Confinement of electrons in size modulated silicon nanowires
CConfinement of electrons in size modulated silicon nanowires
S. Cahangirov and S. Ciraci
1, 2, ∗ UNAM-Institute of Materials Science and Nanotechnology, Bilkent University, Ankara 06800, Turkey Department of Physics, Bilkent University, Ankara 06800, Turkey (Dated: December 4, 2018)Based on first-principles calculations we showed that superlattices of periodically repeated junc-tions of hydrogen saturated silicon nanowire segments having different lengths and diameters formmultiple quantum well structures. The band gap of the superlattice is modulated in real space asits diameter does and results in a band gap in momentum space which is different from constituentnanowires. Specific electronic states can be confined in either narrow or wide regions of superlattice.The type of the band lineup and hence the offsets of valence and conduction bands depend on theorientation of the superlattice as well as on the diameters of the constituent segments. Effects ofthe SiH vacancy and substitutional impurities on the electronic and magnetic properties have beeninvestigated by carrying out spin-polarized calculations. Substitutional impurities with localizedstates near band edges can make modulation doping possible. Stability of the superlattice structurewas examined by ab initio molecular dynamics calculations at high temperatures.
PACS numbers: 73.63.Nm, 73.22.-f, 75.75.+a
I. INTRODUCTION
Rod-like Si nanowires (SiNW) have been synthesizeddown to ∼ They are attractive one-dimensional (1D) materials because of the well-knownsilicon fabrication technology that make them directly us-able on the Si-based chips. Even if unsaturated danglingbonds on the outer surface usually attribute a metal-lic character to SiNWs, they become insulator (or semi-conductor) upon saturation of these dangling bonds byhydrogen atoms. SiNWs display diversity of electronicproperties depending on their diameter, as well as theirorientation. In particular, the band gap of semiconduc-tor SiNWs varies with their diameters. They can beused in various electronic, spintronic and optical appli-cations, such as field effect transistors , light emittingdiodes , lasers and interconnects. The conductance ofthese semiconductor nanowires can be tuned easily bydoping during the fabrication process or by applying agate voltage. Recent studies have shown that 3 d transi-tion metal doped Si nanowires become half-metallic. This letter demonstrates that SiNWs of different diam-eters can form stable superlattices. The electronic bandstructure of the superlattice is different from the con-stituent SiNWs and is modulated in real space leading toa multiple quantum structure and/or to a series of quan-tum dots. In these size induced quantum wells, specificstates are confined. One dimensional multiple quantumwell structures generated by compositional modulationof nanowires were examined previously. For example,superlattices of GaAs/GaP , InAs/InP and Si/SiGe nanowires were fabricated. Moreover, superlattices of Siand Ge were investigated theoretically. In these struc-tures multiple quantum wells are formed because of thedifferent nature of materials that constitute the nanowiresuperlattice. In the present work however, multiple quan-tum wells are formed because of the quantum size effect,which is a diversification of the electronic structure of the same material with a change in its size.Our results are obtained by performing first-principlesplane wave calculations within Density Functional The-ory (DFT) using ultrasoft pseudopotentials. Theexchange correlation potential has been approximatedby Generalized Gradient Approximation using PW91functional . A plane-wave basis set with kinetic energycutoff of up to 250 eV has been used. All atomic positionsand lattice constants are optimized. The convergence forenergy is chosen as 10 − eV between two steps, and themaximum force allowed on each atom is 0.02 eV/˚A. Moredetails about calculations can be found in Ref[8]. II. SIZE MODULATED SILICON NANOWIRES
Here Si nanowires (i.e. Si N with diameter ∼ D and having N Si atoms in the primitive unit cell) arefirst cut from the ideal bulk crystal along desired direc-tion. Then in every alternating segment comprising l unit cell the diameter D is kept fixed, but in the adja-cent segment comprising l unit cell (each having N Siatoms) the diameter is reduced to D . The latter partcan be identified as the segment of Si N nanowire. At theend, the segments of Si N and Si N have made a smoothjunction and hence formed an ideal superlattice Si N sothat its diameter is modulated in real space. Note that N ≤ l N + l N , because some surface atoms attachingwith a single bond to the surface were removed at the be-ginning. Subsequently we relaxed the atomic structureof this bare Si N . Upon relaxation, the dangling bondson the surface are saturated by H atoms and Si N H M superlattice is further relaxed for final atomic structureand the lattice constant. The resulting superlattice canbe described by a Si rod with alternating diameters or ananowire with alternating wide and narrow parts. Herewe consider two superlattice structures, namely Si H and Si H , which are grown in [111] and [100] direc- a r X i v : . [ c ond - m a t . m e s - h a ll ] J u l FIG. 1: Energy band structure of the superlattice, Si H ,formed from periodically repeated junctions consisting of oneunit cell of SiNW with D ∼
11 ˚A and two unit cell of SiNWwith D ∼ D is defined as the largest distance between two Si atomsin the same cross-sectional plane. Isosurface charge densi-ties of specific bands and their planarly averaged distributionalong the superlattice axis together with their confinement inpercentages are also shown. Large and small balls indicate Siand H atoms. Zero of energy is set at the Fermi level shownby dash-dotted line. tions, respectively. For the latter we investigate the sur-face defect and also boron, B and phosphorus, P substi-tutional impurities. A. Si H Superlattice
To form the superlattice, Si H , we took l = 1unitcell of Si and l = 2 unitcell of Si cut in [111] di-rection of the bulk silicon crystal. The combined struc-ture had six Si atoms attached to the surface by onlyone bond. These atoms were removed at the beginningand the structure was optimized. After the relaxation,all Si atoms were at least triply coordinated. The dan-gling bonds of triply coordinated Si atoms were satu-rated with hydrogen atoms and the structure was relaxedagain. The band structure and isosurface charge densitiesof the resulting superlattice is shown in Fig. 1. Propa- FIG. 2: Band structures of infinite and periodic Si H ,Si H and Si H nanowires, which constitute Si H structure. The front and side view of these relevant structuresare shown on the top of the band structure plots. Numeralsas subscripts of Si and H indicate the number of Si and Hatoms in the unit cell. Energy band gaps are shaded. gating states, with dispersive bands have charge densitiesdistributed everywhere in the superlattice rod. For ex-ample, a state near ∼ -1 eV is a propagating state. Thestates of flat degenerate band at the top of the valenceband are confined in the narrow segment. The integralof the planarly averaged charge, (cid:82) l | Ψ( z ) | dz = 0 .
96 in-dicates a rather strong confinement. The states of thelowest conduction band with low dispersion are localizedin the wide segment with relatively weaker confinement.This situation implies the staggered band alignment withconfined hole states in the narrow part and confined elec-trons in the wide part. Confinement may increase withthe size (length) of the confining segment since the en-ergy of the state relative to the barrier varies. For perfectlocalization, i.e. ∼
100 % confinement in l or l , wherecoupling between confined states are hindered, the seg-ment l or l behaves as a quantum dot. Confinementand band edge alignment in the present superlattice arereminiscent of the 2D pseudomorphic or commensuratesemiconductor superlattices. B. Pristine Si H Superlattice and Vacancy
Here we presents the superlattice formation and con-fined states in a different structure, namely Si H . Toform this structure we took l = 2 unitcell of Si and l = 3 unitcell of Si cut in [100] direction of the bulk FIG. 3: (a) Electronic structure and isosurface charge densities of selected states for pristine Si H and defected Si H (with one Si-H on the surface removed) nanowire superlattices. Arrows pointing at the same isosurface charge density plotindicates that those states are not affected by the formation of the defect and have nearly the same charge density profile. Fermilevel, represented by dashed-dotted (red) line, was shifted to the valence band edge and set to zero. Solid (blue) and dashed(green) lines in the right hand side box represent the majority and minority spin bands, respectively. (b) The distribution ofinteratomic distances of relaxed superlattice. The arrows indicate the ideal bulk 1st, 2nd and 3rd nearest neighbour distances. silicon crystal. The combined structure had eight sili-con atoms that were attached to the surface by only onebond. We have removed these atoms at the beginningand relaxed the resulting structure. Then we have pas-sivated the dangling bonds of the final structure and re-laxed the whole structure again. As a result, the wide part of the nanowire superlattice had l = 2 unitcellof Si H with D ∼
14 ˚ A and the narrow part had l = 3 unitcell of Si H and Si H mixture with D ∼ A . Figure 2 presents the band structure of the cor-responding infinite periodic nanowires which constitutethe Si H superlattice structure. To compare with the FIG. 4: (a) Energy per unitcell of Si H superlattice as itstemperature is increased from 500 K to 1400 K (green/light)and kept constant at 1300 K (red/dark). Trajectories of atomsat the surface of (b) wide and (c) narrow parts and at thecenter of (d) wide and (e) narrow parts of the superlattice,projected on a plane perpendicular to the wire axis, as thesystem is kept at 1300 K. band structure of the superlattice (presented in Fig. 3as pristine), the band structure of these nanowires werefolded l + l = 5 times. One can see that the band gapof the superlattice is close to the lowest band gap of itsconstituents. This is expected for the superlattice struc-tures having normal band lineup. The band gaps of struc-tures shown in Fig. 2 do not obey the well-known trend, E G = C/D α + E G [ bulkSi ], occurring due to the quantumconfinement effect at small D . We ascribe this resultto the surface reconstructions which differ in accordancewith the procedure of atomic relaxation. Quantum con-finement trend occurs when Si nanowires, cut from idealbulk, are directly passivated with H atoms and subse-quently relaxed. In the present study, however, the idealstructures are first relaxed, then passivated with H atomsand then relaxed again to obtain the final structure. Such Si nanowire structures with D ∼ /D α behaviour has been also reportedfor other types of SiNWs and was attributed to surfaceeffects. Figure 3(a) presents the band structure and the pro-jected charge density isosurface plots of pristine Si H and defected Si H nanowire superlattices. Flat mini-bands of Si H structure near the edge of conduc-tion and valence bands are distinguished. In contrastto propagating states of dispersive bands near 1.5 eV,the states of these flat mini bands are confined in thenarrow segments of the superlattice. This situation alsoimplies a normal band lineup. We are aware of the factthat fabrication of nanowire superlattices having ideallytapered as in our case is impossible. So we examined theeffect of an imperfection on the behavior of the systemby removal of a Si-H atom pair, which was normally at-tached to the surface. The location of this defect andstructural change after relaxation is given in Figure 3(a).The resulting structure has odd number of electrons, sowe made spin polarized calculation. One can see thatthe valence and conduction band edges are not affectedby defect formation, while one empty and one filled statelocalized around the defected area appear near the va-lence band edge. The magnetic moment induced by thedefect formation is 1 µ B . The vacancy states in bulk Siare located deep in the band gap. In the superlatticestructure, this vacancy state is located near the valenceband, since the defect is formed on the surface at the in-terface between two segments having different diameters.In Figure 3(b) the distribution of interatomic distancesshows significant reconstruction and deformation. In par-ticular, owing to increased surface/volume ratio the de-viation of the second and third nearest neighbour dis-tances from the bulk crystal is rather large at the nar-row and interface regions. We note that in both shortperiodicity superlattices discussed above the electronicstructures are significantly different from the constituentSiNWs. As l and/or l increases, the superlattice effectsand confined states may become more pronounced, andband-offset converges to a well-defined value. When thelengths of l and l > λ B (Broglie wavelength), the elec-tronic properties in each segment becomes close to thosein constituent nanowires. In addition to l and l , D and D and also the direction of growth are also criti-cal parameters, since the electronic structure of the con-stituent SiNWs is strongly dependent on these parame-ters. As D and D increases, the surface effects decreaseand the density of states becomes more bulk like. Thediscontinuities of valence and conduction bands of the su-perlattice is expected to diminish, since the band gap ofthe constituent nanowires converge to the bulk Si value.However for a significant ∆ D = D − D the superlatticeeffects, in particular confinement of specific states are ex-pected to continue. In contrast, smooth interfaces lead-ing to hornlike profiles can be used for focused electronemission. Recently, transport properties of size modu-lated SiNWs have been shown to depend on the growthdirection. The stability of the Si H structure was analyzed us-ing the finite temperature ab initio molecular dynamicscalculations with time steps of 2 × − seconds. Fig- FIG. 5: Atomic and electronic structure and charge density isosurfaces of Si H substitutional doped by B and P atoms(namely Si P(B)H ). Top panel presents a view of the superlattice structure, where the doped site is represented bylarge red(dark) ball. Arrows between the bands of the undoped (pristine) superlattice and the bands of the B and P dopedsuperlattices indicate the states, which preserved their character even after doping. The charge density isosurfaces of specificstates are linked to the corresponding bands by arrows. Fermi level, represented by dashed-dotted (red) line, was shifted tothe valence band edge and set to zero. Solid (blue) and dashed (green) lines represent the majority and minority spin states,respectively. Small panel at the left hand side presents the charge density isosurface plot of B-Si bonds. ure 4(a) presents the total energy difference of the systemwith respect to the ground state energy at 0 K as a func-tion of time. In the first calculation we have raised thetemperature of the system from 500 K to 1400 K in 2 ps.During this calculation there was no major change in thestructure and the interface region was not destroyed bydiffusion. In the second calculation we took atomic posi-tions and velocities of the system from the first calcula-tion when the system was at 1300 K and kept the tem-perature constant for 9 ps. This calculation also resultedin no major change in the structure of the superlattice. Figure 4(b) and (c) present the path of Si atoms duringthe second calculation at the surfaces of wide and narrowparts respectively. The range of the thermal fluctuationsare nearly the same and span a diameter of about 1 ˚ A .Figure 4(d) and (e) present the path of Si atoms at thecenter of the wide and narrow parts respectively. Asseen the range of thermal fluctuations in the narrow part( ∼ . A ) is larger than that in the wide part ( ∼ . A ).It should be noted that, the temporal trajectories of 9picoseconds in the present ab initio molecular dynamicscalculations are rather long as compared to ones usuallycarried out in the stability tests, but may not be longenough to accumulate necessary statistics. However, thetemporal trajectories at temperature as high as 1300 Kin the present study are long enough to eliminate the ex-istence of any atomic configuration in a weak local min-imum of the Born-Oppenheimer surface, which is proneto structural instability. Therefore, based on the accu-rate structure optimization and temporal trajectories aslong as 9 picoseconds carried out at 1300 K, we believethat superlattice structure under study is stable at leastat a temperature higher than the usual operation tem-perature ( ∼
425 K) of a device.
C. B and P doping of Si H Superlattice
Next we examine the effect of doping of the Si H superlattice by substitutional B and P impurities. Bothstructures have odd number of electrons and hence spinpolarized calculations have been carried out. Neverthe-less, B doped structure resulted in a nonmagnetic statewith half filled band edge. P doped structure resultedin a magnetic state with a magnetic moment of 1 µ B . InFig. 5 the electronic structure of pristine and doped su-perlattices are shown to follow the effect of the dopingon the states at the band edges. Owing to the small di-ameter of the superlattice, the band structure is affectedby the impurity. Since the band gap of the wide partis larger than that of the narrow part, the state asso-ciated with the impurity, which occurs normally eithernear the valence or conduction band, falls in the bandsof the narrow part.It is known that substitutional B doping of bulk siliconcrystal results in an acceptor state ∼
45 meV above thevalence band edge. In the substitutional doping of thesuperlattice by B similar situation occurs and the accep-tor state is located above the valence band edge corre-sponding to the wide part, but falls in the valence bandof the narrow part. The filled first and the lower lyingthree valence bands at about ∼ -0.4 eV undergo changes:The filled first valance band having states confined inthe narrow part becomes half-filled. Three lower lyingvalence bands of the pristine superlattice are raised to-wards the edge of the valence band edge. The statesof these bands mix with the tetrahedrally coordinated sp hybridized orbitals of B and form a triply degener-ate state, just below the half filled valence band edge.Charge density isosurfaces of these states around boronatom presented in Fig. 5 clearly indicates sp hybridiza-tion in tetrahedral directions. As a result, the ”hole likestates” becomes confined in the narrow part. As for theconduction band is slightly shifted down, but its statespreserve their character.In P doping of the superlattice at the center of thewide part, the impurity state localized at P atom occursin the conduction band of the superlattice. This state,in fact, the donor state occurring ∼
200 meV below theconduction band corresponding to the wide part. This energy is in good agreement with the energy calculatedrecently for the ionization energy of P donor states inthe Si nanowires. Below this donor state, all conduc-tion band states are confined either in the narrow part orat the interface, but not in the wide part. Energy shiftsof the band edge states are more pronounced in structurehaving substitutional P doping. Conduction band edge isshifted down by 0.75 eV, but the charge density profile re-mains the same. One spin-up and one spin-down bandssplit by ∼ III. CONCLUSION
In conclusion, we considered two specific superlatticesgrown in different crystallographic directions as a proof ofconcept. Each superlattice can be viewed to form the pe-riodically repeating junctions of two hydrogen saturatedsilicon nanowires of different diameters. Since the bandgaps of the constituent Si nanowires are different, theband gaps of these superlattices are modulated in realspace. As a result, one dimensional multiple quantumwell structures with states confined in different (wide ornarrow) parts are formed. At strong confinement or lo-calization, the superlattice structures can be viewed as aseries of quantum dots. We believe that the bad gapmodulation and resulting quantum well structure andconfinement can occur as long as the band gap differencesare maintained between adjacent region at both sides ofthe junction. Furthermore, we showed that the electronicstructure resulting from the substitutional doping of thesuperlattice interesting and give rise to modulation dop-ing. We finally note that, the band gaps of SiNWs atboth sides of junction forming the superlattice structureare underestimated by DFT/GGA calculations. Sincethe systems under study are too large, these bands can-not be improved by GW calculations. Nevertheless, ourwork is just a proof of concept. According to our results,if the band gaps are different in different parts of the het-erostructure, confined states can occur and modulationdoping of these quantum structures can be achieved. De-tails of energy band structure and location of impuritystates in the gap should only be taken qualitatively.Various types of electronic devices, such as resonanttunnelling double barriers generated from these super-lattices, can be arranged on a single rod, where they canbe connected by the metallic segments of unsaturated Sinanowires. The character of these devices can be engi-neered by varying their growth direction and structuralparameters, such as l , l , D , D . Even if the diametersof superlattices we treated in this study are small ( ∼ IV. ACKNOWLEDGEMENT
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