NANO LETTERS
Modeling Transport in Ultrathin Si Nanowires: Charged versus Neutral Impurities
2008 Vol. 8, No. 9 2825-2828
Riccardo Rurali,*,† Troels Markussen,‡ Jordi Sun˜e´,† Mads Brandbyge,‡ and Antti-Pekka Jauho‡,§ Departament d’Enginyeria Electro`nica, UniVersitat Auto`noma de Barcelona, 08193 Bellaterra, Spain, Department of Micro and Nanotechnology, NanoDTU, Technical UniVersity of Denmark, DK-2800 Kongens Lyngby, Denmark, and Laboratory of Physics, Helsinki UniVersity of Technology, P.O. Box 1100, FI-02014 HUT, Finland Received May 16, 2008
ABSTRACT At room temperature dopants in semiconducting nanowires are ionized. We show that the long-range electrostatic potential due to charged dopants has a dramatic impact on the transport properties in ultrathin wires and can virtually block minority carriers. Our quantitative estimates of this effect are obtained by computing the electronic transmission through wires with either charged or neutral P and B dopants. The dopant potential is obtained from density functional theory (DFT) calculations. Contrary to the neutral case, the transmission through charged dopants cannot be converged within a supercell-based DFT scheme, because the system size implied by the long-ranged electrostatic potential becomes computationally unmanagable. We overcome this problem by modifying the DFT potential with finite element calculations. We find that the minority scattering is increased by a factor of 1000, while majority transmission is within 50% of the neutral dopant results.
Semiconducting nanowires are, like bulk semiconductors, typically doped to make them useful as nanoelectronic devices. By definition, a dopant is an ionized impurity at the device operating temperature. Thus, scattering against charged impurities should be a central issue in modeling nanowire devices. Nevertheless, all the first-principles studies reported so far on electron transport in semiconducting nanowires1-4 (as well as in carbon nanotubes5-7) bypass this important point and consider neutral defects. While this approach is justifiable in the zero-temperature limit, it is clearly inadequate for real device operation. The reason for the simplified standard approach is easy to understand: charged impurities are difficult to treat within first-principles approaches which invariably use periodic boundary conditions (PBC). In a well-defined simulation, the computational cell must be large enough to avoid the interaction between the periodic defect images. This requirement is especially demanding for semiconductors where the screening lengths are much larger than those in metals. For a smaller simulation cell an incompletely screened point charge leads to a divergent Coulomb energy, because of the interaction with its periodic images. A partial remedy for this problem is to use a neutralizing jellium background and correct a posteriori † ‡ §
Universitat Auto`noma de Barcelona. Technical University of Denmark. Helsinki University of Technology.
10.1021/nl801409m CCC: $40.75 Published on Web 08/02/2008
2008 American Chemical Society
the spurious terms that affect the total energy.8,9 This approach, however, is not sufficient to treat transport problems, as discussed below. In this paper we address the influence of the dopant charge on the electron transport properties of a Si nanowire (SiNW), showing that it can lead to a dramatic suppression of minority carriers transmission. We propose an approach to abridge the gap between the short length scale of a few nanometers, which are feasible in first-principles calculations, and a longer length scale imposed by the weak screening in semiconducting nanosystems.10 Our main idea is to correct the calculated DFT potential far from the impurity using a finite-element (FEM) electrostatic calculation and then use this corrected potential in the subsequent transport calculation for a longer system. We shall show that the charged impurities often lead to a qualitatively different physical picture compared to the neutral case. In particular we find that while majority transmission is within 50% of the neutral dopant results, minority carriers are strongly scattered since they have the same charge as the dopant. The rationale behind our approach is the following: In the case of neutral dopants the potential in a DFT calculation with PBC converges to the bulk values a few unit cells (∼2 nm) away from the dopant. This calculation can then be combined with a pristine wire calculation in a straightforward manner to obtain the electronic transmission.3 Such single-
Figure 1. (a) The change in Hartree potential averaged over the wire cross section between a charged impurity and a pristine wire DFT calculation (∆V ) Vq - V0), i.e., corresponding to B- and clean SiNW. Red diamonds and blue crosses are for a 5- and 9-UC SiNW, respectively. The continuous black line is the result of a FEM calculation in the case of 9-UC SiNW. The DFT electrostatic potential is locally converged in the region around the impurity, inside the dashed line; far from the dopant the potential can be reliably calculated from a FEM calculation. Inset: FEM results for increasing cell sizes. For clarity, the different potentials have been shifted to have a common maximum. (b) Transmission functions, T(E), with the same supercells as used in the upper panel: the variations in the electrostatic potential cause variations in the transmission. The full black line correspond to the clean wire.
impurity calculations have recently been shown to be useful in a number situations.4 However, this single-dopant setup fails for charged dopants: here the potential converges very slowly toward the bulk value. To address this problem, we consider in the following the change in DFT potential between the charged and corresponding clean wire system, ∆V ) Vq - V0. We note that the extra charge is localized around the dopant, therefore the change in DFT potential, ∆V, can, already a few bond lengths away, be approximated by a simple electrostatic model (Figure 1a) of the wire with point charges. Since we now can obtain the asymptotic dopant potential from a FEM calculation, this also means that we may introduce other boundary conditions and obtain the asymptotic behavior, e.g., for a longer PBC unit cell, or even metallic contacts. We have studied hydrogen-passivated SiNWs oriented along the 〈100〉 axis and with a diameter of approximately 2 nm, of the order of the thinnest wires grown so far. The atomic and electronic structures were obtained from DFT calculations using the Siesta code.11 With this we calculated the conductance with recursive Green’s function techniques.3 We have used a single-ζ polarized basis set which, despite its reduced computational cost, provides a very good agreement with results obtained with a double-ζ polarized basis.12,13 The electronic structure of the wires is qualitatively very well reproduced, and the bulk lattice parameter of 5.489 Å is within 2% of the experimental value. We have used supercells of increasing size, containing up to 11 unit cells (UCs) of SiNW, amounting to 1023 atoms. The purpose was 2826
to check the convergence of different relevant parameterss though we were mainly interested in the conductanceswith respect to the cell size. The atomic positions around the substitutional impurity in the central cell as well as in the two neighboring unit cells were relaxed until all forces were below 0.04 eV/Å; outside this central region the atoms were kept in the relaxed positions of the pristine H-terminated wire. We have considered B and P substitutional dopants as typical choices to achieve n- and p-type doping.14-16 It is known that the conductance will depend on the radial position of the dopant and that some impurities favor surface segregation.1 In this work, however, as the purpose is to study the effects of the charged impurities, we only consider substitutional impurities at the innermost region of the wire. We consider first a substitutional B- dopant and study the effect of the supercell size. While the 5-UC system seems to be sufficient to describe the localized acceptor state (which appears as a reasonably flat band), a major difference emerges when one considers the electrostatic potential and the way it decays away from the impurity. Figure 1a shows the electrostatic potential along the wire axis for a 5- and 9-UC system, respectively. As can be seen, the potentials have not decayed properly. Unfortunately, this spurious effect persists for all computationally manageable sizes: even the 11-UC wire (∼60 Å, >1000 atoms), which is the largest cell that we can treat, still exhibits this behavior. We have also performed FEM calculations of the electrostatic potential17 for even larger supercells; these confirm the trend of the DFT results (inset of Figure 1a). There are two very interesting observations to make: (a) within a central region around the dopant the DFT potential is well converged and does not change significantly when increasing the cell size; (b) far from the dopant the change in DFT potential, ∆V, is excellently reproduced by a cheap FEM calculation (see Figure 1). Note also that around the dopant the FEM results are poor, and thus an accurate quantum mechanical description of the local dopant potential is necessary. Figure 1b shows how the variations in the electrostatic potential lead to a nonconverged conductance. Figure 1a thus suggests that one can calculate the DFT potential around the impurity and then extend it based on FEM calculations to describe other boundary conditions.18 This approach is further justified by the results shown in Figure 2. Here we plot the transmission function for a Bdopant calculated with three different Hamiltonians describing the scattering region. The red line is T(E) obtained from the pure DFT Hamiltonian of a 11-UC SiNW, while the red squares have been obtained using the DFT potential of the five middlemost UCs and adding the FEM values in the tails. This result confirms that the DFT potential a few nanometers away from the dopant can be well approximated from a FEM calculation as also indicated in Figure 1a. If we describe the five central unit cells around the dopant by a 9 UC DFT calculation, i.e., a smaller supercell, plus a FEM correction,19 and use the same FEM potential for the tails, we obtain almost the same transmission. The two main dips in the T(E) are captured, although they are shifted about 0.01 eV, which Nano Lett., Vol. 8, No. 9, 2008
Figure 2. Three different ways to calculate the transmission function T(E) for a SiNW with a B- dopant: pure DFT 11-UC Hamiltonian (continuous red line); DFT Hamiltonian of the five middlemost UC from a 11-UC calculation plus FEM tails (squares); DFT Hamiltonian of the five middlemost UC from a 9-UC DFT calculation plus FEM potential in the central region and in the tails (circles). In all thee cases the contacts are described by the same pristinewire Hamiltonian, H0.
given the crude FEM model (circular cross section, effective dielectric constant, no exchange-correlation effects, etc.) is more than reasonable. Hence, although the FEM potential is not calculated self-consistently with the DFT charge density, the results show that this approximation can be used to obtain the transmission function. Notice that the longrange decay of the Coulomb potential induces spurious electrostatic interactions also in the lateral directions. We have found that at least 30 Å vacuum is required to converge T(E).20 Following the combined DFT and FEM model approach, we have calculated the T(E) in the case of two different separations between dopants, 8 and 14 nm (corresponding to a 15- and 25-UC supercell). The local scattering potential is the DFT potential of the five middlemost UC of a 11-UC supercell SiNW and the long-range tails have been obtained on the basis of FEM calculations.19 Thus, ∆V obtained in the FEM calculation of the model wire with PBC cell length corresponding to the 15 or 25 UC-supercell is added to the pristine wire DFT potential as tails outside the middlemost region.19 We stress that none of these boundary conditions could have been easily implemented within an all-DFT approach (the 25-UC supercell would contain 2325 atoms, beyond our present computational means). The transmissions, shown in Figure 3a, qualitatively change as the doping concentration is reduced, i.e., when the supercell is enlarged, once again underlining the crucial importance of the boundary conditions when dealing with charged defects. Most importantly, both transmissions differ significantly from the neutral B impurity transmission (green continuous line).2,4 A characteristic feature is the appearance of resonant dips in the conductance (for instance around -0.15 eV). These dips are due to an enhanced density of states associated with quasi-bound states at the impurity, a charge-related analogue to the dips found for neutral defects in ref 2 and, more recently, in ref 21. As it can be seen in Figure 3, considering the appropriate charge state of the scattering center is important and can lead to significant deviations from the neutral transmissions. While a P+ scatterer qualitatively reproduce the trend of the Nano Lett., Vol. 8, No. 9, 2008
Figure 3. Transmission of majority carriers for (a) a B- and (b) a P+ impurity for two supercell lengths/dopant-dopant distances: 8 nm/15 UCs (diamonds) and 14 nm/25 UCs (crosses). The transmissions associated to the neutral B and P impurities are shown for comparison (continuous green line).
Figure 4. Transmission of minority carriers for (a) a P+ and (b) a B- impurity for an average separation between impurities of 8 nm. Left axis refers to the transmission of the pristine wire and the wire with the neutral impurity (continuous line), while right axis refers to the wire with the charged impurity wires (crosses). The ionized impurities produce an effective total backscattering (notice the magnified T(E) scale for the wires with charged dopants), completely absent in the case of the neutral impurity.
neutral impurity (panel b), in the case of B- the transmission can be reduced up to half (panel a). The qualitative picture extracted for majority carriers changes dramatically, when the scattering of minority carriers is considered, shown in Figure 4. The ionized impurities give a total backscattering, especially around the band-edges. For the minority carriers, the dopant constitutes an effective barrier in the potential landscape. When the energy of the electron is less than that of the barrier height, it must tunnel through the potential and the transmission is therefore exponentially suppressed for lower (higher) energies for electrons (holes). This fact has far-reaching consequences for nanodevices that rely on minority carriers transport, such as nanowire solar cells. At the same time, however, having a scattering rate which is 3 orders of magnitude more efficient for minority than for majority carriers can be exploited for reducing the off-current in Schottky barriers, thereby improving the performance of metal-semiconductor junctions. In summary, we have shown that the charge state of doping impurities can lead to a qualitatively different picture of scattering in silicon nanowires. This leads to a computational challenge, because allowing a reasonable screening of the electrostatic potential of a charged impurity leads to supercell sizes which are prohibitive for modeling at the first-principles 2827
level of electron transport. We suggest that a combination of an accurate local DFT potential around the impurity and a long-range FEM treatment of the Coulomb potential is a reasonable and computationally affordable approach. We have discussed the cases of B- and P+ showing the dependence of the conductance on the dopant concentration and the qualitative difference with the scattering of the neutral impurity. The difference is dramatic when dealing with minority carriers, where considering the proper charge state of the dopant can lead to complete backscattering. The suppression of minority carrier conductance can be used in the design of efficient Schottky contacts in nanowire-based field effect transistors. Financial support from Spain’s Ministerio de Educacio´n y Ciencia Ramo´n y Cajal program (R.R.) and Departament d’Universitat, Recerca i Societat de la Informacio´ de la Generalitat de Catalunya (J.S.) and funding under Contract No. TEC2006-13731-C02-01 (R.R. and J.S.) are acknowledged. References (1) Ferna´ndez-Serra, M. V.; Adessi, C.; Blase, X. Phys. ReV. Lett. 2006, 96, 166805. (2) Ferna´ndez-Serra, M.-V.; Adessi, C.; Blase, X. Nano Lett. 2006, 6, 2674. (3) Markussen, T.; Rurali, R.; Brandbyge, M.; Jauho, A.-P. Phys. ReV. B 2006, 74, 245313. (4) Markussen, T.; Rurali, R.; Jauho, A.-P.; Brandbyge, M. Phys. ReV. Lett. 2007, 99, 076803. (5) Latil, S.; Roche, S.; Mayou, D.; Charlier, J.-C. Phys. ReV. Lett. 2004, 92, 256805. (6) Gómez-Navarro, C.; De Pablo, P.; Gómez-Herrero, J.; Biel, B.; GarcíaVidal, F.; Rubio, A.; Flores, F. Nat. Mater. 2005, 4, 534. (7) Adessi, C.; Roche, S.; Blase, X. Phys. ReV. B 2006, 73, 125414. (8) Makov, G.; Payne, M. C. Phys. ReV. B 1995, 51, 4014.
2828
(9) Gerstmann, U.; Dea´k, P.; Rurali, R.; Aradi, B.; Frauenheim, T.; Overhof, H. Physica B 2003, 340-342, 190. (10) Trani, F.; Ninno, D.; Cantele, G.; Iadonisi, G.; Hameeuw, K.; Degoli, E.; Ossicini, S. Phys. ReV. B 2006, 73, 245430. (11) Soler, J. M.; Artacho, E.; Gale, J. D.; Garcı´a, A.; Junquera, J.; Ordejo´n, P.; Sa´nchez-Portal, D. J. Phys.: Condens. Matter 2002, 14, 2745. (12) Rurali, R.; Lorente, N. Phys. ReV. Lett. 2005, 94, 026805. (13) Rurali, R.; Poissier, A.; Lorente, N. Phys. ReV. B 2006, 74, 165324. (14) Cui, Y.; Duan, X.; Hu, J.; Lieber, C. J. Phys. Chem. B 2000, 104, 5213. (15) Cui, Y.; Zhong, Z.; Wang, D.; Wang, W.; Lieber, C. Nano Lett. 2003, 3, 149. (16) Zheng, G.; Lu, W.; Jin, S.; Lieber, C. M. AdV. Mater. 2004, 16, 1890. (17) We use FEMLAB 3.1 package by COMSOL (http://www.comsol. com/). We have placed a unit charge on the wire axis. The boundary conditions in the z-direction are periodic with period Lsc and the potential is set to zero at an outer radius, R . RWIRE. Since for R . Lsc the potential approaches that of an infinite homogeneous line charge distribution: V(r) ) F/2πε0(log R-log r), with the charge density F ) e/Lsc. (18) In the case of the neutral defect the extent of the electrostatic potential is much shorter and convergence can be achieved within DFT (see for instance ref 4), without resorting to the approach discussed here. (19) Outside the central region, the potential is obtained only from FEM and the matrix-elements are Vij ) (Ei + Ej)Sij/2, where Sij is the overlap matrix, and Ei, j are the FEM potential values at the atomic positions ri,j. The total Hamiltonian is then H0 + V. Inside the central region, the potential (for a, say 17-UC calc.) is found as Vij ) ∆Hij + (∆Ei + ∆Ej)Sij/2, where ∆Ei,j is the difference in FEM potential between a 11-UC calculation and a 17-UC calculation, and ∆Hij is the difference between the DFT Hamiltonians for the 11-UC and the pristine wire, ∆H ) H11- H0. (20) The computational cost of adding vacuum in our calculations is moderate, due to the localized nature of the basis functions; thus the convergence is easily attained. This would be limiting factor in the plane-wave formalism where the number of basis functions is determined by the size of the simulation cell. (21) Blase, X.; Fernández-Serra, M.-V. Phys. ReV. Lett. 2008, 100, 046802.
NL801409M
Nano Lett., Vol. 8, No. 9, 2008
