Abstract
Linear transverse magnetoresistance is commonly observed in many material systems including semimetals, narrow bandgap semiconductors, multilayer graphene and topological insulators. It can originate in an inhomogeneous conductor from distortions in the current paths induced by macroscopic spatial fluctuations in the carrier mobility and it has been explained using a phenomenological semiclassical random resistor network model. However, the link between the linear magnetoresistance and the microscopic nature of the electron dynamics remains unknown. Here we demonstrate how the linear magnetoresistance arises from the stochastic behaviour of the electronic cycloidal trajectories around lowmobility islands in highmobility inhomogeneous conductors and that this process is only weakly affected by the applied electric field strength. Also, we establish a quantitative link between the island morphology and the strength of linear magnetoresistance of relevance for future applications.
Introduction
Linear transverse magnetoresistance (MR) is a phenomenon commonly observed in many material systems, including semimetals^{1}, narrow bandgap semiconductors^{2,3}, multilayer graphene^{4} and topological insulators^{5}. Linear MR can arise from either quantum^{6} or classical effects^{7,8}. In particular, it can originate in an inhomogeneous conductor from distortions in the current paths induced by macroscopic spatial fluctuations in the carrier mobility; this type of nonsaturating linear MR has been observed in several materials and explained in terms of a phenomenological semiclassical random resistor network model^{7,8}. However, the link between the linear MR and the microscopic nature of the electron dynamics remains still unexplored. Existing theories of classical linear MR apply to systems with ohmic conductivity, whereas the effects of hot electron dynamics on the linear MR are largely unknown. On the other hand, a significant enhancement of MR has been observed in the nonomhic spacecharge limited transport regime^{9} and under avalanche breakdown^{10}, thus raising problems of fundamental interest and technological relevance, that is, what is the specific carrier dynamics giving rise to the linear MR, is it preserved in the presence of a large applied electric field and can it be controlled and exploited in sensitive Hall sensors or magnetoresistors^{11}?
Here, we examine the physical origin of the classical linear MR in a highmobility inhomogeneous conductor. Our measurements and Monte Carlo simulations focus on the ntype compound semiconductor InAs and consider an extended range of applied electric (E) and magnetic (B) fields, and different sample geometries. By modelling our data with Monte Carlo simulations of the electron dynamics, we demonstrate that the linear MR arises from multiple scattering of the currentcarrying electrons by lowmobility islands within the conducting layer. This process is only weakly affected by the details of the electronic band structure and electric field strength. Also, it persists at magnetic fields as high as ~50 T corresponding to a magnetic length as small as 4 nm, significantly smaller than the electron mean free path (~100 nm) and island size. The link between the linear MR and the stochastic behaviour of the electronic cycloidal trajectories around lowmobility islands is a finding of general interest as it is relevant to all highmobility inhomogeneous conductors. The identification of a quantitative link between the island morphology and the strength of linear MR is also potentially useful for tailoring the linear MR in future applications.
Results
Material systems and devices
Our InAs and In(AsN) epilayers were grown by molecular beam epitaxy (MBE) on a semiinsulating (100)oriented GaAs substrate. Owing to the mismatch between the substrate and epilayer lattices, inhomogeneities can arise due to the presence of dislocations and defects; also, the Natoms act as scattering centres for the conduction electrons^{12,13}. Thus with increasing Ncontent from 0 to 1%, the electron Hall mobility decreases from μ_{H}=3.3 m^{2} V^{−1} s^{−1} (3.1 m^{2} V^{−1} s^{−1}) to 0.6 m^{2} V^{−1} s^{−1} (0.5 m^{2} V^{−1} s^{−1}) at T=300 K (4.2 K). Correspondingly, the electron density increases from n_{e}=4×10^{16} cm^{−3} in InAs to n_{e}=1.4×10^{17} cm^{−3} in In(AsN), with both values being very weakly affected by temperature.
The presence of inhomogeneous disorder combined with the high electron mobility makes these InAs and In(AsN) layers particularly well suited to the study of classical linear MR: even in the lowmobility layers the high values of μB≫1, which are necessary for observing a large MR, can be achieved with available conventional (superconducting) solenoids. To investigate the influence of sample geometry and electric field on the MR, twoterminal planar devices with InAs channels of length L=2, 5 and 10 μm and of width W=1, 2 and 5 μm were also studied.
High magnetic field studies
In our experiments, the magnetic field was applied parallel to the growth axis z, that is, B=[0, 0, B_{z}]. The transverse MR ratio defined as Δρ_{xx}/ρ_{xx}=[ρ_{xx}(B_{z})−ρ_{xx}(0)]/ρ_{xx}(0) was studied with magnetic fields, B_{z}, up to 14 T, generated by a superconducting magnet, or in a pulsed solenoid at B_{z} up to 47 T with a total pulse length of 100 ms. Figure 1a shows the B_{z}dependence of Δρ_{xx}/ρ_{xx} for an Hallbar based on ntype InAs. A positive linear MR is observed at B_{z}>1 T and is weakly affected by temperature in the range T=4.2–300 K. A weak oscillatory component of ρ_{xx} is observed at low temperature (T<20 K), which we attribute to the Shubnikov–de Haas effect in the twodimensional electron gas that accumulates on the surface of InAs due to the bending of the conduction band by charged native surface defects^{12}. The resistance of the surface twodimensional electron gas at B_{z}=0 T is at least one order of magnitude higher than ρ_{xx} and, at all values of B_{z} and T, the amplitude of the magnetooscillations is much smaller than the measured monotonically increasing MR.
Linear MR can occur when the electron distribution enters the socalled quantum limit, in which only the lowest Landau level is occupied, that is, when the cyclotron energy, , exceeds the Fermi energy, ε_{F}, and when , ε_{F}≫k_{B}T^{6} (ref. 6). As shown in Fig. 1b, the transition of the MR to a linear Bdependence occurs at B_{z}~1 T below the value satisfying the quantum limit condition , that is, . This limit corresponds to fields >3 T for an electron density n_{e}=4×10^{16} cm^{−3}, which we derive from the gradient of the Hall resistivity, ρ_{xy}, in the limit B_{z}→0. Also, we note that the condition for single occupancy of the lowest Landau level requires that . This condition is satisfied for temperatures <50 K at B_{z}=1 T and for the electron cyclotron mass of InAs (m_{e}=(0.025±0.01) m_{0}, where m_{0} is the electron mass in vacuum)^{14}. However, the linear MR is observed well above T=50 K (see Fig. 1), indicating that it does not have a quantum origin, but is a classical effect.
Classical linear MR has been explained using a phenomenological model in which the distorted current paths arising from disorderinduced ihnomogeneities and macroscopic variations, Δμ, in the electron mobility are simulated by a network of random resistors. Each resistor element has four terminals to take into account not only the local resistive voltage drop but also the Hall voltage associated with the local mobility, μ, of a particular region of the sample^{7,8}. The crossover of the MR from quadratic to linear B_{z}dependence is predicted to occur at a characteristic magnetic field , or at B_{z}^{t}=Δμ^{−1} for Δ μ / > 1, where corresponds to an average of the spatially varying mobility. A comparison of the B_{z}dependence of Δρ_{xx}/ρ_{xx} in our InAs and In(AsN) Hallbars at T=300 K (see Fig. 1b) shows that both the value of 1/B_{z}^{t} and the slope of the MR follow the systematic decrease of μ from 3.3 m^{2} V^{−1} s^{−1} in InAs to 0.6 m^{2} V^{−1} s^{−1} in In(AsN). The characteristic magnetic field B_{z}^{t} provides an estimate of the average mobilities = 1/B_{z}^{t} ~ 1 m^{2} V^{−1} s^{−1} and 0.5 m^{2} V^{−1} s^{−1} in the InAs and In(AsN) samples, respectively, which are comparable with those measured by the Hall effect. Also, the MR tends to follow the temperature dependence of μ: in the InAs sample, an increase of T > 4.2 K and up to T ~100 K leads to an increase of μ_{H} from 3.1 to 3.8 m^{2} V^{−1} s^{−1} followed by a decrease at T>150 K. This variation is caused by the interplay between impurity and phonon scattering and is accompanied by a similar Tdependence of the MR (see Fig. 1a). Linear MR is also observed in the twoterminal small channel devices, see Fig. 1c.
Monte Carlo simulations
As our InAs and In(AsN) epilayers are grown on highly latticemismatched GaAs, threading dislocations tend to form at the epilayer/substrate interface. The macroscopic (>0.1 μm) inhomogeneities observed by transmission electron microscopy of our samples^{15} are likely to induce variations in the electronic properties. We model the effect of such inhomogenities on the electron dynamics by Monte Carlo simulations^{16}. As the magnetic field is applied along the zdirection, the magnetic field affects the components of the electron motion in the x–y plane. In our simulation of the twodimensional electron trajectories, inhomogeneity is introduced by placing lowmobility islands randomly within the x–y plane (Fig. 2). The random spatial profile of the islands, Δ(x,y), is generated from the power spectrum of the autocorrelation function ‹Δ(r)Δ(r′)›=Δ^{2}exp[−r−r′^{2}/Λ^{2}], where Λ is the correlation length, r=(x, y) and r′=(x′, y′). The lowmobility regions R_{low} are then defined according to the relation . This condition determines the fractional area, f, of the sample with low mobility in which electrons undergo scattering at a rate higher than in the remaining highmobility part of the sample. In a threedimensional model, the value of f would result from an average of the fvalues from various x–y planes of the threedimensional conducting channel. In the lowmobility regions, we use a scattering rate r_{island}=r_{i}+r_{e}, where r_{i}=1.5×10^{12} s^{−1} and r_{e}=2.5×10^{14} s^{−1} are the elastic and inelastic phonon scattering rates, respectively; elsewhere, the electron drift is mainly limited by inelastic scattering with r~r_{i}. The nonparabolicity of the electron energy dispersion of InAs is modelled as , where m_{e}=0.026m_{0} and α=2.2 eV^{−1}. This form of the energy dispersion occurs in numerous semiconductors, including the narrow gap InSb for which observations of linear MR have been also reported^{2,3}. The electron velocity, given by , and the wave vector, k, change with time under the action of the Lorentz force, , where E=[E_{x}, E_{y}, 0] and B=[0,0, B_{z}]^{17}. The Hall field component, E_{y}, is set so that the ycomponent of the average electron velocity is zero, that is, , consistent with the current flow for our Hall bars. The main findings of our simulations are not changed if we consider a different geometry in which the Hall field is effectively 'shorted out', that is, E_{y}=0. Thus, our analysis applies not only to Hallbars but also to short channels and Corbino geometries, in which the Hall electric field is fully 'shortcircuited' by the contact electrodes.
Figure 2a–d illustrates the calculated electron trajectories at E_{x}=0.5 kV cm^{1} and B_{z}=0, 1, 10 and 20 T. Here, we have assumed a correlation length Λ=0.5 μm, f=5% and lowmobility regions of average size s=0.4 μm and interisland separation l=8 μm. It can be seen that in the vicinity of the lowmobility islands, the electron undergoes multiple scattering events, which act to deflect the cycloidal motion (see inset of Fig. 2b). This islandinduced scattering is observed at low (Fig. 2b) and high (Fig. 2d) applied magnetic fields. It is absent in a homogenous system and is responsible for the linear magnetic field dependence of the inverse mobility component, μ_{xx}^{−1}, whose strength increases with f, as shown in Fig. 3a.
Discussion
We interpret the results of the Monte Carlo simulations by considering a simplified Drude model of the average electron velocity v_{x} and average hopping distance travelled by the conduction electrons along the xdirection. In a uniform system, that is, with no islands, and for E_{y}=0 and v_{y}≠0, the average velocity along x is v_{x=}ΔX/τ, where ΔX m_{e}E_{x}/(eB_{z}^{2}) is the typical hopping distance and τ is the scattering time. In the presence of inhomogeneity, an electron travels an additional distance ΔX′~s/2 around a lowmobility island. As the electron hits an island every time interval τ_{island}~l/v_{H}, where v_{H}=E_{x}/B_{z} is the electron Hall velocity along y and l is the distance between the islands, the average electron velocity v_{x} and mobility component μ_{xx} increase as v_{x}=ΔX/τ+ΔX′/τ_{island} and μ_{xx}=m_{e}/(τe^{2}B_{z}^{2})+s/(2lB_{z}), respectively. Thus, if we compare this result with the case of an homogenous system for which μ_{xx}=m_{e}/(τe^{2}B_{z}^{2}) and v_{x}=m_{e}E_{x}/(τeB_{z}^{2}), we conclude that at high B_{z} island scattering tends to facilitate electron drift along the xdirection, perpendicular to the cycloidal motion along y (see electron scattering around island A in Fig. 3b).
For the geometry in which E_{y}=0 and v_{y}≠0, the resistivity component ρ_{xx} is given by . Using the dependence of μ_{xx} on B_{z} at large fields, that is, μ_{xx}~ΔX′/τ_{island}=s/(2lB_{z}), we derive a linear MR, that is, Δρ_{xx}/ρ_{xx}~B_{z}μ*, where μ*=μ_{0}(s/2 l) is and effective mobility and μ_{0} is the zero field mobility. A similar conclusion can be drawn for the Hall geometry, that is, for E_{y}≠0 and v_{y}=0. In this case the resistivity component ρ_{xx} is given by , where μ_{xy}=B_{z}^{−1}. Using this expression in the limits of high B_{z}, we obtain also in this case that Δρ_{xx}/ρ_{xx} ~ B_{z}μ*. Note, however, that in this geometry, the island scattering acts to decrease the average electron velocity, v_{x}, by deflecting the electron motion along y (see electron scattering around islandB in Fig. 3b).
In summary, the magnetic field dependence of the mobility component μ_{xx} due to island scattering generates a characteristic linear dependence of the MR on magnetic field, which is the same in both geometries, that is, E_{y}≠0 and E_{y}=0. This analysis identifies the physical origin of the random variation of the resistive and Hall effect components of each fourterminal resistive element of the network model^{7,8}: the randomness arises from the stochastic behaviour of the electronic cycloidal trajectories around the lowmobility islands.
From a linear fit to the measured MR data for the large InAs Hallbar of Fig. 1b by the relation Δρ_{xx}/ρ_{xx}=B_{z}μ*, we obtain μ*=0.33 m^{2} V^{−1} s^{−1}. For μ_{0}=μ_{H}=3.3 m^{2} V^{−1} s^{−1}, this corresponds to s/(2 l)=μ*/μ_{0}=0.1 and an average island coverage factor f=s/(s+l)~0.17. Similarly, for the In(AsN) Hallbar of Fig. 1b we find that s/(2 l)=μ*/μ_{0}=0.23 and f=s/(s+l)~0.32. Thus, in both cases the average island separation, l, is significantly larger than the average island size, s. Although we cannot deduce the exact value of l or s, we can infer a reasonable estimate from our MR experiments in small channel devices, see Fig. 1c. Here, it can be seen that devices with small values of the channel length L (2–10 μm) and/or width W (1–5 μm) exhibit much smaller values of Δρ_{xx}/ρ_{xx} and in the smallest devices the MR tends to saturate at high B_{z}. The linear MR decreases systematically as the channel width W and length L decrease to values of the order of 1 μm. Thus, we infer that the dimensions of these small devices are comparable to the average island separation.
The Monte Carlo simulations indicate that different values of the band nonparabolicity parameter α produce similar linear MR behaviour. Thus, the specific details of the electronic band structure do not have a critical role in the observation of linear MR. Perhaps surprisingly, although μ_{xx}^{−1} tends to increase with increasing E_{x}, its linear dependence on B_{z} is found to be weakly affected by electric field (Fig. 3c) and only at lowmagnetic fields (<1 T) and high E_{x} does the value of μ_{xx}^{−1} tend to decrease with decreasing B_{z}, thus leading to a weak negative MR. This is caused by the interplay between electron cyclotron motion and islandinduced scattering, which tends to be increased by E_{x}, that is, electrons hit a lowmobility islands more frequently at higher speeds. However, overall this effect tends to be weak and, at large B_{z}, the deflection of the cycloidal motion by the lowmobility islands and corresponding linear MR are preserved.
The weak dependence of the linear MR on applied electric field is confirmed by our MR experiments in the short channel InAs samples. Figure 4a shows the dependence of the current density J on E_{x} at low temperature (T=4.2 K) and at various B_{z} (up to 14 T) for an InAs channel of length L=10 μm and width W=2 μm. Similar dependences of J on E_{x} and B_{z} were found in devices with different values of L and W. At B_{z}=0 T, the J(E_{x}) curve is approximately linear at low bias. The ohmic behaviour is then followed by a slight sublinear bias dependence and, at larger electric fields, by a sharp rise of the current that is characteristic of systems switching from a state of low to high conductivity due to impact ionization and avalanche breakdown^{15,18}. At all values of E_{x}, the magnetic field acts to suppress the current and to shift the threshold electric field for impact ionization to significantly higher values. From Fig. 4b, it can be seen that the MR remains linear over an extended range of E_{x} up to ~3 kV cm^{−1}, just below the threshold electric field for impact ionization. In contrast, in the regime of avalanche breakdown the MR is strongly enhanced and tends to depart from a simple linear dependence due to the effect of the magnetic field in suppressing the avalanche multiplication of carriers^{10}.
To conclude, our data and analysis indicate that the presence of lowmobility islands in a highmobility conductor and their effect on the electron trajectories underpin the observation of classical linear MR. We have established a clear link between the stochastic dynamics of the electron trajectory around lowmobility islands caused by multiple scattering at the islands and the classical linear MR in devices with different geometry and size. Although our observations have focused on the highmobility inhomogeneous ntype InAs and In(AsN) epilayers grown on GaAs, our main findings and conclusions are relevant to a wide variety of structures, device geometries and material systems. Our results are therefore of general interest and will help guiding future research into the manipulation of the linear MR by tuning of the electron trajectories in tailormade inhomogeneous materials and device structures.
Methods
Growth and fabrication
Our InAs epilayer (thickness t=1. 5 μm) was grown by MBE on a semiinsulating (100)oriented GaAs substrate. An In(AsN) epilayer (t=1.3 μm) containing a small concentration (N=1%) of substitutional Natoms was also grown by MBE using a radio frequency (RF) plasma Nsource. The surface reconstruction was monitored by in situ reflection highenergy electron diffraction, while the substrate temperature was measured using an infrared pyrometer calibrated with the surface reconstruction transitions at a fixed Asflux. A Veeco UNIBulb RF plasma source was used as nitrogen source. It was set at a RF power of 160 W with a nitrogen flux of 5×10^{−6} mbar for all epilayers. The Ncontent in the In(AsN) epilayers and the crystal quality of the layers were optimized by setting the growth temperature, T_{G}, and the growth rate, r_{G}, to given values in the ranges T_{G}=400–460 °C and r_{G}=0.3–1 μm h^{−1}, and by keeping a minimum Asflux. The InAs epilayer was grown at T_{G}=460 °C. Highresolution Xray diffraction measurements using a Bede QC200 diffractometer confirmed a relaxation of the crystal lattice by 98% in the InAs and In(AsN) epilayers.
The wafers were spun with photoresist and patterned by photolithography into Hallbars of length L=1,250 μm and width W=250 μm. Twoterminal planar devices with InAs channels of length L=2, 5 and 10 μm and of width W=1, 2 and 5 μm were also fabricated by electron beam lithography. Each channel was formed by dryetching the epilayer down to the GaAs substrate. Metal contacts consisting of 10 nm of Ti followed by 200 nm of Au were deposited onto the samples to provide Ti–Au ohmic contacts.
Transport studies
Measurements of the current–voltage characteristics were performed in both direct current and pulsed mode. For the pulsed mode of acquisition of the current, a voltage pulse generator was used to generate sawtooth single voltage pulses of duration of 10 μs, and a digital oscilloscope was used for the synchronized and fast temporal recording of the applied bias and current, thus reducing excessive sample heating at highoperating currents. The current–voltage curves in pulsed magnetic field were taken within 1 ms at the top of the magnetic field pulse, during which the magnetic field remains constant to within ±0.5%.
Additional information
How to cite this article: Kozlova, N.V. et al. Linear magnetoresistance due to multipleelectron scattering by lowmobility islands in an inhomogeneous conductor. Nat. Commun. 3:1097 doi: 10.1038/ncomms2106 (2012).
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Acknowledgements
This work was supported by the Royal Society (UK), DFG KO 3743/11, AOBJ: 550341. We acknowledge Dr R. Airey (EPSRC, III–V National Facility, Sheffield) for processing our samples.
Author information
Affiliations
School of Physics and Astronomy, The University of Nottingham, Nottingham NG7 2RD, UK.
 N.V. Kozlova
 , O. Makarovsky
 , L. Eaves
 & A. Patanè
IFW Dresden, Institute for Metallic Materials, PO Box 270116, D01171 Dresden, Germany.
 N.V. Kozlova
Graduate School of Engineering, Osaka University, 21 YamadaOka, Suita City, Osaka 565–0871, Japan.
 N. Mori
Department of Physics, Lancaster University, Lancaster LA1 4YB, UK.
 Q.D. Zhuang
 & A. Krier
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Contributions
A.P., N.V.K. and O.M. designed and performed the experiments; N.M. performed the Monte Carlo simulations; L.E. took part in the discussions of the Monte Carlo studies; A.K. and Q.D.Z. grew the samples; A.P. wrote the paper; and all authors took part in the discussion/analysis of the data.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to A. Patanè.
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