ω2ωc2tan2(2θ). Ecoh = 21ν uHF(Q~)|∆(Q~)|2, (11) The Hamiltonian is given by QX~=0 6 1 where ν is the filling factor of the lowest Landau level. Hˆ = ρˆ(~q)v(~q)ρˆ(−~q), (3) We carry out the self-consistent HF computation on 2L L x y Xq~ a hexagonal lattice with the wave vectors of the order where v(~q) = 4πe2 is the Fourier transformation of parameters as Q~ = [(j + 21)Q0,√23kQ0], where j and k κ0(q~k2+qz2) are integers. Following the procedure in Ref.[2], when the Coulomb interaction. Here ~q is the in-plane mo- NQ0Q0l2 = 2Mπ, with N and M being integers, the mentum and qz is the momentumkperpendicular to the LandxauylBevelsplitsintoN Hofstadterbands. WhenN = quantum well. 6 and M = 1 the WC has the lowest energy. In our FromEq.(1),theelectrondensityoperatorisexpressed calculations, we choose ν = 0.12, at which the ground in the momentum space as state is a Wigner crystal. Figure 1 displays the dependence of the cohesive en- ρˆ(~q)= eiqxXa†X−aX+Fθ(~q), (4) ergy of the electronic crystal on the tilted angle θ for X various orientation angles of the crystal to the in-plane X magnetic field, where φ is the angle between one side of where X = X ±qylB2/2. a†X(aX) creates (destroys) an the hexagonal lattice with the in-plane field. It shows electroni±nthestateφX. HereFθ(~q)=e−γ2/4−α2/4,with thatof the two typicalconfigurationsof orientationwith respect to the in-plane field: the [100] and the [110], the α2 = (q cosθ˜−q sinθ˜)2l2 +q2l4/l2 cos2θ˜ x z y B energy is always lower for [110] direction parallel to the − − γ2 = (q cosθ˜+q sinθ˜)2l2 +q2l4/l2 sin2θ˜. (5) in-plane field. The energy difference increases with the z x + y B + tilted angle. In Fig.2 we plotthe relationof the cohesive Substitute Eq.(4) into Eq.(3) and carry out the usual energy of the hexagonal lattice with the characteristic procedure of the HF decoupling of the Hamiltonian, we frequency Ω for tilted angles θ = 00 and θ = 43.20, re- get spectively. The [110] direction is along the in-plane field n forbothcurves. Thesmallerthecharacteristicfrequency HHF = 2L uHF(~qk)∆(−~qk) e−iqxXa†X+aX−, is, i.e., the wider the quantum well is, the higher the Xq~k XX cohesive energy. The in-plane magnetic field can lower (6) the cohesive energy of the electronic crystal. From Fig.2 wheren =1/2πl2 isthedensityofonecompletelyfilled one can see that the energy difference becomes larger L B LL and for wider quantum wells. Our calculations show that whenthetiltedangleincreasesfurther,the energydiffer- 2πl2 ∆(~qk)= LxLBy e−iqxXha†X+aX−i (7) emnacgenwetililcinficerldeassteasbiiglniziefiscathntelyo,riimenptlaytiinogntohfatthteheelienc-tprolanniec X X crystal more effectively. istheorderparameterofthechargedensitywave(CDW). The HF potential is denoted with u (~q ) = u (~q )− HF H uex(~q ). The Hartree term uH(~q ) is givekn by (in uknits III. TRANSPORT PROPERTY OF THE of e2/kκ l ) k ELECTRONIC SOLID 0 B dq 1 uH(~qk)=Z πlBz ~q2+qz2[Fθ(~q)]2, (8) sulPtinofnibnrgeaokfintgheofWthigenterrancrsylasttiaolnbayl inimvapruiarintcieeshaassabereen- k widely investigated4,16. Sherman7 had studied the an- and the exchange term u (~q ) in the reciprocal space turns out to be proportioenxalkto the real-space Hartree gular pinning and the domain structure of the electronic crystalmediatedbyacoustic-phononinIII−V semicon- potential as,11,14 ductor. Ourcalculationshowsthatthein-planemagnetic dp~ field may serve as a tunable means to probe the orienta- uex(~qk)=−2πlB2 (2πk)2uH(p~k)eip~k×q~kl2B. (9) tionofthecrystal. Belowwewillexploretheimplications Z ofthepreferredorientationoftheelectroniccrystaltothe Allowing the chargedensity waveby making ansatz in transport measurements. In realistic samples a domain the plane structure is formed due to a finite impurity density. The electrons in each domain are ordered as they are in the <a†X−Qyl2B/2aX+Qyl2B/2 >=eiQxX∆(Q~), (10) crystal. Inthe absenceofthe in-planefield, eachdomain 3 orients randomly, just like the domains in ferromagnets. where A=[4h¯vF]1/2. The characteristic temperature T kBξ 0 An ideal electronic crystal is an insulator and the con- above which the fixed range hopping dominates is deter- ductivity σxx ∝ e−∆0/kBT17,18. This thermal activation mined by R¯ =R0, namely form of the conductivity implies that the electrons are hopping with a fixed range mechanism. It has been con- firmed by experiments that ∆0 is of the order 1K19. In 2h¯2l a realistic domain structure, however, the electrons may kBT0 = B(πnI)3/2. (14) m hopbetweentheedgesoftherandomlyorienteddomains. b Now, we discuss the possible effect of the tilted field. Since the experimentalreachabletemperaturemaybe as As we have discussed, the existence of an in-plane field lowas10mK,thevariablerangehoppingmechanismmay deforms the electron wave function. However, this wave work in this temperature regime20. In the following, we function deformation does not qualitatively change the willdiscussthepossibleconsequenceofthetiltedfieldon electronhoppingmechanismatagiventemperature. The the transport properties. major effect of the tilted field would be on the variation In the usual Anderson localizationthe envelope of the of T . As we have shown, the in-plane field lowers the 0 wavefunctionfallsoffexponentiallyasφ ∼e r/ξ,where 0 − cohesive energy of the Wigner crystal and forces the do- ξ is the localization length. With a magnetic field the mains align to the same direction. Thus, the role of the electronicwavefunctionofaperfectsystemisessentially in-planefieldistointegratethedomainsintolargerones. a Gaussian as φm ∼ e−r2/2l2B. In a slightly disordered Inthisway,thein-planefieldcausessomeofimpuritiesto system one can think that some of the states will be beirrelevantandthereforereducesthe effectiveimpurity pinned at certain isolated impurity sites. The mixing of density. TodetermineT from(14),onlytherelevantim- 0 these states due to quantum-mechanicaltunnelling leads purities shouldbe countedin. Hence, one canreplacen I to a simple exponential tail in the wave function18. In by an effective impurity density n (B ). From Eq.(14), I a strong magnetic field, the electrons condense into a we see that T is sensitive to n (B ).kIn a strong mag- 0 I crystal at lower filling factors. When the temperature is neticfieldthedecaylengthiscompakrabletothecyclotron high enough the transport is of the thermal activation radius ξ ∼R 24. For a sample with n ∼1.0×108cm 2, c I − form, which implies that the electrons are hopping with we estimate T ∼ 40 mK. This temperature is experi- 0 a fixed range mechanism17,18. The hopping range is de- mentally reachable. Therefore, it is possible to observe termined by R0 = 1/πnI, where nI is the impurity a change of transport behavior that displays a crossover density. However, localized states may exist along the fromthevariabletothefixedrangehoppingunderproper p edges of the domains of the electronic crystal because of parameters and temperature as the tilted angle rises. the impurities. When the temperature is sufficiently low suchthatthereisnearlynophononwithenergytoassist theelectronmakingthenearesthopping,Mott’svariable hopping mechanism20 allows the electrons hop a larger distance R > R to a state which has a smaller energy 0 difference ∆(R). In turn, the hopping conduction is de- IV. SUMMARY terminedbythetypicaldecayrateofthetailsofthewave function. The hopping probability is then given by We have shown that the WC has a preferred orien- p∝exp[−R/ξ−∆/k T], (12) B tation with respect to the in-plane magnetic field. We argued that there are domains in a realistic sample and where R=|R~ −R~ | and ∆ is the activation energy. i j predictedthat the temperature dependence ofthe trans- As in the quantum Hall effect regime, strong inter- port behavior may be different in different temperature action between electrons leads to the system condens- regime. Moreover,weemphasizedthatthispreferredori- ing into a WC. The Coulomb gap depresses the den- entation of the crystal may lead to an in-plane field in- sity of states near the Fermi surface21,22. Efros et al23 duced crossover from the variable range hopping to the had derived the density of states near the Fermi surface fixed range hopping of the transport mechanism in the N(E) ∝ |∆E| = |E −E |. From these considerations, F 2D electronic solid. We expect future experiments to one can get the conductivity in the variable range hop- verify our prediction. ping as21, ThisworkwassupportedbyagrantfromBeijingNor- σ ∝p∝e A/T1/2, (13) mal University. xx − 1 B. 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Fogler, A.A. Koulakov, angle φ. and B.I. Shklovskii,Phys. Rev.B 54, 1853(1996). 15 S.J. Yang, Y. Yu, and Z.B. Su, Phys. Rev. B 62, Figure 2 The cohesive energy of the hexagonal lattice 13557(2000). with the characteristic frequency Ω for tilted angle θ = 16 A.J. Millis and P.B. Littlewood, Phys. Rev. B 50, 00 and θ = 43.20, respectively. The energy difference 17632(1994). increases with decreased Ω. -0.1520 -0.1525 -0.1530 h -o0.1535 cE -0.1540 -0.1545 -0.1550 -0.1555 -0.1560 30 -0.1565 20 -0.15700 10 0 10 20 30 -10 j 40 -20 q 50 60 -30 -0.130 0 -0.135 q=0 0 q=43.2 -0.140 h Eco-0.145 -0.150 -0.155 -0.160 0.0 0.5 1.0 1.5 2.0 2.5 3.0 W/w c