Doping-Induced Universal Conductance Fluctuations in GaN

Letter pubs.acs.org/NanoLett

Doping-Induced Universal Conductance Fluctuations in GaN Nanowires Matthias T. Elm,*,†,‡ Patrick Uredat,† Jan Binder,‡ Lars Ostheim,† Markus Schaf̈ er,† Pascal Hille,† Jan Müßener,† Jörg Schörmann,† Martin Eickhoff,† and Peter J. Klar† †

Institute of Experimental Physics I, Justus-Liebig University, Heinrich-Buff-Ring 16, D-35392 Giessen, Germany Institute of Physical Chemistry, Justus-Liebig University, Heinrich-Buff-Ring 17, D-35392 Giessen, Germany

‡

S Supporting Information *

ABSTRACT: The transport properties of Ge-doped single GaN nanowires are investigated, which exhibit a weak localization effect as well as universal conductance fluctuations at low temperatures. By analyzing these quantum interference effects, the electron phase coherence length was determined. Its temperature dependence indicates that in the case of highly doped nanowires electron−electron scattering is the dominant dephasing mechanism, while for the slightly doped nanowires dephasing originates from Nyquist-scattering. The change of the dominant scattering mechanism is attributed to a modification of the carrier confinement caused by the Ge-doping. The results demonstrate that the phase coherence length can be tuned by the donor concentration making Ge-doped GaN nanowires an ideal model system for studying the influence of impurities on quantum-interference effects in mesoscopic and nanoscale systems. KEYWORDS: GaN, nanowires, universal conductance fluctuations, weak localization, phase coherence length, Ge-doping

T

negative magnetoresistance effect.22 By studying these interference effects fundamental transport parameters such as the phase-coherence length can be obtained. Nonintentionally doped (n.i.d.) and Ge-doped GaN NWs were grown by plasma-assisted molecular beam epitaxy (PAMBE) on Si(111) substrates at a temperature of 795 °C in nitrogen-rich conditions. The NWs have a length L of 1.5 to 2.0 μm and a diameter d between 60 and 100 nm. For n-type doping, a germanium beam equivalent pressure (BEPGe) of 5.0 × 10−10, 7.5 × 10−10, and 10.0 × 10−10 mbar was used. By combining time-of-flight secondary ion mass spectroscopy (ToF-SIMS) and energy dispersive X-ray spectroscopy in a transmission electron microscope (TEM-EDX) the average Ge concentrations [Ge] in the NW ensembles were determined to about 1.0 × 1020 cm−3, 1.4 × 1020 cm−3, and 1.6 × 1020 cm−3 respectively.23 An overview of the NWs investigated in this study as well as the corresponding parameters taken from refs 23 and 24 are listed in Table 1. For the transport measurements, electrical contacts were prepared on single NWs by a combination of photo- and electron beam lithography. As electrode material, Ti (25 nm) and Au (200 nm) were used. To obtain ohmic contacts the NWs were annealed for 60 s at 550 °C in vacuum. The temperature dependence of the resistance was investigated in a

he III−V nanowires (NWs) grown by bottom-up approaches have recently attracted much attention as building blocks for different nanodevices.1−6 Especially, GaN nanowires are an interesting material system for short wavelength optoelectronic nanodevices and high-power nanoelectronics,7 such as nanoscaled field-effect transistors,8 ultraviolett light-emitting diodes,9 photodetectors, intersubband optoelectronics,10 or even nanowire lasers.11 Beside their technological importance semiconducting NWs are also ideal structures to study quantum transport phenomena in mesoscopic and nanoscaled systems. Universal conductance fluctuations (UCF) present a prominent example of quantum interference effects and have been observed in Inbased or GaAs-based NWs.12−18 They occur in small systems like nanowires, whose dimensions are comparable to the electron phase-coherence length lϕ, that is, the characteristic length over which the electron phase is maintained. In such systems, the electron wave functions extend coherently throughout the system and different transport paths may interfere. By varying an external magnetic field or the Fermienergy the electron interference is varied resulting in UCFs with an average amplitude of the order of e2/h.19−21 In the absence of a magnetic field, the interference of the electron wave functions can lead to weak localization (WL), where an increased backscattering of the electrons due to constructive interference of time-reversed electron paths results in an additional resistance contribution. Applying a magnetic field breaks the time-reversal symmetry of the electron paths and thus reduces the probability of backscattering resulting in a © XXXX American Chemical Society

Received: June 12, 2015 Revised: October 20, 2015

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DOI: 10.1021/acs.nanolett.5b02332 Nano Lett. XXXX, XXX, XXX−XXX

Letter

Nano Letters Table 1. Parameters of the Different NWs Investigated in This Studya

NW00a NW00b NW05 NW07a NW07b NW10a NW10b

d

L

BEPGe

[Ge]

n

μ

le

D

lT at 2 K

(nm)

(nm)

(10−10 mbar)

(1020 cm−3)

(1019 cm−3)

(cm2/(V s))

(nm)

(cm2/s)

(nm)

59 93 77 47 86 104 76

497 310 171 121 1452 552 552

n.i.d n.i.d 5.0 7.5 7.5 10.0 10.0

0.0 0.0 1.0 1.4 1.4 1.6 1.6

0.9 0.9 2.0−2.6 3.1* 3.7* 4.1−5.5 4.1−5.5

10−43 10−43 34−81 48 51 40−86 40−86

1.1 1.1 2.6 3.1 3.5 4.5 4.5

2.0 2.0 6.8 8.6 10.3 14.9 14.9

27.5 27.5 51.1 57.5 62.8 75.4 75.4

The values for [Ge] are taken from ref 23, while those for the carrier concentration n and the mobility μ are taken from ref 24. Values marked with an asterisk (*) are calculated using eq 1. The elastic scattering length le, the electron diffusion coefficient D, and the thermal length lT were estimated assuming a 3D free electron gas. a

Figure 1. (a) Comparison of the temperature-dependent resistivity for NWs doped with different Ge-concentration. (b) Determined activation energy plotted versus n1/3. The solid line represents a linear fit according to eq 1.

range between 1.6 and 280 K and magnetotransport measurements were performed at temperatures below 40 K using a He4 flow cryostat in a superconducting magnet system yielding external fields up to 10 T. The magnetic field was oriented perpendicular to the NW axis. For the resistance measurements, DC currents between 100 nA and 1 μA were used. The resistivity was calculated assuming the NW to be of cylindrical shape with the corresponding diameter d and a length L being the distance between the two voltage probes. The corresponding parameters are also listed in Table 1. Figure 1a shows the temperature-dependent resistivity of n.i.d. and Ge-doped NWs in an Arrhenius plot. For the n.i.d. and the slightly doped NWs, a semiconducting behavior with a linear increase of the resistivity with decreasing temperature can be observed at higher temperatures (T > 50 K). At low temperatures, that is, T < 25 K, a strongly reduced activation energy of a few microelectronvolts is found, revealing the existence of an impurity band. The activation energy determined for T > 150 K (see Supporting Information) varies between 30 ± 5 and 19 ± 4 meV for different n.i.d. NWs investigated, which is in good agreement with reported ionization energies of Si in GaN thin films of 17−28 meV.25−29 The formation of an impurity band in the n.i.d. NWs can therefore be attributed to diffusion of Si from the substrate into the NW during the growth process.23,24

With increasing BEPGe, that is, increasing [Ge] in the NW, the resistivity of the NWs decreases. For the highest BEPGe of 10.0 × 10−10mbar metallic transport behavior with a decreasing resistance for decreasing temperature is observed. Figure 1 b) shows the extracted activation energy as a function of [Ge]. As Ge acts as a donor with an activation energy of approximately 20 meV,27 that is, similar to Si, the incorporation of Ge results in further broadening of the impurity band and a reduction of the activation energy. For high doping concentrations, the impurity band overlaps with the conduction band leading to the observed metallic transport behavior. It is worth noting that the activation energy as well as the conductivity of single nanowires vary to some extent even between nanowires of the same ensemble. Thus, it is necessary to individually determine the carrier concentration of each nanowire prior to the analysis of the mesoscopic transport behavior. A detailed discussion of the variance of the nanowires’ electrical properties can be found in ref 24. With the parameters given in ref 24 (Table 1), it is possible to assign the determined activation energies to an effective carrier concentration in the nanowires. According to refs 26 and 30−32, the electrical activation energy of the donor depends on the carrier concentration EA = EA0 − αn1/3

(1)

where n is the effective carrier concentration, E0A is the activation energy of the isolated impurity atom and α is an B


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Figure 2. Typical change of the magnetoconductance ΔG(B) in units of e2/h at different temperatures of four nanowires (a) NW00a, (b) NW05, (c) NW07a, and (d) NW10a.

transport in the weakly disordered regime (dirty metal limit le ≪ d). The values for le are also listed in Table 1. For quasi-one-dimensional transport in the weakly disordered regime lϕ can be obtained from the WL magnetoconductance, which is given by21,36,37

empirical parameter. Using eq 1 the experimental data were fitted with weighting of the estimated measurement errors (Figure 1b). The extracted values for E0A and α are about 32 ± 6 meV and 8.4 × 10−6 meVcm, respectively, which is in reasonable agreement with experimentally reported values of E0A = 21−28 meV and of α = 3.4 to 7.02 × 10−6 meVcm determined for Si donors in GaN thin films.25,26,29 These results show that the transport properties of GaN NWs can be tuned by doping with Ge. With the obtained values, an effective carrier concentration was assigned to NW07a and NW07b for which no experimental values were available. In order to investigate the relation between Ge-doping, conductance fluctuations, and phase coherence length lϕ, the magnetoresistance R(B) of all NWs was measured at low temperatures (T ≤ 40 K) as a function of the external magnetic field. The change of the magnetoconductance with magnetic field ΔG(B) was calculated according to33 ΔG(B) = G(B) − ⟨G(B)⟩B ≅ −

ΔR(B) ⟨R(B)⟩2B

−1/2 2 e 2 ⎡⎢ 1 1 d 2e 2B2 ⎤⎥ ΔG(B) = − + L h ⎢⎣ lϕ2 3 ℏ2 ⎥⎦

(3)

As this expression is only valid in the low-field limit, that is, lB = (ℏ/eB)1/2 ≫ d, the measured magnetoconductance was fitted only in the range from 0 to 0.5 T. The second possibility to determine lϕ is the analysis of the mean fluctuation amplitude of the UCF given by rms (ΔG) = (⟨ΔG2⟩B)1/2. At finite temperature, the effect of thermal averaging has to be considered, which is expressed by the 1 thermal length lT = (ℏD/kBT)1/2, where D = 2 vF2τ is the diffusion constant.20,21 For lT ≤ lϕ, thermal averaging results in a reduction of the fluctuation amplitude, while for lϕ ≪ lT the effect of thermal averaging can be neglected.21 The calculated values of the diffusion constant D and the thermal length lT are also listed in Table 1. Usually, lϕ and lT are comparable in NWs15,17 as is also the case for the Ge-doped GaN NWs here. Then, lϕ can be determined with an accuracy of about 10% using the following expression38

(2)

where ⟨R(B)⟩B is the mean resistance averaged over the entire external magnetic field range and ΔR(B) = R(B) − ⟨R(B)⟩B are the fluctuations of the resistance. Figure 2 shows ΔG(B) of four NWs with different Ge-concentrations. In the case of the n.i.d. NW (Figure 2a), only WL, that is, an increasing conductance with increasing magnetic field, is observed. For the slightly doped NW (NW05, BEPGe = 5.0 × 10−10 mbar) an additional onset of UCF with a peak to peak value of about 0.01 e2/h is found (Figure 2b). Doping the NWs with Ge not only increases the magnitude of the UCF, but also the frequency of the fluctuations, that is, the number of fluctuations in a certain magnetic field range. For the higher doped NW (NW07a, Figure 2c) the fluctuation magnitude is about 0.025 e2/h and for the metallic NW10a the fluctuations are of the order of 0.3 e2/h at 1.6 K (Figure 2d), which is comparable to magnitudes observed in other NW systems.15,16,34,35 The phase coherence length lϕ was determined by three different approaches: by analysis of the weak localization effect, by the magnitude of the conductance fluctuations rms (ΔG), and by the analysis of the correlation field BC. According to theory,21,36 the correct description of the WL and the UCF strongly depends on the dominating transport regime, which in the present case was identified by estimating the elastic scattering length of the electrons le. Using the values for the mobility μ and the effective carrier concentration n listed in Table 1 and assuming a simple Drude model, that is, le = vFτ = ℏ(3π2 n)1/3/e, the elastic scattering length was estimated to vary between 1 and 5 nm. Thus, the nanowires show diffusive

rms(ΔG) = α

3/2 2 −1/2 e 2 ⎛ lϕ ⎞ ⎡ α ⎛ lϕ ⎞ ⎤ ⎜ ⎟ ⎢1 + ⎜ ⎟ ⎥ 2π ℏ ⎝ L ⎠ ⎣ β ⎝ lT ⎠ ⎦

(4)

with α = 6 and β = 4π /3 .38 Finally, the phase coherence length was determined from the correlation field BC, which represents a characteristic field scale for a fluctuation period. It can be interpreted as the magnetic field required to provide a magnetic flux Φ through an area enclosed by phase-coherent electrons that is necessary to lead to an electron phase shift of the order of 2π. BC is defined by the relation F(BC) = 1/2F(0) in which F(ΔB) = ⟨ΔG(B + ΔB)ΔG(B)⟩B is the autocorrelation function.19,20,39 From the correlation field lϕ can be estimated for a quasi-one-dimensional channel with d ≪ L and d ≪ lϕ according to the relation

BC = γ

Φ0 lϕd

(5)

where Φ0 = h/e is the magnetic flux quantum and γ is a numerical parameter depending on the transport regime. The area lϕd can be interpreted as the average area enclosed by phase-coherent electrons in the nanowire. The value γ was C


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NWs.15,16,35,41 According to theory this saturation can be caused by a dimensional crossover, that is, a change of the effective sample dimensionality,42 when lϕ becomes comparable to the NW length L leading to a saturation at an effective length Leff ≈ L/2π.43 The effective length for the metallic nanowire is also indicated in Figure 3a, which is comparable to the saturation value of lϕ, especially when considering the relatively large error for the determination of lϕ. At temperatures above 7 K, a very good agreement between the differently determined values of lϕ is obtained. The double logarithmic plot of lϕ shown in Figure 3a reveals a temperature dependence of lϕ proportional to T−0.6, which is a nontypical behavior for NWs. According to theory,36,44 in one-dimensional structures and at low temperatures quasi-elastic electron− electron scattering (Nyquist-dephasing) is expected to be the dominant phase-breaking mechanism. In that case, the dephasing time τϕ is proportional to T−2/3,36,43,44 leading to a temperature dependence of lϕ = Dτϕ ∝ T −1/3. Such a temperature dependence or a slightly weaker one is in fact often observed in semiconducting NWs in agreement with theory.14,16,35,41 However, in the case of the doped GaN NWs investigated here, Nyquist-dephasing cannot be observed. Also dephasing due to electron−phonon scattering can be excluded as it typically shows a temperature dependence of lϕ ∝ T−1 to T−2.45 As shown by Choi et al. the dephasing in a quasi-onedimensional wire is given by36,43

calculated analytically only for the two asymptotic regimes, lϕ ≪ lT (γ ≃ 0.42) and lϕ ≫ lT (γ ≃ 0.95) at T = 0 K.38 From the analysis of the weak localization and the mean fluctuation amplitude a phase coherence length lϕ longer than lT was determined for the higher doped NWs, that is, NW07b and NW10a, while lϕ of the lower doped NWs NW07a and NW05 was smaller than lT. Thus, a value of γ = 0.95 and γ = 0.42 was used for the higher and lower doped NWs, respectively. It is worth noting that the weak localization effect dominates the magnetoconductance at low magnetic fields. Thus, for the determination of lϕ from the correlation field the magnetoconductance behavior at low magnetic fields was excluded. Figure 3a shows a comparison of the lϕ-values for the metallic NW NW10a determined by the three different analysis

2/3 2 ⎛ E ⎞ π (kBT ) 1 1 ⎛ kBπT ⎞ ⎜ ⎟ = ⎜ ln⎜ F ⎟ ⎟ + τϕ 2 ⎝ m* D d ⎠ 2 ℏE F ⎝ kBT ⎠

(6)

The first expression on the right-hand side represents the above-mentioned Nyquist-dephasing 1/τN calculated by Altshuler et al. in the dirty metal regime,44 while the second term is the inelastic electron scattering rate 1/τ2D ee due to electron−electron scattering calculated by Fukuyama and Abrahams for disordered two-dimensional metals (describing large energy transfer during electron−electron-scattering).46 In order to compare these theoretical predictions to the experimental results the temperature dependence of the phase coherence time, 1/τN and 1/τ2D ee were calculated using the values listed in Table 1. As can be seen in Figure 3b, the phase coherence time τϕ at T ≥ 8 K is limited only by the inelastic electron−electron scattering as it was also observed for GaAs/AlGaAs quantum wires47 at temperatures above 10 K. Although 1/τN is smaller than 1/τ2D ee , Nyquist dephasing seems to have no influence on the dephasing time in the Ge-doped GaN NWs. To investigate the influence of doping on the UCF, rms(ΔG) was determined for all investigated NWs. As can be seen in eq 3, rms(ΔG) scales with the length of the corresponding NWs. For NWs longer than the phase coherence length (L > lϕ) or the thermal length (L > lT), self-averaging of the fluctuations occurs and rms(ΔG) is reduced.20 Thus, for a comparison of rms(ΔG) the normalized UCF amplitude rms(ΔG)L2/3 was calculated. As shown in Figure 4a, the normalized rms(ΔG) is largest for the metallic NW (NW10a) and decreases with decreasing doping concentration, that is, strongly depending on the carrier concentration in the NW. A closer look at Figure 4a also reveals that for the metallic NW an uncommon temperature dependence can not only be found for lϕ, but also for rms(ΔG), which follows a T−3/4 law in contrast to the typically observed T−1/2-dependence. Interestingly, with

Figure 3. (a) Comparison of the phase-coherence length lϕ for the metallic NW (NW10a) determined from the mean fluctuation amplitude (rms (ΔG), blue circles), the weak localization effect (WL, green triangles) and the correlation field BC (red diamonds). The estimated thermal length lT is shown as dashed line. The doublelogarithmic plot of lϕ shows a temperature dependence proportional to T−0.6. (b) Comparison of the estimated phase coherence time and the calculated dephasing times τN (Nyquist dephasing) and τ2D ee (inelastic electron−electron-scattering).

approaches. (Comparable values are obtained from the analysis of the UCFs of the second heavily doped NW NW10b, see Supporting Information). At temperatures below 7 K, the values determined from the correlation field BC (lBC ϕ , red diamonds) are significantly smaller than the results obtained from the analysis of the WL effect (lWL ϕ , green triangles) or the mean fluctuation amplitude (lrms ϕ , blue circles). This discrepancy has often been reported for NW systems16,17,40 and can be attributed to the uncertainty of the γ-value in eq 5, which was calculated only for the two asymptotic regimes in a quasi-onedimensional channel. As discussed by Blömers et al., who determined experimentally a γ-value of 1.4 for InAs NWs, a nonideal 1D transport channel can result in a flux cancelation effect due to additional electron trajectories, which would lead to an increased BC and thus to an increased γ-value.16 Additionally, at temperatures below 10 K, lBC ϕ seems to saturate or slightly decrease with decreasing temperature, while an onset rms of saturation may be observable for lWL ϕ and lϕ below 4 K. The saturation of lϕ at low temperatures is also often observed in D


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Figure 4. (a) Temperature dependence of rms(ΔG)L2/3 of the different GaN NWs. The higher doped NWs (NW10a and NW07b) show a T−3/4 behavior, while for lower doped NWs (NW07a) the temperature dependence is proportional to T−1/2. (b) Surface depletion layer of a slightly doped and a highly doped NW of same diameter. Increasing the doping concentration increases the effective diameter deff of the NW (after ref 48).

diameter. By calculating the width of the space charge region using nextnano3 and assuming a homogeneous distribution of dopants (see Supporting Information)50 the transport channel of NW07a can be estimated to be of about 35 nm, that is, comparable to the thickness of the accumulation layer in InAs or InN NWs. Thus, for NW07a the formation of the space charge region may be responsible for the quasi one-dimensional transport behavior, where Nyquist scattering is dominant. Increasing the doping concentration decreases the depletion region, that is, deff increases, leading to a less confined transport channel, where inelastic electron scattering 1/τ2D ee dominates, as observed for the metallic NW. It is worth noting that the space charge region and thus deff were estimated assuming a homogeneous distribution of dopants, although experimental evidence is missing. Nevertheless, also an inhomogeneous distribution of Ge atoms in the nanowire may have a significant effect on the geometry of the transport channel in the NWs. Here, further investigations are needed in order to confirm this assumption. In order to investigate the influence of doping on lϕ in more detail, lϕ was determined from WL, rms(ΔG), and BC as described above for all investigated NWs. The results are shown in Figure 5a,b, where lϕ is plotted as a function of the activation energy EA of the corresponding NWs for two exemplary chosen temperatures T = 4 K and T = 20 K, respectively. Comparable to the metallic NW, an acceptable agreement between lWL ϕ and lrms ϕ is obtained for all NWs at low activation energies (EA < 5

decreasing doping concentration (that is, with increasing activation energy) the absolute value of the exponent in the temperature dependence of rms(ΔG) decreases. The slightly doped NW NW07a with an activation energy of about 6 meV already shows an exponent of −0.42, which is close to a value of −1/2. Using eq 3 lϕ of this NW shows a temperature dependence of lϕ ∝ T−0.28, close to the expected value of T−1/3 and comparable to values observed in InN nanowire systems, which often also show a weaker temperature dependence than expected from theory.13,41 Thus, for slightly doped nanowires (NW07a) Nyquist scattering seems to be the dominant dephasing mechanism as expected for transport in onedimensional systems. For the even lower doped NW (NW05), almost no temperature dependence of lϕ is observed, indicating that not only the amplitude of the RMS but also the temperature-dependence of lϕ sensitively depends on the carrier concentration of the nanowire. Comparable to the highly doped GaN nanowires investigated here, Frielinghaus et al. also observed a stronger temperature dependence of rms(ΔG) ∝ T−0.84 for n-type doped InN nanowires,13 while for undoped InN nanowires a T−0.5dependence was reported.14 Thus, the stronger temperature dependence of rms(ΔG) may directly be attributed to the doping, which affects the spatial distribution of carriers inside the NW.48,49 In undoped InAs or InN nanowires, an accumulation layer with a thickness of about 20 to 30 nm is formed due to a strong downward surface band bending. This highly conductive surface region is responsible for confined, quasi-one-dimensional charge transport. As the GaN NWs are significantly thicker, a less confined transport behavior is expected. However, in GaN NWs the width of the space charge region due to surface band bending reduces the effective thickness of the conductive channel deff as schematically illustrated in Figure 4b. As a consequence, transport takes place in a transport channel smaller than the nominal NW

Figure 5. Determined values of lϕ as a function of EA at (a) 4 K and (b) 20 K. lϕ decreases with decreasing [Ge], that is, with increasing EA of the corresponding NW. (c) lWL ϕ as a function of D. The dashed and dotted lines are guides to the eye for a D3/2- and D2/3-dependence, respectively. E


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as well as the weak localization effect, lϕ could be determined as a function of temperature and [Ge]. For the metallic NWs, all three approaches provide comparable results, while for the semiconducting NWs with lower [Ge] the analysis of BC shows deviations. The temperature dependence of lϕ suggests inelastic electron−electron scattering to be the dominant dephasing mechanism in highly doped NWs, which is untypical for most NWs. This behavior can be explained by considering the influence of doping on the surface space charge region in GaN, which broadens the effective NW diameter resulting in a less confined transport channel. For low doping concentrations, one-dimensional transport with dominating Nyquist scattering is observed. The analysis of lϕ as a function of the diffusion coefficient supports this assumption. As the diffusion coefficient and thus the transport regime can be controlled by the Ge concentration, Ge-doped GaN nanowires are an ideal model system to study the effect of disorder and dimensionality on the phase coherence length in mesoscopic systems.

meV) in the whole investigated temperature range. At higher activation energies, the agreement becomes worse, as the amplitude of the fluctuations decreases, thus making it more difficult to obtain reliable values of lrms ϕ . For lϕ determined from BC strong deviations are found for all nanowires, especially at higher temperatures. The values of lBC ϕ are about a factor of 2 rms larger compared to lWL ϕ and lϕ . This discrepancy may again be attributed to the uncertainty of the γ-value in eq 5. Only the nanowire NW10a exhibits metallic behavior, while the NWs with a lower carrier concentration show semiconducting behavior. Thus, the transition from metallic to semiconducting behavior seems to be accompanied by a decrease of the γ-value. Nevertheless, the obtained values for lWL and lrms clearly ϕ ϕ demonstrate that with decreasing donor concentration, that is, increasing E A, lϕ decreases, resulting in the observed disappearance of the UCF. Niimi et al. investigated the effect of disorder on the quantum coherence in mesoscopic systems.51 They showed that lϕ is strongly correlated to the diffusion coefficient and that the variation of lϕ as a function of D depends on the dominant dephasing mechanism in the system. As discussed above, according to the observed temperature dependence inelastic electron−electron scattering seems to be the dominant dephasing mechanism in the highly Ge-doped GaN nanowires investigated here, while for the lower Ge-doped NWs Nyquist scattering dominates. Using eq 6 and the relation EF ∝ v2F ∝ D2, for high carrier concentrations lϕ depends on the diffusion coefficient according to lϕ =

Dτϕ ≈

Dτee2D ∝

D·E F ∝ D3/2

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* Supporting Information

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.5b02332. Arrhenius representation of the resistivity for T > 20 K including the linear fit to determine the activation energy, magnetoresistance of NW10b at T = 10 K and calculations of the space charge region using NextNano3.(PDF)

(7)

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while for low carrier concentrations it holds: lϕ =

Dτϕ ≈

DτN ∝

D·D1/3 = D2/3

ASSOCIATED CONTENT

S

AUTHOR INFORMATION

Corresponding Author

(8)

*E-mail: [email protected]. Phone: +49 (0)641 9933147. Fax: +49 (0)641 9933139.

lWL ϕ

Figure 5c shows the change of as a function of the diffusion coefficient in a double logarithmic plot for three different temperatures. The expected behavior according to eqs 7 and 8 is also shown as a dashed and a dotted line, respectively. For all temperatures, a good agreement between the theoretical valuation and the experimental results is obtained. At high carrier concentrations, that is, for large D, lϕ follows the expected D3/2-dependence confirming that inelastic electron− electron scattering is the dominant dephasing mechanism, while for low carrier concentrations (small D) a good agreement can be obtained when Nyquist dephasing is assumed to be dominant. The analysis of lϕ as a function of the diffusion coefficient therefore supports the assumption, that Ge-doping of the GaN nanowires not only results in a transition from a semiconducting to a metallic transport behavior but also modifies the dominant scattering mechanism due to a change in the confinement of the transport path. In summary, we have studied the transport properties of single Ge-doped GaN NWs grown by PAMBE as a function of [Ge]. The nonintentionally doped NWs show activated transport behavior at temperatures above 50 K with an activation energy of about 19 to 30 meV, attributed to Si diffusing from the substrate into the NW. At low temperatures, impurity band transport due to the doping with Si is observed. Additional doping with Ge broadens the impurity band, resulting in a reduction of EA and finally leading to metallic transport for the NWs of the highest [Ge]. The magnetotransport measurements show UCF for the doped NWs only with a magnitude strongly increasing with increasing [Ge]. By analyzing the mean fluctuation amplitude, the correlation field

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge support by the research program LOEWE (project STORE-E) funded by the State of Hessen.

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REFERENCES

(1) Duan, X.; Huang, Y.; Cui, Y.; Wang, J.; Lieber, C. M. Nature 2001, 409, 66−69. (2) Cui, Y.; Lieber, C. M. Science 2001, 291, 851−853. (3) Huang, Y.; Duan, X.; Cui, Y.; Lauhon, L. J.; Kim, K.-H.; Lieber, C. M. Science 2001, 294, 1313−1317. (4) Duan, X.; Huang, Y.; Agarwal, R.; Lieber, C. M. Nature 2003, 421, 241−245. (5) Choi, H.-J.; Seong, H.-K.; Chang, J.; Lee, K.-I.; Park, Y.-J.; Kim, J.J.; Lee, S.-K.; He, R.; Kuykendall, T.; Yang, P. Adv. Mater. 2005, 17, 1351−1356. (6) Tomioka, K.; Yoshimura, M.; Fukui, T. Nature 2012, 488, 189− 192. (7) Li, Y.; Qian, F.; Xiang, J.; Lieber, C. M. Mater. Today 2006, 9, 18−27. (8) Huang, Y.; Duan, X.; Cui, Y.; Lieber, C. M. Nano Lett. 2002, 2, 101−104. (9) Li, S.; Waag, A. J. Appl. Phys. 2012, 111, 071101. (10) Beeler, M.; Hille, P.; Schörmann, J.; Teubert, J.; de la Mata, M.; Arbiol, J.; Eickhoff, M.; Monroy, E. Nano Lett. 2014, 14, 1665−1673. (11) Johnson, J. C.; Choi, H.-J.; Knutsen, K. P.; Schaller, R. D.; Yang, P.; Saykally, R. J. Nat. Mater. 2002, 1, 106−110. F


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Nano Letters

(44) Al’tshuler, B. L.; Aronov, A. G.; Khmelnitsky, D. E. J. Phys. C: Solid State Phys. 1982, 15, 7367−7386. (45) Giordano, N. Phys. Rev. B: Condens. Matter Mater. Phys. 1980, 22, 5635−5654. (46) Fukuyama, H.; Abrahams, E. Phys. Rev. B: Condens. Matter Mater. Phys. 1983, 27, 5976−5980. (47) Noguchi, M.; Ikoma, T.; Odagiri, T.; Sakakibara, H.; Wang, S. N. J. Appl. Phys. 1996, 80, 5138−5144. (48) Calarco, R.; Marso, M.; Richter, T.; Aykanat, A. I.; Meijers, R.; v.d. Hart, A.; Stoica, T.; Lüth, H. Nano Lett. 2005, 5, 981−984. (49) Richter, T.; Meijers, H. L. R.; Calarco, R.; Marso, M. Nano Lett. 2008, 8, 3056−3059. (50) nextnano3 quantum: next generation 3D nanodevice simulator; www.nextnano.de/nextnano3. (51) Niimi, Y.; Baines, Y.; Capron, T.; Mailly, D.; Lo, F.-Y.; Wieck, A. D.; Meunier, T.; Saminadayar, L.; Bäuerle, C. Phys. Rev. B: Condens. Matter Mater. Phys. 2010, 81, 245306.

(12) Hansen, A. E.; Björk, M. T.; Fasth, C.; Thelander, C.; Samuelson, L. Phys. Rev. B: Condens. Matter Mater. Phys. 2005, 71, 205328. (13) Frielinghaus, R.; Estévez Hernández, S.; Calarco, R.; Schäpers, T. Appl. Phys. Lett. 2009, 94, 252107. (14) Alagha, S.; Estévez Hernández, S.; Blömers, C.; Stoica, T.; Calarco, R.; Schäpers, T. J. Appl. Phys. 2010, 108, 113704. (15) Yang, P.-Y.; Wang, L. Y.; Hsu, Y.-W.; Lin, J.-J. Phys. Rev. B: Condens. Matter Mater. Phys. 2012, 85, 085423. (16) Blömers, C.; Lepsa, M. I.; Luysberg, M.; Grützmacher, D.; Lüth, H.; Schäpers, T. Nano Lett. 2011, 11, 3550−3556. (17) Estévez Hernández, S.; Akabori, M.; Sladek, K.; Volk, C.; Alagha, S.; Hardtdegen, H.; Pala, M. G.; Demarina, N.; Grützmacher, D.; Schäpers, T. Phys. Rev. B: Condens. Matter Mater. Phys. 2010, 82, 235303. (18) Wagner, K.; Neumaier, D.; Reinwald, M.; Wegscheider, W.; Weiss, D. Phys. Rev. Lett. 2006, 97, 056803. (19) Washburn, S.; Webb, R. A. Rep. Prog. Phys. 1992, 55, 1311. (20) Lee, P. A.; Stone, A. D.; Fukuyama, H. Phys. Rev. B: Condens. Matter Mater. Phys. 1987, 35, 1039−1070. (21) Beenakker, C.; van Houten, H. In Semiconductor Heterostructures and Nanostructures; Ehrenreich, H., Turnbull, D., Eds.; Solid State Physics; Academic Press: New York, 1991; Vol. 44; pp 1−228. (22) Bergmann, G. Phys. Rev. B: Condens. Matter Mater. Phys. 1983, 28, 2914−2920. (23) Schörmann, J.; Hille, P.; Schäfer, M.; Müßener, J.; Becker, P.; Klar, P. J.; Kleine-Boymann, M.; Rohnke, M.; de la Mata, M.; Arbiol, J.; Hofmann, D. M.; Teubert, J.; Eickhoff, M. J. Appl. Phys. 2013, 114, 103505. (24) Schäfer, M.; Günther, M.; Länger, C.; Müßener, J.; Feneberg, M.; Uredat, P.; Elm, M. T.; Hille, P.; Schörmann, J.; Teubert, J.; Henning, T.; Klar, P. J.; Eickhoff, M. Nanotechnology 2015, 26, 135704. (25) Kumar, M.; Bhat, T. N.; Roul, B.; Rajpalke, M. K.; Kalghatgi, A.; Krupanidhi, S. Mater. Res. Bull. 2012, 47, 1306−1309. (26) Jayapalan, J.; Skromme, B. J.; Vaudo, R. P.; Phanse, V. M. Appl. Phys. Lett. 1998, 73, 1188−1190. (27) Götz, W.; Kern, R.; Chen, C.; Liu, H.; Steigerwald, D.; Fletcher, R. Mater. Sci. Eng., B 1999, 59, 211−217. (28) Götz, W.; Johnson, N. M.; Chen, C.; Liu, H.; Kuo, C.; Imler, W. Appl. Phys. Lett. 1996, 68, 3144−3146. (29) Hacke, P.; Maekawa, A.; Koide, N.; Hiramatsu, K.; Sawaki, N. Jpn. J. Appl. Phys. 1994, 33, 6443. (30) Debye, P. P.; Conwell, E. M. Phys. Rev. 1954, 93, 693−706. (31) Meyer, B.; Volm, D.; Graber, A.; Alt, H.; Detchprohm, T.; Amano, A.; Akasaki, I. Solid State Commun. 1995, 95, 597−600. (32) Lopatiuk-Tirpak, O.; Schoenfeld, W. V.; Chernyak, L.; Xiu, F. X.; Liu, J. L.; Jang, S.; Ren, F.; Pearton, S. J.; Osinsky, A.; Chow, P. Appl. Phys. Lett. 2006, 88, 202110. (33) Scheer, E.; Löhneysen, H. v.; Hein, H. J. Vac. Sci. Technol., B: Microelectron. Process. Phenom. 1994, 12, 3171−3175. (34) Blömers, C.; Rieger, T.; Zellekens, P.; Haas, F.; Lepsa, M. I.; Hardtdegen, H.; Gül, O.; Demarina, N.; Grützmacher, D.; Lüth, H.; Schäpers, T. Nanotechnology 2013, 24, 035203. (35) Lucot, D.; Jabeen, F.; Ramdani, M. R.; Patriarche, G.; Faini, G.; Mailly, D.; Harmand, J.-C. J. Cryst. Growth 2013, 378, 546−548. (36) Lin, J. J.; Bird, J. P. J. Phys.: Condens. Matter 2002, 14, R501. (37) Al’tshuler, B. L.; Aronov, A. G. JETP Letters 1981, 33, 499−501. (38) Beenakker, C. W. J.; van Houten, H. Phys. Rev. B: Condens. Matter Mater. Phys. 1988, 37, 6544−6546. (39) Lee, P. A.; Stone, A. D. Phys. Rev. Lett. 1985, 55, 1622−1625. (40) Doh, Y.-J.; Franceschi, S. D.; Bakkers, E. P. A. M.; Kouwenhoven, L. P. Nano Lett. 2008, 8, 4098−4102. (41) Blömers, C.; Schäpers, T.; Richter, T.; Calarco, R.; Lüth, H.; Marso, M. Appl. Phys. Lett. 2008, 92, 132101. (42) Al’tshuler, B. L.; Aronov, A. G.; Zyuzin, A. Y. JETP Letters 1984, 59, 415. (43) Choi, K. K.; Tsui, D. C.; Alavi, K. Phys. Rev. B: Condens. Matter Mater. Phys. 1987, 36, 7751−7754. G


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