NANO LETTERS
Localization/Quasi-Delocalization Transitions and Quasi-Mobility-Edges in Shell-Doped Nanowires
2006 Vol. 6, No. 1 128-132
Jianxin Zhong*,†,‡ and G. Malcolm Stocks*,† Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, Tennessee 37831, and Department of Physics, Xiangtan UniVersity, Hunan 411105, China Received October 5, 2005; Revised Manuscript Received November 16, 2005
ABSTRACT We propose a novel concept, namely, shell-doping of nanowires, for control of carrier mobility in nanowires. Different from traditional doping, where dopant atoms are distributed uniformly inside nanowires, shell-doping spatially confines dopant atoms within a few atomic layers in the shell region of a nanowire. Our numerical results based on the Anderson model of electronic disorder show that electrons in a shell-doped nanowire exhibit a peculiar behavior very different from that of uniformly doped nanowires. Beyond some critical doping, electron dynamics in a shell-doped nanowire undergoes a localization/quasi-delocalization transition, namely, the electron diffusion length decreases in the regime of weak disorder but increases in the regime of strong disorder. This transition is a result of the existence of quasi-mobility-edges in the energy spectrum of the system, which can be exploited experimentally through control of electron concentration, carrier density, and degree of disorder.
Recent breakthroughs in the growth of semiconductor nanowires1-5 have opened up great opportunities to revolutionize technologies in nanoscale electronics,1-10 optoelectronics,10-12 spintronics,13 and sensors.14 In particular, measurements of electronic transport have shown that semiconductor nanowires remain semiconducting independent of diameters, suggesting that carrier transport in a nanowire may be tailored to desired properties through doping for different applications.15-17 Common wisdom regarding doping in semiconductors is that although dopant atoms can increase carrier density, they also decrease the carrier mobility because ionized dopant atoms act as random scattering centers that lead to carrier localization. The disorder-induced carrier localization is more severe in nanowires. Indeed, the theory of Anderson localization predicts that any amount of disorder in onedimensional systems results in carrier localization. Furthermore, the larger the disorder the stronger the localization.19 Therefore, one would expect from the traditional view of doping that the larger the dopant concentration in a nanowire the lower the carrier mobility. Moreover, because of strong carrier localization, one would expect hopping transport characterized by low conductivities in heavily doped nanowires rather than diffusive transport associated with large conductivities. * Corresponding authors: E-mail:
[email protected];
[email protected]. † Oak Ridge National Laboratory. ‡ Xiangtan University. 10.1021/nl051981m CCC: $33.50 Published on Web 12/09/2005
© 2006 American Chemical Society
This common wisdom notwithstanding, recent experiments on nanowires have yielded complicated behaviors, some of which are in sharp contradiction to the above picture. For instance, Cui et al.15 reported that the carrier mobility in boron-doped and phosphorus-doped silicon nanowires under low dopant concentration is extremely low compared to bulk silicon but the conductance becomes diffusive and about 5 orders of magnitude larger after heavy doping. Yu et al.16 found that the conductance of highly gold- and zinc-doped silicon nanowires are substantially larger than expected. However, Zheng et al.17 reported recently that the average mobility in phosphorus-doped silicon nanowires decreases as the doping level increases. It has been suggested that surface scattering,15 surface states,16 surface reconstructions,18 and contact resistance17 may be responsible for these diverse transport properties. In this Letter, we present results of calculations of the transport properties of a shell-doped nanowire that not only illuminate some of the unusual experimental results discussed above but also suggest general principles for understanding and control of transport properties in nanowires. Different from traditional doping, where dopant atoms are distributed uniformly inside nanowires, shell-doping spatially confines dopant atoms within a few atomic layers in the shell region of a nanowire. To elucidate the nature of carrier motion in a shell-doped nanowire, we have employed an Andersontype tight-binding Hamiltonian. Using surface doping as an example, we show that electrons in a shell-doped nanowire
Figure 1. Schematic illustration of a nanowire with surface disorder induced by shell-doping.
exhibit rather counterintuitive behavior, namely, the larger the disorder the weaker the localization. Specifically, we will show that, beyond some critical doping, electron dynamics in a shell-doped nanowire undergoes a localization/quasidelocalization transition, namely, the electron diffusion length decreases in the regime of weak disorder but counterintuitively it increases in the regime of strong disorder. We will further show that this transition is a result of the existence of quasi-mobility-edges in the energy spectrum of the system with large disorder, which separate the states of large localization length in the spectral center from the strongly localized states in the spectral tails. From these results, we expect different behaviors of mobility in shelldoped nanowires at different doping levels. In the regime of low dopant concentration (small disorder) and in the localized regime of heavy doping (large disorder), mobility decreases as the dopant concentration increases. In the quasi-delocalized regime of heavy doping, however, we expect to have large and rather stable mobility that does not decrease as doping level increases. In addition, a sharp transition of mobility may be induced by additional electron doping as the Fermi energy crosses the quasi-mobility edges. On the experimental side, the foregoing predictions appear to be in rather good agreement with the experimental findings in doped silicon nanowires by Cui et al.,15 Yu et al.,16 and Zheng et al.17 despite the fact that the spatial distribution of dopant atoms is unknown in these experiments. Furthermore, the model points the way to additional experimental studies. Once confirmed experimentally, this theory may have important applications in controlling the transport properties of nanowires through controlling the spatial distribution of dopant atoms and Fermi energy levels. In addition, it can aid the understanding of the effect of fabrication, processing, and measurement-induced surface contamination on transport properties. Figure 1 is a schematic representation of a shell-doped nanowire. In this, the cross section of the nanowire is a square lattice of size na × na, where a is the lattice constant. The length of the nanowire is L ) ma with m . n. We consider the following tight-binding Anderson model: H)
h(i, j)|i〉 〈 j| ∑i (i)|i〉 〈i| + ∑ i,j
(1)
Where h(i, j) ) 1 are the hopping integrals for nearest neighbors and (i) are the site energies in units of h. For atoms in the core region, ≡ 0. Site energies of atoms in the shell (surface) region take random values distributed Nano Lett., Vol. 6, No. 1, 2006
Figure 2. The mean square displacement, d(t), for nanowires with different degrees of surface disorder.
within interval [-W, W], and W describes the degree of the shell disorder introduced by dopant atoms. We note that the above Anderson Hamiltonian is a standard Hamiltonian for studying the effects of electronic disorder. It includes not only the contribution of the potential difference of dopant atoms but also the contribution of the lattice distortion induced by doping. This Hamiltonian is ideal for revealing the general effects of electronic disorder. We also used the alloy-type Anderson Hamiltonian, where degree of disorder is simply described by the concentration of dopants. We found same conclusions numerically by using different models. Our analytical analysis further shows that our finding is a general result for shell-doped nanowires within the framework of independent electrons. To understand electron dynamics in this model of a shelldoped nanowire, we have studied the quantum diffusion of an electron initially located at position b r0 as time increases.20,21 The evolution of the electronic wave packets is given by the time-dependent Schro¨dinger equation, iψ˙ (r bn, t) ) Hψ(r bn, t), where ψ(r bn, t) is the wave function amplitude at position b rn at time t. Different electron dynamics can be identified through the mean square displacement d(t) with ballistic motion and diffusion corresponding to d(t) ≈ t and d(t) ≈ xt, respectively. In the case of localization, one has d(t f ∞) ) d0, where d0 is the dynamical localization length. We also calculated the density of states, D(E), and participation number, P(E), two important quantities for understanding electronic properties.19 P(E) is defined as P(E)-1 ) ∑nφ(E, b rn),4 where φ(E, b rn) is the wave function of state E. P(E) measures the localization length of a specific state19 with P(E) ∝ N for an extended state and P(E) ∝ N0 for a localized state, where N is the total number of sites of the system. Large values of P(E) and D(E) together lead to large values of d0. Figure 2 illustrates the time evolution of d(t) as a function of the strength of the shell disorder W, for a nanowire of size 5a × 5a × 201a and electrons initially located at the central site of the nanowire. To obtain good statistics, we used up to 100 different configurations. For the data shown in Figure 2, the error bars are less than 1%. From the theory of electron localization in one-dimensional systems,19 one expects d(t) to become time-independent at large time, namely, d(t f ∞) ) d0(W), where d0 decreases as W increases. From Figure 2, we can see that d(t) indeed 129
Figure 3. Diffusion distance, d0, as a function of W for nanowires of different diameters.
evolves into different constant values at large time. However, values of d0 display a very unusual behavior. In the regime of weaker disorder, W < Wc, d0 decreases as W increases. However, in the regime of stronger disorder, W > Wc, d0 increases as W increases. The critical value, W ) Wc, which marks the onset of the atypical behavior is found to be Wc ≈ 5. In Figure 3 we plot d0 of a function of W for three different wire diameters; all three wires show the same general behavior with Wc decreasing as the wire diameter increases. To understand this intriguing dynamical transition, we performed spectral analysis focusing on D(E) and P(E). Figure 4 illustrates the behavior of these quantities for a nanowire of size 5a × 5a × 201a for both shell and bulk disordered nanowires. In both cases, D(E) has tail states and their spectral width increases as W increases. In the central spectral region, however, D(E) displays very different behavior in the two cases. For bulk disorder, D(E) of the central states decreases continuously with increasing W. However, in the case of shell disorder, D(E) has a rather stable portion in the central spectral region when W > Wc, which is almost independent of W. Our calculations show that the stable part of D(E) in the central spectral region given by |E| < Ec(Ec ≈ Wc) in Figure 4 resembles D(E) of a perfect
nanowire of size 3a × 3a × 201a without any disorder. This indicates that in the case of strong disorder, the disordered shell has little influence on the states of the inner core of the nanowire. Figure 4 also shows that P(E) displays the same behavior as D(E). For bulk disorder, P(E) decreases continuously with W for all of the states in the spectrum. However, in the case of shell disorder, P(E) decreases with W only for the tail states. Inside the central spectral region |E| < Ec, P(E) is rather stable independent of W. As a result, P(E) undergoes a sharp transition around the critical energy level Ec from large values for the central states (|E| < Ec) to small values for the tail states (|E| > Ec). The ratio of P(E) between the central states and the tail states can be as high as 200 for the system we studied, indicating that the states in the region |E| < Ec have much larger localization lengths than the tail states. We have systematically studied nanowires of different diameters with n ) 3, 4, and 5 and found that the above conclusions hold for all of them. Here it is worth noting that this drastic change in P(E) around a specific energy level is very similar to the behavior of P(E) around the mobility edges in three-dimensional disordered systems, where the mobility edge separates extended states from localized states.19 The difference is that in the case of nanowires with shell disorder, the critical energy level, Ec, separates localized states with enormously different localization lengths. We name Ec the quasi-mobility-edge of nanowires. The localization/quasi-delocalization transition of nanowires can be understood from the unique properties of their energy spectra and states. From Figure 4, we know that P(E) decreases dramatically as W increases for W < Wc. In the regime of W > Wc, however, D(E) and P(E) have large values in the central spectral region that are nearly independent of W. Because large values of P(E) and D(E) lead to large diffusion distances, one has the conclusion that d0 decreases as W increases for W < Wc but keeps almost invariant in the regime of W > Wc.
Figure 4. Density of states, D(E), and participation numbers, P(E), for nanpowires of different degrees of disorder, W. (a) Nanowires of surface disorder. (b) Nanowires of bulk disorder. 130
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A rigorous proof of the existence of mobility edges can be derived by considering a shell-doped nanowire as a coupled system comprising two subsystems. From the Schro¨dinger equation, we have (Hp - δ)φp ) Eφp
(2a)
(Hd - λ)φd ) Eφd
(2b)
δ ) Hpd(Hd - E)-1 Hdp
(3a)
λ ) Hdp(Hp - E)-1 Hpd
(3b)
and
where Hp and Hd are, respectively, the sub-Hamiltonians for the perfect core zone and the disordered shell zone of the nanowire, Hpd and Hdp represent the coupling of the two subsystems, and φp and φd are the corresponding wave functions. Without the interaction between two subsystems, the spectrum is a simple superposition of two independent subspectra given by Hp and Hd. The sub-spectrum of Hp for a perfect nanowire has a bandwidth approximately given by [-6h, 6h] and all of the states are extended. The spectral width of Hd for the disordered zone can be simply estimated as [- W, W] in the case of strong disorder W > Wc ≈ 6h. For the coupled system, eq 2 shows that the spectrum consists of a central spectral region separated by two tails in the regime of W > Wc. The central part of the spectrum is determined by the modified Hamiltonian H /p ) Hp - δ and the tails are given by H /d ) Hd - λ. States in the tail regions are strongly localized because of the strong disorder proportional to W in Hd. States in the central spectral region are also localized because of the disordered scattering term, δ. However, because δ is inversely proportional to W, the strength of disorder in H /p is much smaller than that in H /d, resulting in much weaker localization of states. We note that δ f 0 as W f ∞, which leads to H /p ) Hp. In this case, states in the spectral tails are completely localized while states in the central spectral region are extended, resembling that for a periodic system. It is clear that the spectrum of a nanowire of shell disorder with W > Wc has two critical energies approximately given by Ec ≈ (6h, which separate the central states of much less localization from the tail states of strong localization. The theoretical prediction of the values of Ec and Wc is in good agreement with the numerical results. For the purpose of discussion, we have used shell-doped and bulk-doped nanowires both with the same Anderson parameter, W, which inevitably means that the dopant concentration in the bulk-doped nanowire is higher than the nanowire with only shell doping. However, we emphasize that this is not the reason for larger mobility in shell-doped nanowires. Rather, the intrinsic mechanism is the vanishing of the disordered scattering term, δ, for large disorder. From our theory, we expect that carrier mobility in a shell-doped nanowire will be even larger once its dopant concentration reaches the corresponding level in a bulk-doped nanowire. Nano Lett., Vol. 6, No. 1, 2006
To summarize, we have shown that shell-doping of a nanowire under large dopant concentration can enhance carrier mobility because of the existence of quasi-mobilityedges that separate the stable central states of large localization lengths from the strongly localized tail states. Our theory applies to small nanowires with widths comparable to the Fermi wavelength (tens of nanometers for semiconductors), where traditional doping leads to poor conduction because of strong carrier localization. The theory also applies to larger nanowires with small undoped zones of sizes comparable to the Fermi wavelength. Our finding about the drastic change of localization length around the quasi-mobility-edges is similar to the metal-insulator transition around the mobility edges in the three-dimensional Anderson model of disorder, where the mobility edges separate extended states from localized states. However, mobility edges in the three-dimensional Anderson model vanish at large disorder while the quasimobility-edges of shell-doped nanowires persist. Conductance of a shell-doped nanowire is expected to undergo a sharp transition once its Fermi energy passes through quasimobility-edges. This finding may lead to important applications in manipulation of carrier transport not only for shelldoped nanowires of single species but also for coaxial heterostructured nanowires with modulation doping.5 Acknowledgment. This work was supported by the Materials Sciences and Engineering Division Program of the DOE Office of Science under Contract no. DE-AC0500OR22725 with UT-Battelle, LLC, and partially by the Natural Science Foundation and the Education Bureau of Hunan Province, China. References (1) Morales, A. M.; Lieber, C. M. Science 1998, 279, 208. (2) Lauhon, L. J.; Gudlksen, M. S.; Wang, D.; Lieber, C. M. Nature 2002, 420, 57. (3) Wu, Y. Y.; Fan, R.; Yang, P. D. Nano Lett. 2002, 2, 83. (4) Bjo¨rk, M. T.; Ohlsson, B. J.; Sass, T.; Persson, A. I.; Thelander, C.; Magnusson, M. H.; Deppert, K.; Wallenberg, L. R.; Samuelson, L. Nano Lett. 2002, 2, 87. (5) Yang, P. D. MRS Bull. 2005, 30, 85. (6) Ma, D. D. D.; Lee, C. S.; Au, F. C. K.; Tong, S. Y.; Lee, S. T. Science 2003, 299, 1874. (7) Wu, Y.; Xiang, J.; Yang, C.; Lu, W.; Lieber, C. M. Nature 2004, 430, 61. (8) Doh, Y. J.; van Dam, J. A.; Roest, A. L.; Bakers, E. P. A. M.; Kouwenhoven, L. P.; De Franceschi, S. Science 2005, 309, 272. (9) Friedman, R. S.; MacAlpine, M. C.; Ricketts, D. S.; Ham, D.; Lieber, C. M. Nature 2005, 434, 1085. (10) Gudiksen, M. S.; Lauhon, L. J.; Wang, J. F.; Smith, D. C.; Lieber, C. L. Nature 2002, 415, 617. (11) Duan, X. F.; Huang, Y.; Agarwal, R.; Lieber, C. M. Nature 2003, 421, 241. (12) Huang, M. H.; Mao, S.; Feick, H.; Yan, H. Q.; Wu, Y. Y.; Kind, H.; Weber, E.; Russo, R.; Yang, P. D. Science 2001, 292, 1897. (13) Radovanovic, P. V.; Barrelet, C. J.; Gradecak, S.; Qian, F.; Lieber, C. L.; Nano Lett., in press, 2005. (14) (a) Wang, W. U.; Chen, C.; Lin, K. H.; Fang, Y.; Lieber, C. M. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 3208. (b) Patolsky, F.; Zheng, G. F.; Hayden, O.; Lakadamydi, M.; Zhuang, X. W.; Lieber, C. M. Proc. Natl. Acad. Sci. U.S.A. 2004, 101, 14017. (15) Cui, Y.; Duan, X. F.; Hu, J. T.; Lieber, C. M. J. Phys. Chem. B 2000, 104, 5213. (16) Yu, J. Y.; Chung, S. W.; Heath, J. R. J. Phys. Chem. B 2000, 104, 11864. (17) Zheng, G.; Lu, W.; Jin, S.; Lieber, C. M. AdV. Mater. 2004, 16, 1890. (18) Rurali, R.; Lorente, N. Phys. ReV. Lett. 2005, 94, 026805. (19) Lee, P. A.; Ramakrishnan, T. V. ReV. Mod. Phys. 1985, 57, 287. 131
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