LETTER pubs.acs.org/NanoLett
Electronic Phase Coherence in InAs Nanowires Ch. Bl€omers,*,† M. I. Lepsa,† M. Luysberg,‡ D. Gr€utzmacher,† H. L€uth,† and Th. Sch€apers*,† †
Peter Gr€unberg Institut (PGI-9) and JARA-Fundamentals of Future Information Technology, Forschungszentrum J€ulich GmbH, 52425 J€ulich, Germany ‡ Peter Gr€unberg Institut (PGI-5) and Ernst-Ruska Centre for Microscopy and Spectroscopy with Electrons, Forschungszentrum J€ulich GmbH, 52425 J€ulich, Germany ABSTRACT: Magnetotransport measurements at low temperatures have been performed on InAs nanowires grown by In-assisted molecular beam epitaxy. Information on the electron phase coherence is obtained from universal conductance fluctuations measured in a perpendicular magnetic field. By analysis of the universal conductance fluctuations pattern of a series of nanowires of different length, the phase-coherence length could be determined quantitatively. Furthermore, indications of a pronounced flux cancelation effect were found, which is attributed to the topology of the nanowire. Additionally, we present measurements in a parallel configuration between wire and magnetic field. In contrast to previous results on InN and InAs nanowires, we do not find periodic oscillations of the magnetoconductance in this configuration. An explanation of this behavior is suggested in terms of the high density of stacking faults present in our InAs wires. KEYWORDS: InAs nanowires, phase coherence length, Universal Conductance Fluctuations, magnetotransport measurements, flux cancellation effect, III-V semiconductors
S
emiconductor nanowires have attracted high interest for some time because they allow the study of new quantum transport phenomena and, furthermore, are promising building blocks for future nanoelectronics.1,2 In this context InAs is a material of particular interest as it exhibits a low electronic effective mass, high electron mobilities, and a strong spinorbit interaction. Additionally, due to its surface state distribution with charge neutrality level in the bulk conduction band region, the Fermi level at the surface is usually pinned above the lower conduction band edge.3 Therefore electrons accumulate at the surface and Ohmic contacts can be fabricated easily.46 Due to these intriguing properties, InAs nanowires have been widely used to investigate their potential for conventional field effect transistor applications.710 Furthermore interesting quantum transport phenomena were investigated, e.g., quantum dots were fabricated by introducing axial heterostructures or using topgates to achieve a local confinement.1117 In order to determine fundamental parameters with regard to phase-coherent transport, i.e., phase-coherence length and spin relaxation length, the weak (anti)localization has been studied.1822 As an alternative approach, we analyzed aperiodic universal conductance fluctuations (UCFs)23,24 measured in a magnetic field perpendicular to the wire axis to obtain information on phase coherent transport.25 Compared to previous studies on InAs and InN nanowires22,2629 we performed a systematic analysis of wires of similar diameter and different lengths in a temperature range from 0.5 to 25 K. Thereby we are able to extract lϕ quantitatively. At 0.5 K a phase-coherence length as large as 450 nm is extracted. From the large new data basis presented in this work, we were able for the first time to perform a r 2011 American Chemical Society
detailed quantitative analysis of the correlation field BC of the conductance fluctuations. Deviations of BC from common values established so far in literature lead to the detection of a new flux cancelation effect which is related to the topology of the conduction channel being not strictly one-dimensional. Because of the large measured coherence length, we expected to find evidence for regular magnetoconductance oscillations due to electron interference in a magnetic field parallel to the wire axis, e.g., AharonovBohm (AB) or Al’tshulerAronovSpivak (AAS) oscillations. While these oscillations have the same origin as UCF, AB and AAS oscillations only appear if the electron trajectories are restricted to ringlike structures.3033 In the case of nanowires, the projection of the wire shell to the cross section plane constitutes the ring and indeed regular oscillations have been observed in several wirelike structures with a highly conductive shell.3437 In contrast to these previous results, in the present InAs nanowires such quantum oscillations are less pronounced or are lacking completely. This effect, untypical for low band gap semiconductor nanowires, is discussed and possible explanations are offered. The InAs nanowires investigated in this study were grown by In-assisted MBE on a GaAs(111)B substrate. In this method, indium droplets act as nucleation centers and no extrinsic metal is needed to catalyze the nanowire growth. Prior to growth, the substrate was coated with a hydrogen silsesquioxane (HSQ) layer of 10 nm thickness and subsequently etched in hydrofluoric acid to a thickness of approximately 6 nm. The etching process Received: April 1, 2011 Revised: July 28, 2011 Published: August 17, 2011 3550
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Figure 1. Scanning electron micrographs of the undoped InAs wires after growth (a) and of a contacted wire (b). (c) Histogram of the resistivity distribution for doped and undoped wires and typical temperature dependence of the resistivity (inset). (d) Transmission electron micrograph of a typical undoped InAs wire. The blue dashed line indicates the sidewall facets; the yellow dotted lines mark {111} twinning planes.
causes small, randomly distributed pinholes in the HSQ layer. Above these pinholes In droplets can accumulate and hexagonalshaped nanowires grow via the vaporliquidsolid/vapor solidsolid process along the [111] direction. In the present work we show measurements on undoped and Si-doped InAs wires. Both types of wires were grown at a substrate temperature of 595 C and an InAs layer growth rate of 0.07 A/s. The As4 partial pressure was adjusted to 8 107 Torr for the undoped sample and to 1.2 106 Torr for the doped sample. For n-doping, the Si flux was adjusted to a value that corresponds to a layer doping concentration of 5 1018 cm3. Figure 1a shows a scanning electron micrograph of undoped InAs wires. In order to contact the wires, they were transferred to a highly n-doped Si(100) substrate, covered with 100 nm of SiO2, which was used as a backgate dielectric. Nonannealed Ti/Au contacts were prepared by electron beam lithography and metallization. Prior to the metal deposition an oxygen plasma was used to clean the contact area while the native oxide was removed by an Ar+ plasma. Wires with a diameter d ranging from 80 to 140 nm and an inner contact separation length L ranging from 70 nm to 2 μm have been characterized at room temperature. (We choose the term diameter as it is commonly used for wire-like structures. Strictly speaking, the diameter corresponds to twice of the hexagon side length.) Figure 1b shows a scanning electron micrograph of a typical contacted InAs wire. Four-terminal resistance measurements were performed, and a typically temperatureindependent contact resistance of 12 kΩ was determined. The resistivity at room temperature (F = R(A/L), with the cross section area A = (3/8)31/2d2) showed a wide spread between 0.02 and 0.28 Ω cm with a mean value of 0.06 ( 0.04 Ω cm for the doped wires and 0.09 ( 0.06 Ω cm for the undoped wires
(see Figure 1c). Due to the broad spread and the lack of statistics for undoped wires, we cannot conclude if the dopants are incorporated during growth or if they are electrically active and therefore we do not focus on this issue in the following. A reason for the broad spread in resistivity could be that no surface passivation was performed. The native oxide probably is inhomogeneous, leading to inhomogeneities in the surface state distribution and thus in the surface Fermi level pinning.3841 Additionally inhomogeneities in the density of structural defects (described later) could be a reason for the observed spread. The inset of Figure 1c shows the typical temperature dependence of the resistivity of the doped InAs wires. The resistivity decreases slightly with increasing temperature in the range from 4 K to room temperature. From backgate measurements at room temperature an electron concentration of 5 1017 to 5 1018 cm3 was calculated for both doped and undoped wires, using formulas derived elsewhere.42,43 We emphasize that the obtained values are rough approximations for two reasons. On the one hand the applied backgate voltage charges the interface states between the InAs wire and its native oxide and/or the SiO2, which leads to an increased threshold voltage and an overestimation of the carrier concentration.44 On the other hand, especially for wires with small contact separation lengths, the sourcedrain electrodes screen the applied backgate voltage, leading to an underestimation of the carrier concentration. Therefore we again cannot conclude details about the incorporation of dopants. Using a simple Drude formalism, an elastic mean free path of le = 1030 nm was estimated; thus the transport takes place in the diffusive regime. Due to the weak temperature dependence of the resistivity one can assume, that le is only slightly higher at low temperatures. This estimation is in 3551
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agreement with Roulleau et al.21 and Dhara et al.,19 where le ≈ 3040 nm is estimated for InAs nanowires at liquid helium temperatures. Figure 1d shows a transmission electron microscope picture of a typical undoped InAs wire. The picture was taken in the Æ110æ direction. The structure is cubic zinc blende with a high density of {111} twinning planes perpendicular to the growth direction. On average, the twinning planes have a distance of 2.5 nm. The twinning is correlated with a faceting of the sidewalls of the wire (see inset of Figure 1d). This type of nanowire structure has already been observed in GaP, InP, and InAs nanowires grown by metal organic vapor phase epitaxy.4547 The high density of twin boundaries might be a possible reason for the low elastic mean free path. Elastic scattering is expected to occur because of the symmetry mismatch introduced by the boundary.48 Low temperature magnetotransport measurements were performed in a He-3 cryostat in a temperature range between 0.5 and 25 K. Standard lock-in technique was used to measure the resistance of the devices with a low bias current of 13 nA to prevent Ohmic heating of the samples. We carried out measurements on a set of six wires (A1A6) in perpendicular magnetic field orientation (with respect to the wire axis) and on a set of three wires (B1B3) in parallel magnetic field orientation. The dimensions of the wires are shown in Table 1. Wires A3, Table 1. Wire Dimensions, Maximum rms (G), Minimum BC, and γ Values of the UCF length
diameter
rms (G)
wire
L (nm)
d (nm)
(e2/h)
BC (T)
γ
A1
185
110
0.35
0.27
1.35
A2
220
95
0.29
0.27
1.38
A3
280
100
0.3
0.21
1.46
A4
305
95
0.3
0.22
1.54
A5
420
110
0.31
0.11
1.24
A6
1120
115
0.14
0.1
B1
70
140
0.54
0.19
B2 B3
230 120
105 80
0.27 0.5
0.18 0.45
A4, and A6 are from the doped sample; the others are from the undoped one. In panels a and b of Figure 2 typical magnetoconductance curves G(B) of a short (A1) and a long nanowire (A6) are shown for various temperatures, respectively. Both wires show pronounced conductance fluctuations with a decreasing amplitude for increasing temperature. By comparing the patterns, one finds that the fluctuation of the long wire takes place on a smaller magnetic field scale. The different field scales of the fluctuation in Figure 2 can be explained by the different areas Ld (with the length L and diameter d of the wire), which are penetrated by the magnetic field. The UCF originate from shifts in the electronic phase resulting from the magnetic flux penetrating closed electron trajectories. Due to the diffusive transport, it is likely that in the long wire larger electron loops exist, leading to faster phase shifts with the magnetic field. Thus, the conductance fluctuates with a higher frequency. This explanation implies that phase coherence is preserved during the propagation around closed loops. Later on, we will quantify this result by analyzing the correlation field BC of the UCF. To investigate the decreasing fluctuation amplitude with temperature in more detail, the conductance fluctuations δG(B) were determined from the magnetoconductance G(B), by subtracting the average conductance ÆGæ: δG = G(B) ÆGæ, Æ...æ denoting the average over the magnetic field. The decrease of the fluctuation amplitude is visualized in panels c and d of Figure 2 by means of a color scale plot, showing δG versus magnetic field and temperature. While the fluctuation amplitude of the short wire (A1) remains nearly constant up to temperatures as high as 2 3 K, the amplitude of the longer wire (A6) decreases much faster with increasing temperatures. The average fluctuation amplitude rms(G) = (ÆδG2æ)1/2 quantifies this observation. The temperature dependence of rms(G) for wires A1A6 (A4 is omitted for clarity) is shown in Figure 3. For the short wires A1A3, rms(G) is constant in the temperature range up to 23 K at a value of about 0.30.35 e2/h. The rms(G) value of wire A5 reaches 0.3 e2/h as well, but no clear saturation is visible. The longest wire A6 shows a decreasing rms(G) in the complete temperature range and the maximum rms(G) value is about 0.15 e2/h.
Figure 2. Conductance G versus magnetic field for wire A1 (a) and wire A6 (b) at different temperatures T. Conductance fluctuations δG(B) in units of e2/h of samples A1 (c) and A6 (d) as a function of magnetic field and temperature. 3552
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Figure 3. Root mean square of the conductance fluctuations in units of e2/h as a function of temperature for wires A1A6. The dashed line corresponds to a T1/2 decrease.
For a given sample, rms(G) is determined by both phasecoherence length lϕ and thermal length lT = (pD/kBT)1/2, with D the diffusion constant.24 To interpret rms(G), both length scales have to be related to the sample length. A constant rms(G) at low temperatures is only possible if both lT and lϕ exceed the sample length L.49 If one of the values is smaller than L, an energyaveraging (for lT < L,lϕ) or a self-averaging (for lϕ < L) process takes place, which reduces the fluctuation amplitude.24 The constant values of rms(G) observed for the short wires A1A3 at low temperatures (cf. Figure 3) therefore imply that here both lT and lϕ exceed L, i.e., phase coherence is maintained in the complete wire. For wire A3 this means for example, that lϕ g 280 nm up to a temperature of 1.8 K, because a saturation in rms(G) can be seen up to this temperature. In contrast, wires A5 and A6 show no saturation at low temperatures because lϕ and/or lT is shorter than L. The rms(G) value of wire A5 reaches the same maximum value (of ≈0.3 e2/h) as wires A1A3. This means that lϕ is close to L at the lowest temperature. In contrast, for wire A6, the averaging process leads to a clear reduction of the amplitude to 0.15 e2/h. These findings suggest that the thermal length exceeds the lengths of the short wires at low temperatures. In a quasi-one-dimensional transport channel of width d < lϕ, the dependence of the fluctuation amplitude on lϕ in absence of energy averaging (lT > L) is given by49 e2 lϕ 3=2 rmsðGÞ ð1Þ h L Assuming a temperature dependence of the phase-coherence length of T1/3,50 which is confirmed later in this paper, one obtains a temperature dependence of T1/2 for the rms(G) values. This fits nicely with the measurements, as indicated by the dashed line in Figure 3. In order to obtain quantitative values of the phase coherence length, the correlation field B C was analyzed. It is defined by the autocorrelation function F(ΔB) = ÆδG(B + ΔB)δG(B)æ at zero magnetic field, BC = (1/2)F(0). For a quasi-one-dimensional transport channel of width d and length L with d , L and d , l ϕ the relation between B C and lϕ can be expressed as 24,49 Φ0 BC ¼ γ lϕ d
ð2Þ
Figure 4. (a) Correlation field BC plotted versus temperature for wires A1A6. (b) Temperature dependence of the phase-coherence length resulting from the correlation field shown in (a). The horizontal dotted lines indicate the lengths of the according wires; the dashed line shows a power law decrease with T1/3. (c) Schematic of hypothetical closed electron trajectories in two-dimensional transport channel (left) and in a hexagonal nanowire (right). Due to the different orientations of the enclosed areas with respect to the magnetic field (indicated by the magnetic field arrows), the enclosed flux is reduced or can be canceled out completely.
with the magnetic flux quantum Φ 0 = h/e and γ being a constant between 0.42 and 0.95, depending on the relation between l ϕ and lT . 49 The area l ϕ d can be interpreted as the maximum area enclosed phase coherently by electron partial waves during their diffusion through the sample. The derivation of this formula is based on the idea of taking into account all possible classical (“random walk”) trajectories traversing the disordered region.24 In Figure 4a BC is plotted versus temperature for the wires A1A6. Despite of some fluctuations in the data points, a clear saturation of B C is found for the wires A1A4 at low temperatures. The saturation implies that an increase of lϕ (because of the decrease in temperature) does not increase the area enclosed phase coherently by electron partial waves. This means, that lϕ equals L and the maximum area Ld is enclosed phase coherently. This allows us to directly obtain the parameter γ using eq 2 for the wires A1A4. The values of BC and γ can be found in Table 1. Strikingly, within an error of about 10% we obtained on average the same value of γ ≈ 1.4 for the wires A1A4, demonstrating the consistency of this evaluation method. Once the parameter γ is determined, lϕ can be obtained quantitatively using eq 2. In Figure 4b the phase-coherence length is plotted versus temperature for wires A1A6 using the 3553
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Nano Letters fitted value of γ = 1.4. The dotted horizontal lines in Figure 4b indicate the length of the wires. For wires A5 and A6 one obtains a phase-coherence length of 400500 nm at 0.5 K. Remarkably, in both wires lϕ is close to each other in the complete temperature range, which one would expect for macroscopically identical wires with L > lϕ. The results also suggest that the incorporation of doping atoms—if there is any—has no major influence on the electron phase coherence. The decrease of lϕ with temperature (in the case of lϕ < L) is close to T1/3 (dashed line), which indicates Nyquist dephasing being the major source of decoherence.50 The lϕ values extracted here are comparable to the values reported for InAs nanowires grown by different growth methods.18,19,21,22 The value of γ ≈ 1.4 is considerably larger than the theoretical values of 0.420.95 for a quantum wire structure based on a twodimensional transport channel.49 A reason for this could be that the requirements for one dimensionality in eq 2 (d , L and d , lϕ) are not fulfilled strictly, as we use wires with a length between 185 and 305 nm and diameters around 100 nm to extract the γ value. Unlike the strictly one-dimensional case, where the direction parallel to the magnetic field is irrelevant,24 in our case the exact shape of the sample becomes relevant. In particular the extension of electron trajectories in the direction parallel to the magnetic field is not negligible anymore compared to their extension along the wire axis. As pointed out by Lee, Stone, and Fukuyama,24 the correlation field depends on the average area enclosed by two “random walks” across the sample, which simply means that the flux through all possible areas enclosed by electron trajectories is averaged, in order to determine BC. In the case of our hexagonal wires, new possible trajectories, which could be neglected in the one-dimensional limit, have to be considered. To illustrate this, a sketch of a two-dimensional transport channel of width d = 2a is shown on the left side of Figure 4c. A hypothetical closed electron trajectory, which encloses a flux 2alϕBz, is illustrated. On the right side of Figure 4c a hexagonal nanowire with a diameter d = 2a (with the hexagon side length a) is shown, above the wire the side facets of the nanowire are folded to the plane perpendicular to the magnetic field. Due to the inclination of the facets, the effective magnetic flux enclosed by a trajectory of the same circumference, located at the surface of the wire, is reduced to (3/2)alϕBz. Another possible loop is shown, where due to the inclined sidewall facets the enclosed flux cancels out completely. Trajectories with a reduced effective flux can also be located in the bulk of the wire. The reduction of the flux leads to an increased correlation field BC, as on average a higher magnetic field is required to obtain a phase shift in closed loops. An increased BC leads to a higher γ value in eq 2. This flux cancelation effect is different from the one explained in ref 49, where for quasi-onedimensional wires the boundary scattering in a ballistic transport regime (le > d) leads to the flux cancelation. In literature there are two reports in which eq 2 is used to determine lϕ in InAs nanowires.22,51 In fact, in both reports the authors find comparatively small values for lϕ which could be explained by the flux cancelation effect proposed in this work. In the following the measurements in a parallel magnetic field configuration are presented. Assuming an electron accumulation layer at the surface of the wires, one would expect regular magnetoconductance oscillations, if the phase coherence length exceeds the circumference of the wires.3537 An oscillation frequency of f0 = A/Φ0 is expected if coherent angular momentum quantum states form in the two-dimensional electron gas. For AAS-type oscillations originating from the interference of
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Figure 5. δG(B) at T = 0.8 K and the corresponding fast Fourier transformations (FFT) for wires B1B3. The dotted vertical lines indicate the frequency range of f0 = A/Φ0 and 2f0, respectively.
time-reversed paths, the characteristic frequency 2f0 is expected.31 The measurements were carried out at a temperature of 0.8 K. According to our analysis presented in the previous subsection, lϕ should be around 400 nm; it is therefore smaller than the circumference of wire B1 and it exceeds the ones of wires B2 and B3. In the left part of Figure 5 δG is plotted versus the magnetic field for wires B1B3. In the right part of Figure 5 the corresponding Fourier spectra and the frequency ranges f0 and 2f0 are shown. The ranges result from uncertainties in the diameter determination ((5 nm) and in the alignment between wire and magnetic field ((5). No peaks are observed at the frequencies f0 and 2f0 (not shown in the graph) for wire B1. This can be explained by the fact that lϕ < πd, and thus coherence is not maintained around the circumference. The frequency 2f0 is not visible for the wires B2 and B3 as well, which means that no AAS-type interferences take place in our experiments. However small peaks can be found at f0, although, compared to earlier measurements on InN wires,35 their amplitude is relatively weak. There could be several reasons for the observation that the Fourier frequency bands at 2f0 are not visible and the ones at f0 have comparatively low amplitudes. First of all, the surface Fermi level pinning position above the lower conduction band edge is lower in InAs compared to InN, which means that the conductive shell is less pronounced. Additionally, the electron concentration of the Mott transition in InAs is very low52 (in the order of n = 1015 cm3). Due to unintentional doping it is possible that the nanowire core is degenerate and conductive. This would furthermore reduce the relative contribution of the conductive shell in our interference experiments. As mentioned before, no passivation of the nanowires was performed, which could lead to enhanced elastic scattering on surface states.3941 Another reason could be a disturbance of the surface conductive cylinder by the high density of stacking faults (see Figure 1d). Each stacking fault represents a one-layer wurtzite section embedded between zinc blende segments. Here the wurtzitic sections pose a special problem. As it is well-known from the study of group III-nitride heterostructures,53,54 wurtzite interlayers carry a spontaneous (pyroelectric) polarization, which gives rise to interface charges in the order of 1013 cm2 and therefore to an electric field induced tilt of the electronic band structure. Simulations on a layer stack consisting of a series 3554
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Nano Letters of alternating ZB and WZ layers show that the conduction and valence band edges get a sawtooth shape along the stack.55 In this situation the sawthooth-shaped band profile leads to a modulation of the electron concentration in the bulk as well as at the surface of the nanowire. The effect of the charged surface states with their neutrality level within the bulk conduction band is partly compensated by the wurtzitic (stacking faults) polarization charges. Depending on the specific stacking fault configuration the surface electron accumulation might be reduced. This scenario might also cause the observed reduction (or lack) of regular conductance oscillations. Therefore it might be questionable that the system is well described by a model of a conductive cylinder, on which the explanation of regular oscillations is based. In summary, we determined the phase coherence length in InAs nanowires by comparing the universal conductance fluctuations of wires of different lengths. Using a model for a quasi-onedimensional transport channel, we obtained consistent results. The deviation of the determined γ parameter from the model was explained by the fact that our samples are not strictly onedimensional and thus a flux cancelation effect takes place. The investigated nanowires exhibit a high number of stacking faults. Nevertheless, electronic phase coherence is comparatively well preserved, with phase-coherence lengths around 450 nm at a temperature of 0.5 K. This value is in the order of magnitude which is found for crystallographically perfect InN wires.26 This result is in accord with the fact that defect scattering on stacking faults is essentially elastic and influences the phase-coherence length only indirectly via the diffusion constant. Quantum oscillations in the magnetoconductance (in a parallel field) are less pronounced compared to those in InN nanowires. This observation is attributed to a distortion of the surface accumulation layer as a conducting cylinder due to several reasons mentioned above. It would be of major interest to prepare InAs nanocolumns with little or no stacking faults, to find out if the stacking faults have a major influence on the surface electron accumulation. Future experiments on crystallographically uniform nanowires could substantiate our present interpretation. In the same way, first principle calculations of the electronic (surface) band structure in nanowires with a high number of stacking faults are needed in order to elucidate their effect on surface states and facet formation quantitatively.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
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