Electronic phases, band gaps, and bandoverlaps of bismuth antimony nanowires

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Citation Tang, Shuang, and Mildred S. Dresselhaus. “Electronic Phases,Band Gaps, and Band Overlaps of Bismuth Antimony Nanowires.”Phys. Rev. B 89, no. 4 (January 2014). © 2014 American PhysicalSociety

As Published http://dx.doi.org/10.1103/PhysRevB.89.045424

Publisher American Physical Society

Version Final published version

Citable link http://hdl.handle.net/1721.1/88948

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PHYSICAL REVIEW B 89, 045424 (2014)

Electronic phases, band gaps, and band overlaps of bismuth antimony nanowires

Shuang Tang1,* and Mildred S. Dresselhaus2,†1Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4037, USA

2Department of Electrical Engineering and Computer Science and Department of Physics, Massachusetts Institute of Technology,Cambridge, Massachusetts 02139-4037, USA

(Received 18 November 2013; revised manuscript received 1 January 2014; published 24 January 2014)

We have developed an iterative one-dimensional model to study the narrow band gap and the associatednonparabolic dispersion relations for bismuth antimony nanowires. An analytical approximation has also beendeveloped. Based on the general model we have developed, we have calculated and analyzed the electronic phasediagrams and the band-gap/band-overlap map for bismuth antimony nanowires, as a function of stoichiometry,growth orientation, and wire width.

DOI: 10.1103/PhysRevB.89.045424 PACS number(s): 73.63.Nm, 81.30.Dz, 73.22.Gk

I. INTRODUCTION

The bismuth antimony (Bi1−xSbx) alloy materials areconsidered to be among the best materials for low-temperaturethermoelectrics, supercooling, millivolt electronics, and in-frared applications. A notable number of interesting propertieshave been observed in bulk bismuth materials, such as non-parabolic dispersions and abnormal magnetoresistance [1,2],ultrahigh mobility of carriers [3], and high anisotropy [4,5].In 1993, Hicks et al. suggested that thermoelectric materialscould have enhanced figure of merit if the materials weresynthesized in the form of low-dimensional systems andnanosystems [6,7]. Since then, much more focus has beengiven to bismuth antimony as related to nanoscience andnanotechnology. Lin et al. have synthesized and studied someelectronic properties of Bi1−xSbx nanowires of diametersd = 40, 45, and 60 nm and Sb composition x = 0, 0.05,0.10, and 0.15, which are oriented along the (012) crys-talline axis [8,9]. Bi1−xSbx nanowires with their wire axisoriented in the trigonal direction have been studied by Rabinet al. [10], for the composition range of x = 0–0.30 and thenanowire diameter range d=0–100 nm, below the temperatureof 77 K, where the authors suggested that thermoelectricperformance can be optimized by aligning the carrier pockets.Bi1−xSbx wires of larger diameters on the order of micronshave been studied for their strain effect in the wire directionon their electrical resistivity by Nikolaeva et al. [11–13].Tang and Dresselhaus have given systematic guidance onthe electronic band structure of Bi1−xSbx thin films as afunction of growth orientation, film thickness, stoichiometry,and temperature [14–16]. However, there are no correspondingsystematic studies of Bi1−xSbx nanowires, and researchers,especially experimentalists, are very eager for global guidanceon Bi1−xSbx nanowires as a function of growth orientation,wire diameter, stoichiometry, temperature, etc. The presentpaper aims to provide such guidance on the electronic phasesand band gaps or band overlaps of Bi1−xSbx nanowiresto stimulate the synthesis of such nanowires for differentapplications.

*tangs@mit.edu†millie@mgm.mit.edu

In the current paper, we first develop a model for the mini–band gap and the related nonparabolic dispersion relations atthe L point of bismuth antimony in one dimension. In par-ticular, we use an iterative one-dimensional two-band model,and we have here developed an analytical approximation forthis model. Thereafter, we study the band edges and electronicphases as a function of growth orientation, wire diameter, andstoichiometry, including the semimetal phases, the indirectsemiconductor phases, and the direct semiconductor phases.The band overlap of the semimetal phases, and the band gap ofthe semiconductor phases are then studied as well. The aim ofthis paper is to (1) develop a one-dimensional nonparabolicband model, and (2) provide a guide for the synthesis ofBi1−xSbx nanowires with various crystallographic orientationsthat could be used for different applications.

We first review the crystal and electronic band structure ofbulk materials of Bi1−xSbx . The bulk materials of bismuth,antimony, and their alloys have the same R3̄m symmetry witha rhombohedral lattice structure. There are two atoms per unitcell, as shown in Fig. 1(a). The trigonal axis with a threefoldsymmetry, the binary axis with a twofold symmetry, and thebisectrix axis form a natural Cartesian coordinate frame asshown in Fig. 1(a). A mirror plane is formed by the trigonalaxis and the bisectrix axis, which is perpendicular to the binaryaxis. In the first Brillouin zone, the band edges are located atthe T point, the three L points, and the six H points [17,18],as shown in Fig. 1(b). In the range of x = 0–0.30, the bottomof the conduction band is always located at the L points, whilethe top of the valence band can be located at the T point, the L

points, or the H points, depending on the Sb concentration. Therelation between Sb composition x = 0–0.30 and the relativepositions in energy of the band edges at different locationsin the first Brillouin zone changes with temperature, pressure,external magnetic field, strain, etc. Figure 1(c) shows the casewhere the temperature is below 77 K, and the pressure isat 1 atm with no external magnetic field [19]. Other caseshave similar relations, but different values for the critical Sbconcentration for the various phase transitions [3,17,20,21].The conditions of the sample and measurements using differentexperimental techniques may also result in different values ofx at certain phase transition points [22,23]. In this paper, wewill just concentrate on the case of Fig. 1(c), where the bandstructure does not change notably with temperature. Othercases can be studied in a similar way.

1098-0121/2014/89(4)/045424(7) 045424-1 ©2014 American Physical Society

SHUANG TANG AND MILDRED S. DRESSELHAUS PHYSICAL REVIEW B 89, 045424 (2014)

Mirror Planeror PlaneMirrrr o

BisectrixBinary

Trigonal

Mirror Plane

(a)

(b)

H(6)

(1)L(3)L

(2)L

BinaryBisectrix

TrigonalTTH(1)

H(3)

H(5)

H(4)

H(2)

0 0.05 0.1 0.15 0.2 0.25 0.3−120−100−80−60−40−20

0204060

Ener

gy (m

eV)

Antimony Composition x

(c)

FIG. 1. (Color online) Atomic and band structure of bulk bismuthantimony. (a) The atomic unit cell of bismuth antimony. The twoatoms in each unit cell are marked by green and red. The trigonal,binary and bisectrix axes form a natural three-dimensional Cartesiancoordinates framework. (b) The different carrier pockets at the T

point for holes, the three L points for holes and electrons, and the sixH points for holes. (c) How the band edges at the T point, the three L

points, and the six H points change as a function of Sb compositionx in bulk Bi1−xSbx .

II. METHODOLOGY

The quantum confinement effect of nanowires may changethe symmetry properties of the carrier pockets, the positionsin energy of the band edges, and possibly the shape of thedispersion relations at the T point, the L points, and the H

points. There are two types of quantum confinement effects onthe band edges. One is a trivial quantum confinement effectthat does not change the inverse-effective-mass tensor or theshape of the dispersion; the other is a nontrivial quantumconfinement effect that does change the inverse-effective-masstensor and the shape of the dispersion. For the T pointand the H points, the dispersions of the band edges areparabolic, and the quantum confinement effect is trivial. Thevalence-band edge at the T point will decrease in energyby h2/8 · (d−2

1 · α[T ,Wire]‖,11 + d−2

2 · α[T ,Wire]‖,22 ), where d1 and d2

are the cross-sectional widths of a rectangular nanowire,and α

[T ,Wire]‖,11 and α

[T ,Wire]‖,22 are the corresponding components

of the cross-sectional inverse-effective-mass tensor [24,25].For a square nanowire, the width is d = d1 = d2, and theexpression of energy decrease at the T point is reducedto h2/8d2 · (α[T ,Wire]

‖,11 + α[T ,Wire]‖,22 ). Similarly, the valence-band

edge at an H point will decrease in energy by h2/8 ·(d−2

1 · α[H,Wire]‖,11 + d−2

2 · α[H,Wire]‖,22 ) for a rectangular nanowire,

and by h2/8d2 · (α[H,Wire]‖,11 + α

[H,Wire]‖,22 ) for a square nanowire.

Furthermore, we can still assume that α[T ,Wire]‖ = α

[T ,Bulk]‖

and α[H,Wire]‖ = α

[H,Bulk]‖ as have been validated in previous

reports [8–10,14–16].For the L points, the dispersions of the band edges are non-

parabolic, and the quantum confinement effect is nontrivial, sothat the traditional square-well model is not accurate any more.There coexists an electron pocket and a hole pocket at each L

point. Thus, the conduction band edge and the valence-bandedge for the L point are very close to each other in energy, andare strongly coupled, which results in the nonparabolicity ofthe dispersion for the L-point electrons and holes or possiblyeven linearity if the two bands are touching. Meanwhile, theshape of the dispersion is also correlated with the magnitude ofthe narrow band gap. On the one hand, quantum confinementin the cross-sectional plane will change the narrow band gap,which is associated with the inverse-effective-mass tensor. Onthe other hand, the inverse-effective-mass tensor of Bi1−xSbx

nanowires is changed by the change of the narrow bandgap, which is also different from the inverse-effective-masstensor of a bulk Bi1−xSbx with the same Sb composition.Therefore, the puzzle is that neither the narrow band gap northe inverse-effective-mass tensor is known at this stage for thenanowires case.

Historically, the relation between the narrow band gap andthe dispersion for bulk bismuth is described by a two-bandmodel [26],

p · α[L,Bulk,Bi] · p = E(k)

[1 + E(k)

E[L,Bulk,Bi]g

], (1)

where α[L,Bulk] is the inverse-effective-mass tensor for the L-point carrier pocket of bulk bismuth, and it is assumed thatα[L,Bulk] for both the conduction band edge and the valence-band edge are the same due to the strong interaction betweenthese two bands; this approximation has been shown to bevalid for bulk bismuth samples. The relation between the bandgap and the inverse-effective-mass tensor is seen more clearlyin the form of the second derivative of Eq. (1) [10,27],

α[L,Bulk,Bi] = 2

�2

∂2E(k)

∂k2= 1

m0I ± 1

m20

2

E[L,Bulk,Bi]g

p2 (2)

where m0 is the mass of free electron and I is the identitymatrix. It is assumed that the form of Eq. (2) holds also forBi1−xSbx in the range of x = 0–0.30, though the band gapand the effective-mass tensor may change as a function of Sbcomposition x. Thus, we further have [8,9,28]

α[L,Bulk,Bi1−xSbx ] = E[L,Bulk,Bi1−xSbx ]g

E[L,Bulk,Bi]g

(α[L,Bulk,Bi] − 1

m0I)

+ 1

m0I, (3)

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ELECTRONIC PHASES, BAND GAPS, AND BAND . . . PHYSICAL REVIEW B 89, 045424 (2014)

which is consistent with the experimental results carried outby Mendez et al., where a simpler version of Eq. (3) wasadopted [29] and is given by

α[L,Bulk,Bi1−xSbx ] = E[L,Bulk,Bi1−xSbx ]g

E[L,Bulk,Bi]g

α[L,Bulk,Bi]. (4)

The theoretical validation from Eq. (3) to Eq. (4) isdiscussed in Ref. [14], and these experiences can each beextended to connect the bulk materials and nanowire materials;i.e.,

α[L,Wire,Bi1−xSbx ] = E[L,Wire,Bi1−xSbx ]g

E[L,Bulk,Bi]g

(α[L,Bulk,Bi] − 1

m0I)

+ 1

m0I, (5)

and

α[L,Wire,Bi1−xSbx ] = E[L,Wire,Bi1−xSbx ]g

E[L,Bulk,Bi]g

α[L,Bulk,Bi]. (6)

Now we have two approaches to solving the dispersionrelation and for finding the narrow band gap of the L-point

band edges. One is the iterative way. We set E(0)g = E[L,Bulk,Bi]

g

and α(0) = α[L,Bulk,Bi], and we repeatedly carry out the iterationsteps of

E(n+1)g = E[n]

g + 2h2

8d2Tr(α(n)

‖ ) (7)

and

α(n+1) = E(n)g

E(n+1)g

(α(n) − 1

m0I)

+ 1

m0I (8)

until convergence, where α(n)‖ is the cross-sectional subtensor

of α(n).The other approach is to use the simpler Eq. (6), and derive

a solution in an analytical form, by solving Eq. (6) and

E[L,Wire,Bi1−xSbx ]g = E[L,Bulk,Bi]

g + 2h2

8d2Tr(α[L,Wire,Bi1−xSbx ]

‖ ).

(9)

Thus, the L-point narrow band gap of the nanowire can besolved to be

E[L,Wire,Bi1−xSbx ]g =

E[L,Bulk,Bi1−xSbx ]g +

√(E

[L,Bulk,Bi1−xSbx ]g

)2 + h2

d2 Tr(α

[L,Bulk,Bi]‖

)E

[L,Bulk,Bi]g

2. (10)

III. RESULTS AND DISCUSSION

Figure 2 illustrates the electronic phase diagrams andband gap/overlap of Bi1−xSbx nanowires with d = 100 nm,as a function of growth orientation and stoichiometry. Theillustrations are made for the growth orientation within thebinary plane, the trigonal plane, and the bisectrix plane, inFigs. 2(a), 2(b), and 2(c), respectively. The semimetal (SM)phase regions, indirect semiconductor (ISC) phase regions,and the direct semiconductor (DSC) phase regions are markedfor each orientation. The Sb composition x is denoted by theradius of the circles in Fig. 2. With 100 nm wire width, it can beseen that the electronic phase starts from a semimetal, wherethe top of the valence-band edge is located at the T point, atx = 0. As the Sb composition increases, phase changes occur.Explicitly, the electronic phase changes from a semimetal toan indirect gap semiconductor with the top of the valence-bandedge located at the T point, too. At around x = 0.15, the top ofthe valence-band edge becomes located at an L point, and theelectronic phase becomes a direct gap semiconductor. For yethigher Sb concentration x, the top of the valence-band edge isshifted to an H point, and the band gap of the semiconductorphase becomes indirect again. When x is further increased, theelectronic phase finally is changed back to a semimetal, onlywith the top of the valence-band edge located at an H point.

This is seen more clearly in the maps of the bandgap/overlap as a function of growth orientation and Sbcomposition, in the corresponding subfigures below, i.e., inFigs. 2(d), 2(e), and 2(f), for the binary plane, the trigonalplane, and the bisectrix plane, respectively. A negative value

stands for a band overlap, while a positive value stands fora band gap. A zero value denotes the gapless state. At Sbcomposition x = 0, the band overlap exhibits a negative valuecorresponding to the semimetal phase. As x is increased, themagnitude of the band overlap starts to decrease, and thenbecomes zero, beyond which the band overlap disappears andthe nanowire exhibits a band gap with a positive value, andthe band gap increases with increasing Sb composition x. Ataround x = 0.15, the band gap starts to decrease with increas-ing Sb composition x, until the band gap reaches zero. At yethigher Sb composition x, a band overlap with a negative valueappears again, indicating the onset of a semimetallic phase atthe relatively Sb-rich side of the phase diagram [see Fig. 1(c)].

The anisotropic properties of bulk bismuth antimony phasesare reflected in the symmetry properties of the electronicphase diagrams and the band-gap/overlap diagrams, when thegrowth orientation lies in different crystallographic planes, ornormal to different crystallographic directions. The diagramsof Figs. 2(a)–2(c) have similar profiles, but are actuallydifferent in their detailed shapes at each phase boundary, if theyare examined more closely. The binary crystalline plane hasinversion symmetry but not mirror symmetry. The inversionsymmetry is reflected in Figs. 2(a) and 2(d). However, at thisspecific wire width, the contrast associated with the anisotropyis not strong, and the nonexistence of mirror symmetry is notobvious. We discuss below how quantum confinement helps toenhance the contrast of the anisotropy in the electronic phasediagrams and the band-gap/overlap diagrams, and also howquantum confinement causes the mirror symmetry to disappearfor the diagrams associated with the binary crystalline plane.

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SHUANG TANG AND MILDRED S. DRESSELHAUS PHYSICAL REVIEW B 89, 045424 (2014)

Binary

Trigonal

Bisectrix

Binary

Bisectrix

Trigonal

(a) (b) (c)

(d) (e) (f )

0.300.150.00 -30 -20 4030100-10- - 20-

1.SM2.ISC

5.SM4.ISC 3.DSC

Overall Band Gap/Overlap (meV)

FIG. 2. (Color online) The electronic phase diagrams (a)–(c), and the band-gap/overlap diagrams (d)–(f) of Bi1−xSbx nanowires of 100 nmwire width, as shown for various wire growth orientations and Sb composition x. The illustrations are made for the binary (a) and (d), trigonal(b) and (e), as well as bisectrix (c) and (f) crystalline planes. In each diagram, the direction stands for growth orientation, and the length of radiusfor a point stands for the Sb composition x, which lines up with the origin of each circularly shaped diagram, so that the binary direction in (a),for example, denotes the direction normal to the trigonal and bisectrix directions shown for the square sample faces. At the origin, the value ofx is 0, and at the outermost point, the value of x is 0.30, as scaled in the legend. The region’s semimetal phase (SM), indirect semiconductorphase (ISC), and direct semiconductor phase (DSC) are marked out in (a)–(c) and in the legend. In the legend of the phase regions, the top ofthe valence-band edge is located at the T point for the upper regions of the semimetal (region 1) and indirect semiconductor phases (region 2),and at the H points for the lower regions of the indirect semiconductor phase (region 4) and semimetal phase (region 5). Both the top of thevalence-band edge and the bottom of the conduction band edge are located at the L points for the direct semiconductor phase (Region 3). Inthe diagrams of the band gap/overlap, a positive value stands for a band gap, while a negative value stands for a band overlap. A zero valuestands for a gapless state.

When all six H -point hole pockets, all six L-point half-holepockets, and all six L-point half-electron pockets are projectedonto the trigonal plane, a sixfold symmetry is formed, which isa higher symmetry than the threefold symmetry of the trigonalaxis in the bulk materials. This is reflected in the electronicphase diagram of Fig. 2(b), and in the band-gap/overlap phasediagram of Fig. 2(e). In Fig. 2(b), the shape of each phaseboundary is actually a mixture of a circle and hexagon. Tosee this, it requires a very careful reading of the diagram, andit might not be obvious. However, this circle and hexagonare much more clear in the band-gap/overlap diagram inFig. 2(e), where the shapes of six-petal flowers can best beseen at the edge of the inner yellow DSC (direct semiconductorphase) region. The diagrams of the bisectrix crystalline planehave both mirror symmetry about the binary axis, and mirrorsymmetry about the trigonal axis, which are accompanied byinversion symmetry. These symmetry properties are seen inFig. 2(c), and more obviously in Fig. 2(f).

Now we illustrate the electronic phase diagrams and band-gap/overlap diagrams of Bi1−xSbx nanowires with a muchstronger quantum confinement effect, occurring in nanowireswith a small width, explicitly for a nanowire with a width of

10 nm. The small-width nanowires show how the quantumconfinement effect influences the symmetry properties andthe electronic phases of the nanowires comparatively. Theelectronic phase diagrams and band gap/overlap of Bi1−xSbx

nanowires with d = 10 nm, as a function of growth orientationand stoichiometry, are illustrated in Fig. 3. The changes inthe electronic phase diagrams are more obvious in Fig. 3than in Fig. 2. First, the direct semiconductor phase regionshave disappeared in all of the three cases (a), (b), and (c).The semimetal phase region (dark blue) where the top of thevalence-band edge is located at the T point has significantlyshrunk to a tiny size in both Figs. 3(a) and 3(c), and hasdisappeared in Fig. 3(b). The semimetal phase region wherethe top of the valence-band edge is located at an H point(light blue) has shrunk as well, in all three cases, but stillis present. The dominant phase regions become the indirectsemiconductor phases, which have both expanded remarkablyin Figs. 3(a)–3(c). Such information is very important forthe design of electronic devices using Bi1−xSbx nanowires.The much stronger quantum confinement effect in 10-nm-widenanowires makes the contrast for the anisotropy of all thediagrams much more obvious. The existence of inversion

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ELECTRONIC PHASES, BAND GAPS, AND BAND . . . PHYSICAL REVIEW B 89, 045424 (2014)

(a) (b) (c)

(d) (e) (f )

0.300.150.00 -40 8060200-20 40

1.SM2.ISC

5.SM4.ISC 3.DSC

Overall Band Gap/Overlap (meV)

Binary

Trigonal

Bisectrix

Binary

Bisectrix

Trigonal

FIG. 3. (Color online) The electronic phase diagrams (a)–(c), and the band-gap/overlap diagrams (d)–(f) of Bi1−xSbx nanowires of 10 nmwire width, as a function of wire growth orientation and Sb composition x. All the notations and legends are the same as those defined in Fig. 2.

symmetry and the absence of mirror symmetry is shown clearlyin Figs. 3(a) and 3(d). In Fig. 3(b), it is clearer that the shapesof the boundaries of the electronic phase regions are mixturesof a circle and a hexagon, and are more hexagonal than inFig. 2(b). The outlines of six-petal flowers in Fig. 3(e) arealso much easier to observe than in Fig. 2(e). Furthermore, themirror symmetry about the binary axis, mirror symmetry aboutthe trigonal axis, and the associated inversion symmetry of theorientation-stoichiometry phase diagram of the trigonal-binarycrystallographic plane normal to the bisectrix are furtherclarified in Figs. 3(c) and 3(f).

The above discussions show that Bi1−xSbx nanowires oflarger wire width show a much richer variation of electronicphases, but the contrast of the anisotropy for different growthorientations is less obvious, while for the Bi1−xSbx nanowiresof larger wire width, the richness of the variation of electronicphases is reduced, but the contrast of the anisotropy fordifferent growth orientations is much enhanced. In order toshow how the electronic and symmetry properties change withwire width and growth orientation, we have calculated theband-gap/overlap diagrams as a function of wire width andgrowth orientation for both a small value of Sb composition ofx = 0.05 and a medium value of Sb composition x = 0.13, asshown in Fig. 4. The case of large values of Sb compositionsturns out to be similar with the case of small values, onlythe location of the top of the valence-band edges for thesemimetal phases and the indirect semiconductor phases areat the H points for the median value of x = 0.13, instead ofat the T point which applies to x = 0.05. In order to makethe values of band-gap/overlap comparable in one plot, we

choose to use a logarithm scale, and in order to avoid thedivergent values of band shift when the wire width gets closeto 0, we choose to ignore the cases where the wire width issmaller than 100.5 � 3 nm.

For the small Sb composition x = 0.05 cases in Figs. 4(a)–4(c), we see that there are mainly three regimes for eachdiagram: the inner semimetal regime when the wire widthis small, the semiconductor regime when the wire width ismedium, and the outer semimetal regime when the wire widthis large. The outer semimetal regime is easy to understand,because the bulk bismuth antimony material at this Sb com-position is in the semimetal phase region, with the top of thevalence-band edge located at the T point, as shown in Fig. 1(c).The quantum confinement effect has induced an increase ofthe direct band-gap region around the L points, and leadsto a semiconductor phase at the medium wire width regime.That is also why the band overlap increases and saturatesat a certain value with increasing wire width. However, themechanism governing the inner semimetal regime is different.In our model, the components of the inverse-effective-masstensor and the direct band gap at an L point are negativelycorrelated. When the width of nanowire is large, the nontrivialquantum confinement effect near an L point shifts the bandedge more quickly as the width decreases than does the trivialquantum confinement effect near an H or a T point. However,when the width of a nanowire is very small, the direct band gapat an L point is large enough to induce a significant reductionof the interband coupling, which changes the nonparabolicdispersions at the L point into parabolic dispersions with largermass components. For this situation, the valence-band edges at

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SHUANG TANG AND MILDRED S. DRESSELHAUS PHYSICAL REVIEW B 89, 045424 (2014)

(a) (b) (c)

(d) (e) (f )

103.0101.5100.0

Wire Width (nm)

10-2.0 102.0

Overall Band Gap/Overlap (meV)

Binary

Trigonal

Bisectrix

Binary

Bisectrix

Trigonal

101.010-1.0

FIG. 4. (Color online) The band-gap/overlap diagrams of Bi1−xSbx nanowires of small Sb composition (x = 0.05) (a)–(c) and medium Sbcomposition (x = 0.13) (d)–(f), as a function of wire growth orientation and wire width. All the notations and legends are the same as thosedefined in Figs. 2(d)–2(f), except that the length of the radius in Fig. 4 stands for the wire width scaled logarithmically, which lines up with theorigin of each circularly shaped diagram.

(a) (b) (c)

(d) (e) (f )

Binary

Trigonal

Bisectrix

Binary

Bisectrix

Trigonal

103.0101.5100.0

Wire Width (nm)

1.SM2.ISC

5.SM4.ISC 3.DSC

FIG. 5. (Color online) The electronic phase diagrams of Bi1−xSbx nanowires of small Sb composition (x = 0.05) (a)–(c) and medium Sbcomposition (x = 0.13) (d)–(f), as a function of wire growth orientation and wire width. All the notations and legends are the same as thosedefined in Figs. 2(a)–2(c), except that the length of the radius in Fig. 4 stands for the wire width scaled logarithmically.

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ELECTRONIC PHASES, BAND GAPS, AND BAND . . . PHYSICAL REVIEW B 89, 045424 (2014)

the T point and at the H points may, in contrast, have a largerquantum confinement effect. For the medium Sb composition(x = 0.13) cases in Figs. 4(d)–4(f), except for a small areanear the center, where the wire width is very small, each band-gap/overlap diagram has a positive value, which increases andsaturates to a certain value with increasing wire width. Thisis also because the bulk bismuth antimony material with thisSb composition is in the indirect semiconductor phase region,as shown in Fig. 1(c). The corresponding phase diagrams asa function of wire width and growth orientation, for the smalland medium Sb composition cases (x = 0.05 and x = 0.13) areshown in Fig. 5, which further solidifies the analysis above.

IV. CONCLUSION

In conclusion, we have developed a model to accu-rately describe the quantum confinement in one-dimensional

narrow-band-gap systems, which accurately captures the bandedge shifts and the shape of the nonparabolic dispersionrelations. We have then used this model to study the phasediagrams and band-gap/overlap of the bismuth antimonynanowire materials system. The symmetry properties, theband edge configurations, the electronic phase diagrams, andthe band-gap/overlap diagrams as a function of wire growthorientation, stoichiometry, and wire width are systematicallystudied. Such information provided important guidance for thesynthesis and characterization of bismuth antimony nanowiresand for their use in the design of applications.

ACKNOWLEDGMENT

The authors acknowledge the support from AFOSR MURIGrant No. FA9550-10-1-0533, Subaward No. 60028687. Theviews expressed are not endorsed by the sponsor.

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