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ORNÝ, SMRČKA, AND DEUTSCHMANN Magnetic breakdown Magnetic breakdown in a 1D superlattice 1D superlattice MAGNETORESISTANCE CALCULATIONS FOR A TWO- . . . PHYSICAL REVIEW B 66, 205318 !2002" 2DEG with strong 1D modulation PHYSICAL VOLUME 86, NUMBER 9 PHYSICAL REVIEW B 66 REVIEW LETTERS 1011 cm22 . Each curve is n+ GaAs respect to the inverse field (110)≡z B y 21 SL ponents D of the magne ohmic contacts results are summarized in n+ GaAs frequency components are 50 nm GaAs substrate 50 nm AlGaAs scale, where large and smal blue, respectively. FIG. 1 (color). Sample cross section. The 2DES is field inTo analyze the transport at Alternatively, the GaAs!AlAs in the (110) direction G. 1. Right: 2DES with lateral periodical modulation and the effectively two-point-contactduced geometry. this mayheterointerface be means of ‘‘barriers’’ a positive voltage applied to the n1 GaAs gate. ived as an array of coupled quantum wires !parallel to the x axis". The ‘‘wires’’ !layers ofby GaAs" and the !layers of energy EF of the modulate a1$x As, x"0.32) were 11.9 and 3.1 nm thick in the experiments !Ref. 13" we refer to. Owing to the height of the ‘‘barriers’’ the The (001)-oriented SL provides an atomically precise potential information acquired from a ing between wires is small (E F "4 ! t ! !V 0 ) and a tight-binding model is thus appropriate. Left: Zero field band !along k y ) modulation to structure the 2DES. 11 keeping while i ated by Kronig-Penney model. We keep only the lowest band !thick line" our calculations and we setCalculations 4 ! t ! equal to the and width experiment. of this FIG. 5.in Magnetoresistance. Electron concentrations #from calculation, the top$ 5.1%10 cm-2, 3.8% along the y direction. Her %1011 cm-2. The lowest concentration corresponds to E F )2 ! t ! . that depends on the GaAs layer thickness. For our sample an index n that runs from 2 2 t relevant for these experiments. Other experiments k $ this xvariation of the density, integrated over the z direclyingisminiband to NFor forinte the E ! k x ,k y " R!cϱ $2 !"ϱ " t ! cos k y d. !1" presented by Chowdhury et al.15 who fabricated bilatthe quadratic behavior reconciled. 2m exceeds yy xy10%, as determined by a self-consistent 2Dtion, (001)≡x 200 nm n+ GaAs 100 nm AlAs 30 nm GaAs with 2DES PRB 66, 205318 (2002) ...ato to 1D superlattices superlattices Model of... 1D1D superlattice (strong modulation) Tight-bindingmodel modelfor for strongly strongly modulated modulatedtwo-dimensional two-dimensionalsuperlattices superlattices Tight-binding zz 22 BB Current Current xx yy V(y) VbVb V(y) yy Current Current contact contact Voltage Voltage contact contact Figure 1. 1. Two–dimensional Two–dimensional electron electron gas gas with with periodic periodic unilateral unilateral modulation. modulation. Figure Alternatively,the thesystem systemmay maybe beconceived conceivedas asan anarray arrayofofwires wirescoupled coupledby bytunnelling. tunnelling. Alternatively, 33 1.5 1.5 11 0.5 0.5 -0.5 -0.5 E/2|t| E/2|t| 55 44 33 22 11 00 -1-1 -4-4 -2-2 00 kxkx 22 44 -1-1 11 0.5 0.5 00 k kd/π d/π yy -0.5 -0.5 Figure2. 2. Band Bandstructure structureininzero zeromagnetic magneticfield: field:all allmodulation modulationbands bands(in (iny–direction) y–direction) Figure exceptfor forthe thelowest lowestare arediscarded. discarded. Fermi Fermicontours contours(related (relatedtotoreal realspace spacetrajectories) trajectories) except areclosed closedfor for−2|t| −2|t|<<EEFF <<2|t| 2|t|and andopen openfor forEEFF >>2|t|. 2|t|. are 2 22 2 1. Model Model for for Zero Zero Magnetic Magnetic Field: Field: Semiclassics Semiclassics ~~ kkxx 1. cond-mat/0208224 E(kxx,,kkyy)) = = + 2t 2tcos coskkyydd cond-mat/0208224 E(k + ⇤ Sincethe thepotential potentialin inour oursystem systemisisseparable, separable,zero zeromagnetic magneticfield fieldband bandstructure structure 2m⇤ isisanan Since 2m Experimental determination of Fermi surface 198 Ph. Hofmann / Progress in Surface Science 81 (2006) 191–245 the bulk band structure on different Bi surfaces. The result of the calculation is show Fig. 2, the bulk Brillouin zone is shown in Fig. 3. An inspection of Fig. 2 immediately reveals the semimetallic character of Bi: The b structure can very nearly be described by two filled s bands and three filled p ba Fermi surfaces FIRST-PRINCIPLES SCATTERING MATRICES FOR¼ Reminder: Shubnikov-de Haas Shubnikov - oscillations de Haas oscillations in graphene Potassium binding energy (eV) LETTERS 2.0 1.0 10 8 6 4 2 0 0 35 0.04 0.08 0.12 1/B (T¬1) 0.16 0.5 10 Ig (nA) 30 C (nF cm¬2) 1.5 Landau index m Longitudinal resistance Rxx (B) ¬ Rxx (B = 0 T) (kΩ) Copper 0 2 4 6 10 0 ¬20 25 ¬30 ¬5 20 0 5 10 Vg (V) 15 20 X Dark Ultraviolet illuminated Simulation 0 ¬0.5 T = 300 K Aluminium 10 100 8 B (T) 150 valley/spin degeneracy 200 T (K) 250 L U T 12 14 16 de Haas-van Alphen Organic 2D metal effect Nat. Mat. 10, 357 (2011) 39 PRB 73, 064420 Γ Fig. 2. Bulk band structure of Bi from the tight-binding calculation of Liu and Allan (Ref. [15], green lines) principles calculation by Gonze et al. (Ref. [18], red lines), only in the C–T direction. (For interpretati colours in this figure legend, the reader is referred to the web version of this article.) Bismuth 300 T Figure 4 | Measurement and simulation of the temperature dependent capacitance of sample HD2. The capacitance is calculated using the difference of the charge carrier density at Vg = 0 V and Vg = 3 V. The xx strong increase of the capacitance for T > 260 K indicates the transition from the IPC to the SC regime. The solid line shows the simulated g The inset shows a fan diagram of the SdH oscillations that yields a Berry temperature dependence using an extended Schottky model with phase of . Straight lines are best fits to the data. self-consistent field distribution. By illuminating the sample with ultraviolet SC regime741 can – be744 extended to lower temperatures. The error bars 742 A. Usher et al. / Physica light E 22the(2004) (equivalent to 100INVESTIGATIONS nm of SiO2 dielectric) compared with C(T = are estimated using the errors of the measurements of the Hall resistance DE HAAS-VAN ALPHEN OF THE ORGANIC ETC. 200 K) = 8.5 nF cm 2 . at Vg = 0 V and Vg = 3 V. The inset shows the asymmetric (Schottky-like) 1=2 The SC regime is reached for all samples at elevated gate current at T = 300 K. U X L L U Γ 2DEG B [5], with the addition of a constant background inverse mag. flux quantum temperatures. The higher the implantation dose, the lower is the DOS between LLs [4,5]. to the4SC regime. In addition α-(ET) Θ = 1.8 transition temperature dDLo depends case. Interestingly, as soon as the ultraviolet light is switched off, the 2TlHg(SeCN) 1.5 on the applied gate voltage. As a consequence, the device undergoes a transition back to the IPC regime. This confirms Potts et al. [8] were the !rst to analyse the differential dHvA capacitance is no longer independent of the gate voltage as for the our notion that the two regimes are not kinetically distinguished, oscillations usingbuta itrigorous procedure. They IPC regime, decreases for!tting higher gate voltages. Furthermore but each is stable in time. 1 the leakage the insulating area of at T multi-layer = 300 K shows the For the sample HD3, ⇥(Vg ) and 1/eRH (Vg ) and the associated found that data current takenacross from a range typical asymmetric shape of a Schottky diode (inset Fig. 4), which gate current were measured in the IPC (T = 29 K) and the SC regime 2DEGs !t best to a model DOS which was Lorentzian underlines that for high temperatures the gate has to be described (T = 295 K; Fig. 5a,b). In the SC regime a much higher capacitance, in shapein0.5 with a width a Schottky as in the IPC regime, is observed, resulting in a steeper slope of 0model.that 30 was 60 independent of B. The !the charge To model the transition from the IPC to the SC regime we 1/eR (V ). Notice that in the SC regime neutrality point H g Θ (deg) best !t was obtained with no background. A Gaussian consider a Schottky contact with graphene 0.5 1=2 as the metal and SiC as is already reached at Vg ⇤ 14 V. form with width proportional gave the15 semiconductor. thatan theequally electrostatic The minimal measured charge carrier density nmin = −1 In addition −2toweBincluded gate potential drops partially along the DL, partially along the highly (1/eR H )min is a measure of potential fluctuations in the graphene good !t at low B. resistive semiconductor (Fig. 1c,d). This unusual situation is due layer (electron–hole puddles). Typically, nmin is temperatureImprovements in sample quality and magnetometer to the high activation energy of the vanadium dopand compared independent or increases with temperature if thermally generated with the low injection barrier; for typical contacts design have enabled measurements toSchottky be made onthe charge carriers become important. Here, nmin is much smaller in 0 situation is vice versa. As a consequence, the extension of the DL the high temperature SC regime, indicating a more homogeneous 2 effect is presumably high-mobility single-layer 2DEGs [9–11]. and the associated voltage drop has to be calculatedSaw-tooth self-consistently lateral potential distribution. This unexpected ' locallyBZ 10 15 20 25 30 for different temperatures, were taking into account theatcourse deform the magnetization oscillations observed lowof !the caused by defects in the gate stack, which may (T) the vanadium donor level electrostatic profile in the IPC regime. Improved wafer quality quasi-Fermi level20 , which is well B above from which ˝!c could be determined without !tting. F (103 T) Γ de Haas - van Alphen effect 10 ✓ ◆ Figure 3 | Shubnikov–de Haas oscillations for different gate voltages. 1 2⇡e 1 e f Longitudinal magnetoresistance R of sample HD1 at 6 K. A clear change in = is visible with changing=gate voltages the oscillation frequency (V = 210 V triangles/dotted). BV squares/solid, 0~V circles/dashed, Ae +2102⇡~ p L L electron pocket T hole pocket FIG. 5. %Color& Top row, left-h Prog Surf Sci 81, 191 (2006) Cu; middle panel: majority-spin F viewed along the %111& direction Brillouin zone %BZ& onto a plane p of the two-dimensional BZ. Botto els: projections onto a eplane perp Fig. 3. Bulk Brillouin zone of Bi and a schematic sketch of the Fermi surface (not to scale). The C– corresponds to the C3 axis and the [1 1 1] direction in real space. FIG. 4. Interface conductance G %111& %in units of 10 & m & for an fcc Cu/ Co%111& interface for majority and minority spins plotted as a function of the normalized area element EPL 35, 37 (1996) used in the Brillouin zone summation, )k / A = 1 / Q2. Q, the M (arb. units) 5 ¬10 15 p = 4.7 × 1012 cm¬2 p = 6.1 × 1012 cm¬2 p = 7.3 × 1012 cm¬2 holes spin-orbit gap 0 electrons NATURE MATERIALS DOI: 10.1038/NMAT2988 et al.: PHYSICA ~ Extremal cross section ? B (1/B) / 1/A Obrázek 20: Teplotní pr*b'h vzork* E081#1 a D101#8. Jeho charakter z molekulov!ch svazk%) Extreme quantum limit je dobrou indicií správné funk&nosti vzorku a nakontaktování 2DEGu. ∆ρminim =v ideálních pozicích. L /ρav le<í polohách ( n1 25.813 k , n = 4, 6, 8, . . .), SdHO pom4ry ⌫⇡ 1 ⌫ koncentrace 1 nosi39 nSdHO 11 2 Z jejich polohy byla ur3ena ⇥ 2.5 10 cm . Zúdery Hallovy C cos 2φ + C cos(2φ 4θ) I I ,C ionty povrch vzorku. Napra:ování (Leybold), kde argonové do + slitiny AuGe + vy11 2 atomy a ty :í9í ke vzorku. P9i této metod6 je rychlost atom; v6t:í, sm4rnice potom koncentrace nHall ⇥ 2.4 10 rá>ejí cm a sesv:esm6rov6 ohledem na rozm4ry hallbaru pronikají tedy hloub6ji do materiálu a pokr=vají i mírn6 zakryté prostory (nevytvá9ejí + údaj9m 4θ) + CzUdoby cos(2φ + 2θ 2 +C Cse cos(4φ „stíny@). Titanová vrtsva i vodpovídá tomto p9ípad6 pouze pa9í. (l=1000 µm, d=100 µm) mobilita nosi39 µ ⇥ 1.1 cm /Vs. V8e P9esto>e sou5et v=:ek jednotliv=ch pater sandwiche by m6l b=t 100 nm, na Dektaku r9stu waferu. se zm69ila v=:ka reliéfu 73 nm (pa9ení) a 160 nm (prá:ení). P9esto>e se metody sna>í Shubnikov - de Haas SdH oscillations Plain 2DEG b=t obdobné, srovnatelné se nezdají b=t. Pro to hovo9í i srovnání morfologie na obrázku ní>e. Povrch legendárních kontakt; od pana Melichara vykazovaly charakter podobn= spí:e sou5asnému Leyboldu. D101#8 de Ranieri et al., NJP ’08 Hall bar geometry E081#1 3 10 Rxx Rxy 2.5 Integer Quantum IQHE Hall Effect (IQHE) 6 Rxx Rxy 8 [110] 3 out-of-plane B M 5 4 1 2 0.5 ⌫n=4= 4 ⌫ n=6 =6 0 0 0 0.5 1 1.5 2 B [T] 2.5 3 3.5 4 2 Rxy [kΩ] 1.5 Rxx [kΩ] 6 Rxy [kΩ] Rxx [kΩ] 2 Obrázek 3: Srovnání morfologie ohmick+ch kontakt*. Zleva: napra)ované, napa(ované, pan Melichar, pan Melichar. Rozdílnost metod je zjevná. Autor obrázk*: Z. V+born+. # ! 3 Za>íhání je provád6no v pícce a typické hodnoty jsou 450 C1po 2 minuty. Na2 tyto ohmické kontakty (tedy kontaktu kov-polovi5, kde nedochází k vytvo9ení n=6 materiálu do druhého, tzv. Shottkyho bariéry) bariéry p9i pr;chodu nosi5; z jednoho je posléze nanesena dostate5n6 velká plocha n=4 :icího kovu (nej5ast6ji zlata), na n6j> lze 1 0 zlata je t9eba plochu p9ipevnit ultrazvukovou fixací st9íbrn= vodi5. V p9ípad6 vyu>ití 0 0.5 1 1.5 2 2.5 3 3.5 nejprve pokr=t titanem, kter= zvy:uje adhezi Au. B [T] Obrázek 21: Magnetotransport pro vzorky E081#1 a D101#8. Zatímco p7ede8l; vzorek fungoval2skv4le, D101#8 se skute3n4 choval nekulturn4 – e signál 2⇡~ byl zna3n4 za8um4n;, pravd4podobn4 vlivem (ostré Obrázek 4:neideálního P(íklad kontaktování naokontaktování zlaté )icí desky. Autor obrázk*: Z. V+born+. ⇢xyúzké = struktury 2nco jsem rychle pro8el kolem vodi39. . . ). Oscila3ní chaxy =poté, se objevily 2 e ⌫ 2⇡~ Courtesy of L. Nádvorník (MFF/FZU rakter sice dob7e viditeln; je, nicmén4 Hall9v odpor má p7ibli<n4 t7ikrát men8í sm4rnici,AV) 10 ne< bylo o3ekáváno, zcela neprochází nulou a nevykazuje <ádná plata. Z SdH oscilací "