Phase-coherent transport in InN nanowires of various sizes
C. Blomers, Th. Schapers, T. Richter, R. Calarco, H. Luth, M. Marso
PPhase-coherent transport in InN nanowires of various sizes
Ch. Bl¨omers, Th. Sch¨apers, ∗ T. Richter, R. Calarco, H. L¨uth, and M. Marso Institute for Bio- and Nanosystems (IBN-1) and JARA J¨ulich-Aachen Research Alliance,Research Centre J¨ulich GmbH, 52425 J¨ulich, Germany Institute for Bio- and Nanosystems (IBN-1), JARA J¨ulich-Aachen Research Alliance,and Virtual Institute of Spinelectronics (VISel), Research Centre J¨ulich GmbH, 52425 J¨ulich, Germany Institute for Bio- and Nanosystems (IBN-1) and JARA J¨ulich Aachen Research Alliance,Research Centre J¨ulich GmbH, 52425 J¨ulich, Germany (Dated: November 4, 2018)We investigate phase-coherent transport in InN nanowires of various diameters and lengths.The nanowires were grown by means of plasma-assisted molecular beam epitaxy. Information onthe phase-coherent transport is gained by analyzing the characteristic fluctuation pattern in themagneto-conductance. For a magnetic field oriented parallel to the wire axis we found that thecorrelation field mainly depends on the wire cross section, while the fluctuation amplitude is gov-erned by the wire length. In contrast, if the magnetic field is oriented perpendicularly, for wireslonger than approximately 200 nm the correlation field is limited by the phase coherence length.Further insight into the orientation dependence of the correlation field is gained by measuring theconductance fluctuations at various tilt angles of the magnetic field.
Semiconductor nanowires fabricated by a bottom-up approach have emerged as very interesting sys-tems not only for the design of future nanoscale de-vice structures but also to address fundamental ques-tions connected to strongly confined systems. Regardingthe latter, quantum dot structures, single electronpumps, or superconducting interference devices havebeen realized. Many of the structures cited above werefabricated by employing III-V semiconductors, e.g. InAsor InP. Apart from these more established materials,InN is particularly interesting for nanowire growth be-cause of its low energy band gap and its high surfaceconductivity.
At low temperatures the transport properties of nanos-tructures are affected by electron interference effects, i.e.weak localization, the Aharonov–Bohm effect, or univer-sal conductance fluctuations.
The relevant length pa-rameter in this transport regime is the phase coherencelength l φ , that is the length over which phase-coherenttransport is maintained. In order to obtain informa-tion on l φ , the analysis of conductance fluctuations isa very powerful method. In fact, in InAsnanowires pronounced fluctuations in the conductancehave been observed and analyzed, recently. Here, we report on a detailed study of the conduc-tance fluctuations δG measured in InN nanowires of var-ious sizes. Information on the phase-coherent transportis gained by analyzing the average fluctuation amplitudeand the correlation field B c . Special attention is drawnto the magnetic field orientation with respect to the wireaxis, since this allowed us to change the relevant probearea for the detection of phase-coherent transport.The InN nanowires investigated here were grown with-out catalyst on a Si (111) substrate by plasma-assistedMBE. The measured wires had a diameter d rang-ing from 42 nm to 130 nm. The typical wire length was1 µ m. From photoluminescence measurements an over-all electron concentration of about 5 × cm − was TABLE I: Dimensions and characteristic parameters of thedifferent wires: Length L (separation between the contacts),wire diameter d , root-mean-square of the conductance fluctu-ations rms( G ), correlation field B c . The latter two parameterswere determined for B parallel to the wire axis.Wire L d rms(G) B c (nm) (nm) ( e /h ) (T)A 205 58 1.35 0.38B 580 66 0.58 0.22C 640 75 0.52 0.21D 530 130 0.81 0.15 determined. For the samples used in the transport measurements,first, contact pads and adjustment markers were definedon a SiO -covered Si (100) wafer. Subsequently, the InNnanowires were placed on the patterned substrate andcontacted individually by Ti/Au electrodes. Four wireslabeled as A, B, C, and D will be discussed in detail,below. Their parameters are summarized in Table I. Inorder to improve the statistics, additional wires whichare not specifically labeled, were included in part of thefollowing analysis. A micrograph of a typical contactedwire is depicted in Fig. 4 (inset).The transport measurements were performed in a mag-netic field range from 0 to 10 T at a temperature of 0.6 K.In order to vary the angle between the wire axis and themagnetic field B , the samples were mounted in a rotat-ing sample holder. The rotation axis was oriented per-pendicularly to the magnetic field and to the wire axis.The magnetoresistance was measured by using a lock-intechnique with an ac bias current of 30 nA.The fluctuation pattern for nanowires with differentdimensions are depicted in Fig. 1(a). Here, the nor-malized conductance fluctuations δG for wires A to Ccomprising successively increasing diameters are plotted a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b FIG. 1: (a) Conductance fluctuations normalized to e /h forwires with different length and diameter. The curves are offsetfor clarity. As illustrated by the sketch, the magnetic field isaxially oriented. (b) Conductance fluctuations of wire C witha magnetic field oriented perpendicularly to the wire axis. as a function of the magnetic field B . The field was ori-ented parallel to the wire axis. The measurements wereperformed up to a relatively large field of 10 T. This isjustified, since even at 10 T the estimated cyclotron di-ameter of 70 nm just begins to become comparable to thewire diameter. The conductance variations were deter-mined by first subtracting the typical contact resistanceof (330 ±
50) Ω, and then converting the resistance vari-ations to conductance variations. It can clearly be seenin Fig. 1(a), that for the narrowest and shortest wire,i.e. wire A, the conductance fluctuates with a consider-ably larger amplitude than for the other two wires withlarger diameters and length. The parameter quantifyingthis feature is the root-mean-square of the fluctuationamplitude rms( G ) defined by (cid:112) (cid:104) δG (cid:105) . Here, (cid:104) ... (cid:105) rep-resents the average over the magnetic field. For quasione-dimensional systems where phase coherence is main-tained over the complete wire length it is expected thatrms( G ) is in the order of e /h . As one can in-fer from Table I, for the shortest nanowire, i.e. wire A,rms( G ) falls within this limit. For the other two wiresthe rms( G ) values are smaller than e /h (cf. Table I).Thus, for these wires it can be concluded that the phasecoherence length l φ , is smaller than the wire length L .Beside rms( G ), another important parameter is thecorrelation field B c , quantifying on which field scale theconductance fluctuations take place. The correlation fieldis extracted from the autocorrelation function of δG de-fined by F (∆ B ) = (cid:104) δG ( B +∆ B ) δG ( B ) (cid:105) . The magneticfield corresponding to half maximum of the autocorrela-tion function F ( B c ) = F (0) defines B c . The B c valuesof the measurements shown in Fig. 1 are listed in Table I.Obviously, for wire A, which has the smallest diameter, one finds the largest value of B c . In a semiclassical ap-proach it is expected that B c is inversely proportionalto the maximum area A φ perpendicular to B which isenclosed phase-coherently: B c = α Φ A φ . (1)Here, α is a constant in the order of one and Φ = h/e the magnetic flux quantum. As long as phase coherenceis maintained along the complete circumference, A φ isequal to the wire cross section πd / B c ∝ /d . The B c values given in Table I followthis trend, i.e. becoming smaller for increasing diame-ter d . As can be recognized in Fig. 2 (inset), F (∆ B ) alsoshows negative values at larger ∆ B . This behavior can beattributed to the limited number of modes in the wires,as it was observed previously for small size semiconduc-tor structures. However, as discussed by Jalabert etal. , at small fields F (∆ B ) and thus B c being calcu-lated fully quantum mechanically correspond well to thesemiclassical approximation.In order to elucidate the dependence of B c on the wirediameter in more detail, a larger number of wires wasmeasured. As can be seen in Fig. 2(a), B c systemati-cally decreases with d . Leaving out wire D which has thelargest diameter, the decrease of B c is well described bya 1 /d -dependence. As mentioned above, for short wires( L ≈
200 nm) we found that phase coherence is main-tained over the complete length. This length correspondsto a circumference of a wire with a diameter of about 64nm. Except of wire D, d is in the order of that value, sothat one can expect that phase coherence is maintainedwithin the complete cross section. For the parameter α we found a value of 0.24, which is by a factor of 4 smallerthan the theoretically expected value of 0.95. Choos-ing α = 0 .
95 would result in lower bound values of B c being larger than all corresponding experimental values,which is physically unreasonable. We attribute the dis-crepancy to the different geometrical situation, i.e. forthe latter a confined two-dimensional electron gas with aperpendicularly oriented magnetic field was considered, while in our case the field is oriented parallel to the wireaxis. In addition, an inhomogeneous carrier distributionwithin the cross section, e.g. due to a carrier accumu-lation at the surface, can also result in a disagreementbetween experiment and theoretical model. As can beseen in Fig. 2(a) (inset), the data point of the wire withthe largest diameter of 130 nm, i.e. wire D, is foundabove the calculated curve. This indicates that presum-ably for this sample, A φ is slightly smaller than the wirecross section.Next, we will focus on measurements of δG with a mag-netic field oriented perpendicular to the wire axis. Asa typical example, δG of wire C is shown in Fig. 1(b).Here, a correlation field of 0 .
17 T was extracted, whichis smaller than the value of corresponding measurementswith B parallel to the wire axis [c.f. Fig. 1(a) and Ta-ble I]. The smaller value of B c can be attributed to the ef- FIG. 2: (a) Correlation field B c as a function of the wirediameter d . As illustrated in the schematics the magneticfield B was oriented axially. The solid lines corresponds tothe calculated correlation field. The inset shows F (∆ B ) /F (0)for wire C. (b) B c as a function of the maximum area A = Ld (see schematics) of the wire. The magnetic field is orientedperpendicular to the wire axis. The solid lines represents thecalculated lower boundary correlation fields assuming α =0 .
95 and 0.24, respectively. fect that now the relevant area for magnetic flux-inducedinterference effects is no longer limited by the relativelysmall circular cross section but rather by a larger areawithin the rectangle defined by L and d , as illustrated bythe schematics in Fig. 2(b).In Fig. 2(b) the B c values of various wires are plottedas a function of the maximum area A max = Ld pene-trated by the magnetic field. As a reference, the cal-culated curve using Eq. (1) and assuming A φ = A max are also plotted. It can be seen that the B c values oftwo wires with small areas, including wire A, match tothe theoretically expected ones if one takes α = 0 .
95, asgiven by Beenakker and van Houten. This correspondsto the case of phase-coherent transport across the com-plete wire, as it was, in case of wire A, already concludedfrom the rms( G ) analysis. For all other wires the B c values are above the theoretically expected curve, cor-responding to the case A φ < A max . At this point, onemight argue that for B oriented along the wire axis abetter agreement is found for α = 0 .
24. However, ascan be seen in Fig. 2(b), if one assumes α = 0 .
24 allexperimental values are above the calculated curve, i.e. A φ < A max . This does not agree with the observationthat for short wires rms( G ) is in the order of e /h . Weattribute the difference between the appropriate α valuesfor different field orientations to the different characterof the relevant area penetrated by the magnetic flux, e.g.due to carrier accumulation at the surface.Beside B c we also analyzed the fluctuation amplitudefor five different wires with B oriented perpendicular to FIG. 3: rms( G ) for wires with a diameter of (75 ±
5) nmas a function of wire length L (square). The magnetic fieldis oriented perpendicular to the wire axis. The calculateddecrease of rms( G ) proportional to L − / is plotted as solidline. The inset shows B c vs. L for wires with d ≈
75 nm.The dashed line corresponds to the calculated value of B c assuming l φ = 430 nm. the wire axis. Only wires with comparable diameters of(75 ±
5) nm were chosen, here. It can be seen in Fig. 3that rms( G ) tends to decrease with increasing wire length L . From the previous discussion of B c it was concludedthat for long wires, as it is the case here, l φ < L . In thisregime rms( G ) is expected to depend on L as rms( G ) = β e h (cid:18) l φ L (cid:19) / , (2)with β in the order of one. The above expression is validas long as the thermal diffusion length l T = (cid:112) (cid:126) D /k B T ,is larger than l φ . Here, D is the diffusion constant. Fromour transport data we estimated l T ≈
600 nm at T =0 . l φ = 430 nm and β = 1.For the limit l φ < L , a correlation field according to B c = 0 . /dl φ is expected. As confirmed in Fig. 2(b),most experimental values of B c are close to the calculatedone.If one compares the rms( G ) values for wires with d ≈
75 nm and B oriented axially (not shown here) withthe corresponding values for B oriented perpendicularly,one finds, that both are in the same range. Thus it canbe concluded that the fluctuation amplitude does not sig-nificantly depend on the magnetic field orientation. Thisis in contrast to the correlation field, where one finds asystematic dependence on the orientation of B .In order to discuss the latter aspect in more detail thecorrelation field was studied for various tilt angles θ ofthe magnetic field. Figure 4 shows B c of sample D if θ is FIG. 4: Correlation field B c of wire D as a function of theangle θ between the wire axis and B . The solid line representsa linear fit. The broken line corresponds to the theoreticallyexpected B c if phase-coherent transport is assumed in thecomplete wire. The left-hand-side inset shows a schematics ofthe geometrical situation. The right-hand-side inset shows amicrograph of a 580-nm-long wire with a diameter of 66 nm. increased from 0 ◦ to 90 ◦ . The inset in Fig. 4 illustrateshow θ is defined. Obviously, B c decreases with increasingtilt angle θ . As explained above, the value of B c is a mea-sure of the maximum area normal to B , which is enclosedphase-coherently by the electron waves in the wire [seeFig. 4 (schematics)]. As long as θ ≤ arctan( L/d ), thismaximum area is given by A ( θ ) = πd / θ . The ex-pected θ -dependence of the correlation field is then given by B c ( θ )= B c (0) cos( θ ), with B c (0) the correlation fieldat θ = 0. As can be seen in Fig. 4, the calculated cor-relation field B c , corresponding to fully phase-coherenttransport, decreases much faster with increasing θ thanthe experimentally determined values. The experimentalsituation is better described by a linear decrease. As itwas discussed above, at θ = 0 one can assume that thearea enclosed phase-coherently is equal to A (0). How-ever, if the tilt angle is increased the maximum wire crosssection A ( θ ) presumably becomes larger than A φ , result-ing in a much smaller decrease of B c than theoreticallyexpected for fully phase-coherent transport. In addition,as pointed out above, the different tilt angles result inan angle-dependent parameter α . This is supported bythe measurements of B c for B parallel and perpendic-ular to the wire axis, where different values for α weredetermined, respectively.In conclusion, the conductance fluctuations of InNnanowires with various lengths and diameters were in-vestigated. We found that for an axially oriented mag-netic field the correlation field B c and thus the areawhere phase-coherent transport is maintained is limitedby the wire cross section perpendicular to B . In con-trast, rms( G ) decreases with the wire length, since thisquantity also depends on the propagation of the electronwaves along the wire axis. If the magnetic field is ori-ented perpendicularly we found that for long wires B c islimited by l φ rather than by the length L . Our inves-tigations demonstrate that phase-coherent transport canbe maintained in InN nanowires, which is an importantprerequisite for the design of quantum device structuresbased on this material system. ∗ Electronic address: [email protected] C. Thelander, P. Agarwal, S. Brongersma, J. Eymery,L. Feiner, A. Forchel, M. Scheffler, W. Riess, B. Ohlsson,U. G¨osele, et al., Materials Today , 28 (2006). W. Lu and C. M. Lieber, J. Phys. D: Appl. Phys. , R387(2006). K. Ikejiri, J. Noborisaka, S. Hara, J. Motohisa, andT. Fukui, J. Cryst. Growth , 616 (2007). M. T. Bj¨ork, B. J. Ohlsson, C. Thelander, A. I. Persson,K. Deppert, L. R. Wallenberg, and L. Samuelson, Appl.Phys. Lett. , 4458 (2002). T. Bryllert, L.-E. Wernersson, T. Lowgren, and L. Samuel-son, Nanotechnology , 227 (2006). Y. Li, J. Xiang, F. Qian, S. Gradecak, Y. Wu, H. Yan,D. Blom, and C. M. Lieber, Nano Letters (2006). S. D. Franceschi, J. A. van Dam, E. P. A. M. Bakkers,L. Feiner, L. Gurevich, and L. P. Kouwenhoven, Appl.Phys. Lett. , 344 (2003). C. Fasth, A. Fuhrer, M. T. Bjork, and L. Samuelson,Nanoletters , 1487 (2005). A. Pfund, I. Shorubalko, R. Leturcq, and K. Ensslin, Appl.Phys. Lett. , 252106 (2006). A. Fuhrer, C. Fasth, and L. Samuelson, Appl. Phys. Lett. , 052109 (2007). J. A. van Dam, Y. V. Nazarov, E. P. A. M. Bakkers,S. D. Franceschi, and L. P. Kouwenhoven, Nature ,667 (2006). C. H. Liang, L. C. Chen, J. S. Hwang, K. H. Chen, Y. T.Hung, and Y. F. Chen, Appl. Phys. Lett. , 22 (2002). C.-Y. Chang, G.-C. Chi, W.-M. Wang, L.-C. Chen, K.-H.Chen, F. Ren, and S. J. Pearton, Appl. Phys. Lett. ,093112 (2005). R. Calarco and M. Marso, Appl. Phys. A , 499 (2007). C. W. J. Beenakker and H. van Houten, in
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