Electron Transport in SiGe Alloy Nanowires in the Ballistic Regime

Letter pubs.acs.org/NanoLett

Electron Transport in SiGe Alloy Nanowires in the Ballistic Regime from First-Principles Michele Amato,†,‡,§,⊥ Stefano Ossicini,‡,† and Riccardo Rurali*,§ †

Dipartimento di Scienze e Metodi dell’Ingegneria, Centro Interdipartimentale En&Tech, Universitá di Modena e Reggio Emilia, Via Amendola 2 Pad. Morselli, I-42100 Reggio Emilia, Italy ‡ ”Centro S3”, CNR-Istituto di Nanoscienze, Via Campi 213/A, 41125 Modena, Italy § Institut de Ciència de Materials de Barcelona (CSIC), Campus de Bellaterra, 08193 Bellaterra, Barcelona, Spain S Supporting Information *

ABSTRACT: Silicon−germanium alloying is emerging as one of the most promising strategies to engineer heat transport at the nanoscale. Here, we perform first-principles electron transport calculations to assess at what extent such approach can be followed without worsening the electrical conduction properties of the system, providing then a path toward highefficiency thermoelectric materials.

KEYWORDS: SiGe nanowires, electron transport, DFT, thermoelectrics

T

conductance is not affected because the electronic structure does not change, while phonons transmission can suffer additional scattering, leading to a drop in σT. The use of different isotopes is not practical, however, and similar schemes have been explored in SiGe systems,15 where a significant difference in the mass offers the possibility of engineering phonon scattering, while the similar electronic structure allows expecting a limited degradation of the electrical conductivity. In this Letter, we study from first-principles the electrical conductance of Si1−xGex alloy nanowires in the ballistic regime for increasing values of the Ge concentration x. Our calculations are based on density-functional theory (DFT) as implemented in Siesta.16,17 Core electrons are accounted for by means of norm-conserving pseudopotentials and the oneelectron wave function of valence electrons is described by a linear combination of single-ζ polarized basis functions. The exchange-correlation energy is calculated within the generalized gradient approximation (GGA).18 We consider a 1 × 1 × 9 supercell of the primitive cell of a ⟨110⟩ NW and sample the Brillouin zone with 2 k-points along the wire periodic direction. Previous works have shown that a supercell of this size yields converged values of the formation energy of the single impurity19 and the conductance of neutral defects.20 We have optimized the axial lattice parameter of a pristine Si and Ge NW, adjusting it according to Vegard’s rule21 in the alloyed sections.

he possibility of achieving large thermoelectric efficiencies in one-dimensional conductors was first predicted in 1993 by Hicks and Dresselhaus.1 Recently, reports of a figure of merit ZT approaching 1 in Si nanowires (NWs)2,3 boosted a resurgent interest in this field, especially in view of the control of the structural and electronic properties of these systems that can be now achieved.4 Since these works, many theoretical studies attempted to either explain their experimental results or propose novel approaches to further increase those efficiencies. The thermoelectric efficiency of a semiconductor (where the electron contribution to thermal transport can be neglected) is given by the dimensionless figure of merit σ ZT = S2T e σT (1) where S is the Seebeck coefficient, T is the temperature, and σe and σT are electrical conductivity and thermal conductivity, respectively. Hence, large values of ZT are provided by materials that are good electrical conductors (high σe) and poor thermal conductors (low σT). Unfortunately, these quantities are normally correlated and it is difficult to increase one and decrease, or even leaving unaffected, the other at the same time. All the recent theoretical proposals to increase ZT of nanowires focused on the decrease of the thermal conductivity, either through nanostructuring,5,6 alloying,7 surface/interface scattering,8,9 or forming ordered superlattices,10,11 and neglect the effects on the electrical conductivity (a noticeable exception is the work of Shelley and Mostofi on SiGe superlattices12). Such an approach is rigorous only in the idealized case of isotope disorder13 or isotope superlattices,14 where different isotopes of the constituent elements are mixed on purpose and in a controlled way. In these systems, the ballistic electrical © 2012 American Chemical Society

Received: December 7, 2011 Revised: April 20, 2012 Published: April 30, 2012 2717

dx.doi.org/10.1021/nl204313v | Nano Lett. 2012, 12, 2717−2721

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Figure 1. (a) Transmission of a Si NW with a single Ge impurity substituting at a Si lattice site (red circles), at an interstitial site in the core of the wire (blue squares) and at an interstitial site close to the surface (green diamond). All the substitutional defects considered give a very similar conductance, thus one single curve is plotted [defect 1, see panel b]. To resolve the difference between the lowest and highest case among the eight substitutional configurations, a magnified view is shown in the inset. The conductance of the pristine Si NW is shown by the continuous black line. (b) Cross-section view of the Si NW where the substitutional and interstitial Ge defects considered are shown in yellow and light orange spheres (blue and white spheres representing Si and H atoms, respectively). (c) Total density of states (continuous black line) and projected density of states of a Ge substitutional in position 1 (dashed red line). μ is the Fermi level and it will be determined by the doping conditions.

impurity is in the core region of the NW (defect a, blue squares). This is mostly a configurational effect and is not related to the chemical nature of the impurity; interstitial Si produces qualitatively similar scattering. However, the formation energy23 of these defects is 3.2−3.7 eV larger than that of substitutionals, thus the interstitial/substitutional ratio

To estimate the conductance, we have partitioned the system into three regions, as it is common in this kind of calculations:22 a left lead, a right lead, and a central scattering region containing the Ge impurities. We have used the nonequilibrium Green’s function formalism, where the open boundary conditions imposed by the electrodes are accounted for through the left (right) self-energy Σ̂L,R(E). The zero-bias transmission T(E) is calculated as r

a

T (E) = Tr[GĈ (E)ΓL̂ (E)GĈ (E)ΓR̂ (E)]

s i N int ≃ e Ef − Ef / kT sub N

(2)

Esf

where is the formation energy of the substitutional (interstitial) defect, k is the Boltzmann constant, and T is temperature, tends to be zero. Hence, virtually no interstitial defect is present in the system.24 Scattering is larger in the valence band, although at this scale is difficult to see for substitutional defects, because most of the weight of the Ge states falls in the Si NW valence band edge region, as can be seen in the projected density of states shown in Figure 1c. In other words, the potential of the pristine wire is less affected by the presence of a Ge impurity at energies immediately above the bottom of the conduction band than at energies immediately below the top of the valence band. This represents a general property of SiGe systems, which has been found in superlattices25,26 as well as different types of SiGe NWs.27−29 However, some care should be taken before generalizing these observations. We discuss this point in depth in the Supporting Information where we show cases where scattering is stronger in the conduction band. From the transmission plots of Figure 1 the current can be calculated as

̂ r,a ̂ r,a ̂ ̂ where Ĝ r,a C (E) = [ESC − HC − ΣC (E) − ΣR (R)] is the retarded (advanced) Green’s function of the scattering region and Γ̂L,R(E) = i(Σ̂rL,R − Σ̂aL,R). The conductance G(E) is then calculated through the Landauer formula as G (E ) =

2e 2 T (E ) h

(4)

(Eif)

(3)

This expression accounts for all mechanism of carrier scattering in the elastic regime. Inelastic scattering, negligible in the ballistic regime, is not described. We begin our analysis with the study of the single Ge impurity. We have considered eight substitutional and two interstitial defects (see Figure 1 for configurations and labeling). We have found that the presence of substitutional defects has a minimal influence on the electron transport of the pristine wire, whose transmission channels close to the band edges are almost entirely preserved. The conductances associated to the investigated substitutional defects are almost indistinguishable, thus we plot only the case of the substitutional defect labeled as 1 in Figure 1b. The difference between the lowest and highest transmitting case among the eight substitutional configurations can be appreciated in the inset, where a magnified view of the first conduction channel of the conduction band is shown. On the other hand, interstitial defects yield a large suppression of the conductance, especially when the Ge

I=

2e 2 h

+∞

∫−∞

⎛ ∂f (ε , 0) ⎞ T (ε − μ)⎜ − ⎟d ε ⎝ ∂ε ⎠

(5)

where f is the Fermi distribution function. μ is the Fermi energy which will be determined by the doping of the system.30 These results suggest that the assumption that Ge impurities do not worsen the conductive properties of Si NW seems 2718


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reasonable because (i) only interstitial defects act as efficient scattering centers, but their concentration is expected to be negligible; and (ii) substitutional defects are easily incorporated in the Si lattice (Esf ∼ 0) and the transport channels of the pristine wire are only marginally affected. Yet, in SiGe alloy NWs Ge concentrations of up to 70% can be reached31 and the study of Si1−xGex NWs has to be addressed directly. Additionally, it has been shown that in the limit of diluted doping the scattering resistances add classically according to Ohm’s law32 and the validity of this result should be checked for lightly to heavily alloyed NWs. Therefore, our next step consisted in the study of Si NWs with a Si1−xGex central scattering region, with x ranging from 0.1 to 0.7. Because of the intrinsic stochastic nature of the alloys here considered, we have generated six random configurations for each value of x and below we discuss only averaged properties. The results are shown in Figure 2. Again, the

Figure 3. Comparison of the transmission of the following three cases: (i) n impurities concentrated in a small region (symbols), (ii) full Ge substitution in the same scattering region (blue continuous line), and (iii) n impurities distributed over a much longer portion of NW so that scattering events can be considered independent from each other and resistances add up classically following eq 6 (dashed line). Left, Si0.9Ge0.1; right, Si0.3Ge0.7. The conductance of the pristine Si NW is shown by the continuous black line. Top panel: schematic sketch of the three situations in the case of n = 3.

substitutionals, but well-spaced along its length, estimated feeding eq 6 with the single-impurity results; (ii) full Ge substitution, that is, a segment of pure Ge NW with abrupt interfaces is inserted into the Si NW. These three situations are summarized in the sketch of Figure 3a, while Figure 3b displays the results of the lowest and highest Ge concentration considered, Si0.9Ge0.1 and Si0.3Ge0.7. The calculated conductances show that in both cases the concentration turns out to be sufficiently high so that the alloy NW behaves in practice as an abrupt Ge NW inclusion. This is an important result because abrupt junctions, if needed, are difficult to obtain avoiding interface diffusivity,35,36 while (random) concentration gradients can be obtained in a easier way. At low Ge concentrations, the conductance can be still roughly approximated by means of eq 6 (Figure 3b, left panel), although underestimating it. As the Ge content increases, nonetheless, a description in terms of single impurity becomes more and more unsuitable to predict the conductance of the alloy NW. What is unexpected is that distributing the impurities along a longer wire section yields a larger scattering. Or, which is the same, lower concentrations results in a lower conductance. What happens can be summarized as follows. In the limit of very low Ge concentration, the conductance is high, as it results from a limited number of individual scattering events. As the concentration increases, the conductance starts decreasing and reaches a minimum value; the resistance is made up by many individual scattering occurrences that add up classically according to eq 6. Increasing further the Ge concentration leads toward another low resistance state, the abrupt Si−Ge junction where scattering occurs mostly at the interface, thus the conductance increases again. These results have obvious and important consequences for the use of SiGe NWs in thermoelectric applications: if alloying is an efficient path toward the reduction of the thermal

Figure 2. Transmission of a Si/Si1−xGex/Si NW with x = 0.1, 0.3, 0.5, and 0.7. The conductance of the pristine Si NW is shown by the continuous black line. A side view of the NW in the case of a Si0.5Ge0.5 scattering region is also shown. The inset shows a magnified view of the first transmission channel of the conduction band.

conductance in the conduction band is extremely well-behaved with minimal scattering and almost insensitive to the concentration of Ge (see the magnified view in the inset of Figure 2). At the valence band edge this happens only for the lower concentrations considered, x = 0.1 and 0.3, while nonnegligible scattering appears at x = 0.5 and, especially, x = 0.7. To get a better insight on this unusually low scattering we compare the resistance of the alloy NW and that of a NW in the limit of diluted doping.33 Markussen et al.32 showed that in this regime the resistance of a wire with N impurities is given by ⟨R(E)⟩ = R c(E) + ⟨R s(E)⟩N

(6)

⟨Rs(E)⟩ is the average scattering resistance of the different substitutional configurations which can be estimated from the single impurity conductances derived from Figure 1, G(E), as ⟨Rs(E)⟩ = 1/⟨G(E)⟩ − 1/G0(E), where Rc(E) = 1/G0(E) is the contact resistance of the pristine wire. Equation 6 holds if the impurities are sufficiently spaced to allow considering the total resistance as the result of a series of individual scattering events.34 This condition is obviously not satisfied by the Si1−xGex NW, thus it is interesting to compare these two cases. In Figure 3, we plot the conductance of the alloy NW, obtained from a direct calculation, with that of two limiting cases: (i) a wire with the same number of Ge 2719


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for thermoelectrics applications where disorder is one of the most used approach to suppress phonon transmission. In summary, we presented first-principles calculations of the electron transport of SiGe nanowires in the ballistic regime. Our most important result consisted in showing that for typical Ge concentrations alloyed SiGe NWs scatter carriers like abrupt inclusion of an all-Ge NW segment. Hence, the exact value of the Ge content has no direct effect on the conductance of the system and can be chosen to maximize heat transport suppression in thermoelectrics applications. It would be tempting to conclude that n-type devices, where the current is carried by electrons, should be preferred over p-type devices where the charge carriers are holes, as previous reports based on the Boltzmann transport equation suggested.37 However, our results also indicate that the length of the alloyed segment plays an important role, thus providing an additional degree of freedom for the optimization of these systems. This situation, whose physical grounds have been elucidated by Akman et al.,38 is briefly discussed in the Supporting Information. We also show that scattering is much lower than if the same number of Ge impurities would be distributed along a much longer length with an average spacing between impurities that allows the consideration of the the overall resistance as the sum of individual scattering events. These results are expected to be of interest for applications in the field of thermoelectricity, as well as for a general understanding of the physics of electron transport in nanosystems.

conductivity, the concentrations that minimize thermal transport can be pursued, as the electrical conductivity is almost insensitive to the exact value of the Ge content. Notice that here we have considered a rather small scattering region to illustrate the main mechanisms, while keeping the computational load at a manageable level. For a final evaluation of the performances of a SiGe alloyed NW for thermoelectric applications, the specific geometry should be taken into account. In the Supporting Information, we discuss the dependence of the conductance on the length of the alloyed region, while a discussion of the case of multiple heterostructures can be found in ref 12. For the sake of generality, we conclude our analysis considering an all-SiGe NW. Now the semi-infinite electrodes are made of Si0.7Ge0.3 NW and the scattering region of a Si1−xGex NW inclusion with x = 0.1, 0.3, 0.5, and 0.7 as in the previous case. We take as a reference system an all Si0.7Ge0.3 NW with the same distribution of Ge impurities in the scattering region and in the electrode. In this perfectly periodic NW, the incoming transport eigenstates propagates without scattering and the conductance is strictly quantized. It is noteworthy that the electron conductance is roughly the same as the one of the pristine Si NW, while the hole conductance is rather different, which is a further indication of the valence nature of Ge states. Also here, for each value of x we generate six different random configurations and average the transmission curves. The conclusions already discussed qualitatively still hold: the conductance is well preserved in the conduction band with no dependence around the band edge on the Ge content; some scattering appears in the valence band when the Ge concentration exceed 0.5% (see Figure 4). The curve

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ASSOCIATED CONTENT

S Supporting Information *

Additional information and figures. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Present Address ⊥

Laboratoire de Physiques des Solides (LPS), Université Paris Sud, CNRS UMR 8502, F-91405, Orsay, France

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS M.A. greatly acknowledges the Transnational Access Programme of the HPC-EUROPA2 Project and the computer resources, technical expertise and assistance provided by the Red Españ ola de Supercomputación (RES), Centro de Supercomputación de Galicia (CESGA) and the CINECA award under the ISCRA initiative (No. HP10BQNB3U) for the availability of high-performance computing resources and support. S.O. and M.A. acknowledge the European Community’s Seventh Framework Programme (FP7/2007-2013) under Grant 245977, MIUR-PRIN 2007, Ministero Affari Esteri, Direzione Generale per la Promozione and Cooperazione Culturale and Fondazione Cassa di Risparmio di Modena. Funding under Contract Nos. TEC2009-06986, FIS200912721- C04-03, and CSD2007-00041 are greatly acknowledged. We thank Mads Barndbyge and Nicolás Lorente for useful discussions.

Figure 4. Transmission of a Si0.7Ge0.3/Si1−xGex/Si0.7Ge0.3 NW with x = 0.1, 0.3, 0.5, and 0.7. The conductance of the perfectly periodic Si0.7Ge0.3 NW is shown by the continuous black line. The inset shows a magnified view of the first transmission channel of the conduction band.

corresponding to x = 0.3 is especially interesting, because the Ge concentration does not change moving from the semiinfinite electrodes to the scattering region, and the resulting scattering is entirely due to the disorder. The possibility of isolating disorder-induced scattering, assessing the negligible degradation of the conductance, has a potentially high impact 2720


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(36) Perea, D. E.; Li, N.; Dickerson, R. M.; Misra, A.; Picraux, S. T. Nano Lett. 2011, 11, 3117−3122. (37) Shi, L.; Yao, D.; Zhang, G.; Li, B. Appl. Phys. Lett. 2010, 96, 173108. (38) Akman, N.; Durgun, E.; Cahangirov, S.; Ciraci, S. Phys. Rev. B 2007, 76, 245427.

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Electron Transport in SiGe Alloy Nanowires in the Ballistic Regime

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