Constructing Anisotropic Single-Dirac-Cones in Bi1–xSbx Thin Films

Mar 20, 2012 - M.S.D.: e-mail: [email protected]; telephone, 1-617-253-6864; fax: ... made available by participants in CrossRef's Cited-by Linking s...
3 downloads 0 Views 4MB Size
Download PDF
Letter pubs.acs.org/NanoLett

Constructing Anisotropic Single-Dirac-Cones in Bi1−xSbx Thin Films Shuang Tang*,† and Mildred S. Dresselhaus*,‡ †

Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States ‡ Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States ABSTRACT: The electronic band structures of Bi1−xSbx thin films can be varied as a function of temperature, pressure, stoichiometry, film thickness, and growth orientation. We here show how different anisotropic single-Dirac-cones can be constructed in a Bi1−xSbx thin film for different applications or research purposes. For predicting anisotropic single-Diraccones, we have developed an iterative-two-dimensional-twoband model to get a consistent inverse-effective-mass-tensor and band gap, which can be used in a general two-dimensional system that has a nonparabolic dispersion relation as in the Bi1−xSbx thin film system. KEYWORDS: Anisotropic, single-Dirac-cone, bismuth antimony, thin film

M

a topological insulator was made in bulk Bi0.9Sb0.1.23 Some 2D experiments on Bi1−xSbx thin films grown normal to the trigonal direction have already been carried out.24−26 However, possible Dirac-cone materials of 2D Bi1−xSbx thin films have not yet been studied. It is meaningful to start from the cryogenic physical properties of 2D Bi1−xSbx, because the band structure does not change over temperature when T is below the liquid nitrogen boiling point (77 K), which significantly simplifies the analysis. In addition, because many experiments have been carried out for bulk Bi and Bi1−xSbx at cryogenic temperatures, the experimental techniques are relatively mature. Thus, it would be easy to experimentally realize the theoretical predictions in 2D Bi1−xSbx films in this temperature range. In this Letter, we focus on the cryogenic temperature range. We have systematically studied how the properties of the 2D Lpoint single-Dirac-cones in Bi1−xSbx thin films, such as their anisotropy and linearity, change with film thickness and growth orientation. We have also discovered that different varieties of Dirac cones can be constructed in Bi1−xSbx thin films, such as single-Dirac-cones with different anisotropies. Regarding the methodology, we have developed an iterative-two-dimensionaltwo-band model, which could be used as a general model to study systems with nonparabolic 2D dispersion relations or Dirac cones. We recall that the band gap and band-overlap of bulk Bi can be changed by adding Sb to form Bi1−xSbx alloys. 3D Dirac points may form under proper conditions at the three L

aterials with two-dimensional (2D) Dirac cones in their electronic band structures have recently attracted considerable attention. Many important novel physical studies have been carried out on both massless and massive 2D Dirac fermions, including studies of the half-integer quantum Hall effect,1,2 the anomalous absence of back scattering, the Klein paradox effect,3 high temperature superconductivity,4 and unusual microwave response effects.5 The ultrahigh carrier mobility of the Dirac fermions in graphene offers new opportunities for a variety of electronics applications.6 Recently, 2D Dirac cones observed in topological insulators have identified this class of materials as candidates for quantum computation, spintronics, novel superconductors, and promising thermoelectrics.7−11 Materials with 2D single-Dirac-cones, especially 2D anisotropic single-Dirac-cones, are of special interest. Simulations with ultracold atoms trapped on optical lattices have been used to study general 2D single-Dirac-cones.12 It is believed that graphene superlattice materials, which are described by anisotropic 2D single-Dirac-cones, could potentially be developed for use in nanoelectronic-circuits without cutting processes, and in table-top experiments that simulate highenergy relativistic particles propagating in anisotropic space.13 Among all materials candidates that may possibly make anisotropic single-Dirac-cones, 2D Bi1−xSbx thin films are especially attractive. First, it has been proved that the band structure of three-dimensional (3D) bulk Bi1−xSbx can be varied as a function of Sb composition x, temperature, pressure, and strain, and the Fermi level can be adjusted to change the electronic properties.14−19 Second, not only have bulk state Dirac points been studied in bulk Bi1−xSbx alloys20−22 but also the first observation of a surface state 2D single-Dirac-cone for © 2012 American Chemical Society

Received: January 6, 2012 Revised: March 6, 2012 Published: March 20, 2012 2021

dx.doi.org/10.1021/nl300064d | Nano Lett. 2012, 12, 2021−2026

Nano Letters

Letter

points,20−22 where the valence band and the conduction band exhibit a band-crossing. At each L point, E(k) becomes linear in the limit of band-crossing (Eg → 0), because E(k) = ±((v·p)2 + Eg2)1/2, where v is the carrier velocity, p is the carrier momentum, and Eg is the L-point direct band gap.23,27 Historically, the 3D two-band model28 has been successful in describing the nonparabolic dispersion relation E(k) in 3D bulk Bi materials by ⎛ ⎞ E(k) ⎟ p·α·p = E(k)⎜⎜1 + Eg ⎟⎠ ⎝

film. The Hamiltonian for Bi and Bi1−xSbx based on k·p theory in this model is equivalent to a Dirac Hamiltonian with a scaled canonical conjugate momentum.27 Thus, eqs 5 and 6 are also good approximations to describe the Dirac cones. The band parameters we use in the present calculations are values that were measured by cyclotron resonance experiments.32−34 There are three symmetrical L points in the first 3D Brillouin zone of Bi1−xSbx as shown in Figure 1. To get a single-Dirac-

(1)

where α is the inverse-effective-mass-tensor, and we assume here that α is the same for both the conduction band and the valence band within the context of a two-band model and strong interband coupling. Generally, the relation between α and Eg around an L point is described as 2

Figure 1. 3-fold degenerate L points and the T point in the first Brillouin zone of bulk Bi1−xSbx.

∂ 2E(k)

1 1 2 2 α= 2 = ·I ± ·p 2 m ℏ ∂(k − kL) m 0 2 Eg 0

(2)

cone, we need to grow the Bi1−xSbx thin film normal to a low symmetry direction so that we can have a single L point that differs from the other two by breaking the 3-fold symmetry occurring in 3D. Moreover, two quantities in eq 4 need to be minimized, namely the bulk-band-gap term Egbulk(Bi1−xSbx) and [0] /4·lz2 for this the quantum-confinement-induced term h2α33 special single L point. Thus, growing a film normal to the bisectrix direction is a good way, where the 3-fold symmetry is broken and the α33 inverse mass component is near its minimum,32−34 which is discussed below. Minimizing the value of the bulk term Eg bulk(Bi1−xSbx) by varying temperature T, pressure P, and Sb composition x has been discussed in the literature,23,29−31,35 especially for the case of x = 0.04, where Egbulk(Bi1−xSbx) decreases to 0 at cryogenic temperatures under atmospheric pressure. Thin films with the composition Bi0.96Sb0.04 have already been synthesized in experiments by researchers in the Ukraine26 for films grown normal to the trigonal direction. The model that we have developed above is valid for all values of Sb composition x (≤0.15). However, for the convenience of experimentalists to use our theoretical predictions, in the present Letter, we illustrate simple examples of single-Dirac-cones that can be found in Bi0.96Sb0.04. More generally, Bi1−xSbx thin films of other Sb compositions x can be modeled in a similar way. On the basis of our model, a single 2D Dirac cone can be constructed in a Bi0.96Sb0.04 thin film normal to the bisectrix axis.36 For this thin film the 3-fold symmetry of the L points in reciprocal space is broken. According to our calculations, the (1) L(1)-point band gap EgL will not exceed ∼1 meV until the film is thinner than 80 nm. However, the L(2)- and L(3)-point band (2 ) ( 2) (3 ) gaps EgL (EgL = EgL ) will open up and increase significantly (2) when the film thickness decreases. When lz = 300 nm, EgL is (1) already larger than 7 meV, whereas EgL is still smaller than 0.1 meV. This means that a single-Dirac-cone is formed at the L(1) point as shown in Figure 2a. Up to now, we have only discussed the band structure and the dispersion relations at the L points. Another important aspect is the Fermi level Ef, which determines the carrier density and transport properties. The Fermi level for a Bi1−xSbx thin film changes with film thickness, temperature, external gate voltage, and impurity doping. For further discussion of how the Fermi level influences the physical properties of the single-

under the k·p approximation,29−31 where I is the identity matrix and m0 is the free electron mass. We assume here that a twoband model also applies to Bi1−xSbx alloys, where the influence on α of adding Sb atoms up to a Sb concentration of x = 0.1 to bulk Bi31 is then α(Bi1 − xSbx ) =

⎛ 1 ⎞ 1 ·⎜α(Bi) − ·I⎟ + ·I Eg (Bi1 − xSbx ) ⎝ m0 ⎠ m0 Eg (Bi)

(3)

We have developed an iterative-two-dimensional-two-band model for Bi1−xSbx thin films, where both αfilm and Egfilm of a Bi1−xSbx thin film differ from αbulk and Egbulk in the 3D bulk case, and they are both unknown at the start. The conduction band and the valence band at an L point are first separated by Eg[0] = Egbulk (Bi) and the inverse-effective-mass-tensor is α[0] = αbulk(Bi). In a Bi1−xSbx thin film, the L-point gap will be influenced by both the added Sb atoms and the quantum confinement effect, so that the L-point gap to the lowest order approximation becomes Eg[1] = Egbulk (Bi1 − xSbx ) + 2·

h2 α[0] 33 8 · lz 2

(4)

The change in the L-point gap will lead to a change of the Lpoint inverse-effective-mass-tensor according to α[n] =

⎛ ⎞ Eg[n − 1] ⎟ 1 ⎜ [n − 1] ·α + · 1− ⎟·I m0 ⎜ Eg[n] Eg[n] ⎠ ⎝

Eg[n − 1]

(5)

where n denotes the step number in the iteration. Equation 5 is a general equation for iterations, which is just a consequence of eq 3. Thereafter, the L-point band gap can be updated with the new inverse-effective-mass-tensor by 2 [n]

h α33 Eg[n + 1] = Eg[n] + 2· 8 · lz 2

(6)

The iterative procedure is repeated until α and become self-consistent, and then we get an accurate solutions for α[n] = αfilm(Bi1−xSbx) and Eg[n] = Egfilm(Bi1−xSbx) for the Bi1−xSbx thin [n]

Eg[n]

2022

dx.doi.org/10.1021/nl300064d | Nano Lett. 2012, 12, 2021−2026

Nano Letters

Letter

Figure 2. How the L(1)-point single-Dirac-cone is formed. (a) shows the band structure of a Bi0.96Sb0.04 thin film grown normal to the bisectrix axis and how the band structure changes over different film thicknesses. The green curves show the lowest conduction band (upper one) and the highest valence band (lower one) at the L(2) and L(3) points. The blue curves are for the L(1) point. The dashed red curve is the highest valence band at the T point. The L(1)-point band gap remains less than the order of ∼1 meV until the film thickness lz is very small. The L(2) and L(3) points have the same band gap, which is largely opened up. Thus, an anisotropic single-Dirac-cone is formed at the L(1) point when the film thickness lz is large enough to retain the L(1)-point mini-gap essentially zero, i.e., less than ∼1 meV. (b) shows the thermal smearing (−∂f 0/∂E) of the Fermi-Dirac distribution as a function of cryogenic temperature. For comparison between (a) and (b), the Fermi level Ef is aligned with E = 0, i.e., the middle point of the L-point band gap. Cases where the Fermi level Ef is at other positions can be discussed in a similar way. Only carriers within the smearing region will get excited and contribute to the transport phenomena. (c) If no doping is added and no gate voltage is applied, the lz dependence of the intrinsic Fermi levels is shown at 77 K (red curve) and at 4.2 K (blue curve). Carrier concentration of Bi0.96Sb0.04 vs film thickness lz and Fermi level Ef are shown at (d) 77 K and (e) 4.2 K.

respectively, show the total carrier concentrations for a Bi0.96Sb0.04 film at the liquid nitrogen boiling point (77 K) and the liquid helium boiling point (4.2 K). The overall carrier concentration is very low (∼1017 cm−3) at 77 K and much lower (∼1016 cm−3) at 4.2 K. There are two kinds of Dirac fermions associated with a Dirac cone that researchers are interested in, the massless Dirac fermions and the massive Dirac fermions. The massless Dirac fermions are the fermions that are right at the apex of a Dirac cone, whose effective mass m* is 0 because of the singularity of E(k) at the apex. Other Dirac fermions that are very near the Dirac cone but not at the apex are massive, whose effective mass m* is ∞ because E(k) is linearly dispersed. In experiments, a Dirac cone is usually not perfect. A mini-gap often exists that induces a mini-mass at the apex of the Dirac cone. Such an effect also occurs in single layer graphene.1,2 Therefore, practically, there are two main features that characterize the quality of a Dirac cone, the mini-mass for the “massless” Dirac fermions occurring at the apex, and the fermion velocity v(k) as a function of k for the massive Dirac fermions. Because the fermions are linearly dispersed near a Dirac cone, v(k) should only be a function of the direction of k. The anisotropy of the Dirac cone can be characterized by the ratio between the maximum and the minimum v(k). The L(1)-point anisotropic single-Dirac-cone in a 300 nm thick Bi1−xSbx thin film grown normal to the bisectrix axis has a

Dirac-cone, we assume that the Fermi level for a specific Bi1−xSbx thin film can be varied freely within the range 0−25 meV37 without destroying the single-Dirac-cone. How the carrier concentration changes with film thickness and temperature will be discussed later. The carriers that contribute to transport are the ones that are within the smearing of the Fermi-Dirac distribution (−∂f 0/∂E), where f 0 = (1 + exp[(E − Ef)/kB·T])−1. The quantity of (−∂f 0/ ∂E) is very sharp over E at cryogenic temperatures as shown in Figure 2b and has a width on the order of ∼kB·T. For comparison to Figure 2a, we have aligned the Fermi level in Figure 2b with E = 0 to indicate the absence of carrers due to dopants. In this case, the Fermi level is at the apex of the L(1)point single-Dirac-cone. Then the Dirac fermions that contribute to transport will only come from this L(1)-point single-Dirac-cone, and not from the L(2) or L(3) points, at cryogenic temperatures. Figure 2c shows the Fermi level for intrinsic Bi0.96Sb0.04 films without doping or gate votage. At the cryogenic temperatures, the intrinsic Fermi level starts to drop with film thickness when the film is thinner than ∼40 nm, which reveals the semimetalsemiconductor transition, where the T-point valence band falls below the L(1)-point conduction band, consistent with the prediction of Figure 2a. How the carrier concentration for a Bi0.96Sb0.04 film changes as a function of film thickness and Fermi level is calculated next. Parts d and e of Figure 3, 2023

dx.doi.org/10.1021/nl300064d | Nano Lett. 2012, 12, 2021−2026

Nano Letters

Letter

transport properties are dominated by the L-point carriers, which have ultrahigh electron and hole mobilities,38−40 because of the ultralarge carrier group velocities of the L(1)-point carriers. This is not difficult to understand and can be explained in a very simple manner. For example, the electronic conductivity of the carriers for a specific carrier pocket is σij = e 2D Ef ·vi ·∑ (τ Ef )jl ·vl l

(7)

where τEf and DEf are, respectively, the anisotropic relaxation time tensor and the density of states for carriers of this specific carrier pocket at the Fermi level Ef, and i, j, and l denote components of the various vector and tensor quantities. We take the principal axis along kx as an example, i.e., i = j = x and σxx = e2DEf(τEf)xx·νx2. In Bi and Bi1−xSbx, the relaxation time (τEf)xx = λxxνx−1, where λxx is the mean free path and λxx ∝ DEf−1. Thus, we have σxx ∝ vx(Ef), where vx(Ef) is the carrier group velocity component along the kx direction for this specific carrier pocket at the Fermi level Ef. For the L-point Dirac [Dirac] fermions, νx(E ∼ 10−2·clight, whereas for the T-point f) [T] parabolically dispersive fermions, νx(E ∼ 103 m/s, thereby f) [Dirac] [T] explaining why σxx ≫ σxx . The difference of electrical conductivity between the thin film Bi1−xSbx and the bulk Bi1−xSbx can be explained by the classical confinement effect and the quantum confinement effect.26,41−50 The quantum confinement effect is accounted for by the band structure itself.41 The classical confinement effect comes from the film boundary scattering mechanism, which can be described by the empirical relation σlz/σbulk = C1·lz·(C2 −ln lz),26,42−46 where σlz is the electrical conductivity for the thin film with thickness lz, σbulk is the bulk electrical conductivity, and C1 and C2 are empirical constants that can be measured by experiments. This simple relation is very successful in describing the Bi thin films and Bi1−xSbx thin films.26,42−50 On the basis of the above, we know that the main factor that determines the electrical conductivity in a Bi1−xSbx thin film is also the carrier group velocity, and because the L(1)-point Dirac fermions have much larger group velocities than the T-point parabolically dispersive fermions, we believe that the electrical conductivity is dominated by the L(1)-point Dirac cone. Moreover, the T-point parabolically dispersed carrier pocket can also be moved further down in energy below the L(1)-point single-Dirac-cone, in a 2D Bi0.96Sb0.04 thin film, which is not achieved in bulk Bi0.96Sb0.04. When the film thickness decreases, the top point of the T-point valence band decreases in energy much faster than the L(1)-point valence band, as shown in Figure 2a. When the film thickness is less than 40 nm, the Tpoint valence band is totally below the bottom of the L(1)-point conduction band, indicating a semimetal−semiconductor transition The effective mass at the apex of this single-Diraccone is ∼10−4m0, which is still essentially massless. The corresponding Dirac cone is plotted in Figure 3c, as well as the velocity vs momentum relation v(k) for the massive Dirac fermions as in Figure 3d. The linearity and the anisotropy of the Dirac cone is not notably influenced by film thickness for a film of lz = 40 nm, as can be seen by comparing Figure 3c,d to Figure 3a,b, respectively. In some applications, e.g., in nanoelectronic-circuit design, a higher anisotropy of the Dirac cone could be required. For a Bi0.96Sb0.04 thin film, the L(1)-point E(k) has a smaller minimass at the apex but a lower anisotropy if the film is grown

Figure 3. Different anisotropic single-Dirac-cones in different Bi0.96Sb0.04 thin films: (a) and (b) describe a sharp-apex L(1)-point anisotropic single-Dirac-cone in a 300 nm thick Bi0.96Sb0.04 film grown normal to the bisectrix axis. For convenience, the origin of momentum k is chosen to be at the L(1) point. (c) and (d) describe an L(1)-point anisotropic single-Dirac-cone where the T-point carrier-pocket is totally below the L(1)-point Dirac cone, in a 40 nm thick Bi0.96Sb0.04 film grown normal to the bisectrix axis. (e) and (f) describe a highly anisotropic single-Dirac-cone in a 300 nm thick Bi0.96Sb0.04 film grown normal to the [606̅1] crystalline direction. (a), (c), and (e) show the dispersion relations of these single-Dirac-cones. (b), (d), and (f) show the group velocities v of Dirac fermions over different momenta k. (c) and (d) are not significantly different from (a) and (b), but (e) and (f) are obviously more anisotropic than (a) and (b), and (c) and (d).

linear E(k) behavior with a very sharp apex (Figure 3a). In this film, kx and ky represent the trigonal axis and the binary axis, (1) respectively. EgL for this Dirac cone is smaller than 0.1 meV, and the effective mass at the apex of the Dirac cone is ∼10−5m0, which can be considered as essentially gapless and massless. We have also calculated the v(k) relation of the Dirac fermions for different values of momentum (Figure 3b). For the anisotropy of this single-Dirac-cone, it can be seen that the maximum and the minimum of v(k) are 1.6·clight/300 (along kx) and 1.1·clight/ 300, (along ky), respectively, which differs by a factor of ∼1.5, where clight is the speed of light. The contribution of the L(1)-point single-Dirac-cone fermions to the transport properties is much greater than that contributed by the parabolically dispersed fermions at the T point. In bulk Bi and Bi1−xSbx at cryogenic temperatures, it has been shown both theoretically and experimentally that the 2024

dx.doi.org/10.1021/nl300064d | Nano Lett. 2012, 12, 2021−2026

Nano Letters

Letter

normal to the bisectrix axis, whereas the L(1)-point E(k) has a larger mini-mass at the apex but a higher anisotropy if the film is grown normal to the trigonal axis. This gives us an idea that a film grown normal to a crystalline direction between the trigonal axis and the bisectrix axis, would have both a small mini-mass and a high anisotropy. As an example, we illustrate in Figure 3e,f a film grown normal to a low-symmetry crystalline direction n̂, where n̂ is in the trigonal-bisectrix plane, making an angle of 14° to the bisectrix axis and 76° to the trigonal axis. In the hexagonal notation system for the rhombohedral Bi1−xSbx, the trigonal axis is [0001] and the bisectrix axis is [101̅0]. Thus, n̂ can be denoted as [606̅1]. Here n̂ is just a randomly chosen crystalline direction within the trigonal-bisectrix plane to be an example, and other crystalline directions can be discussed in a similar way. Parts e and f of Figure 3 illustrate a 300 nm thick film grown normal to n̂, where ky is still along the binary axis, whereas kx is in the direction perpendicular to both the binary (1) axis and n̂. Then EgL for this Dirac cone (Figure 3e) is smaller than 0.46 meV, and the mini-mass at the apex of the Dirac cone is still negligible (∼10−4m0). The anisotropy for this singleDirac-cone (Figure 3e) is much higher. The maximum and minimum values of v(k) for the massive Dirac fermions are 1.65·clight/300 and 0.55·clight/300, respectively, which differ by a factor of ∼3 (Figure 3f). Generally, the anisotropy of the L(1)-point single-Dirac-cone can be controlled by the film growth orientation. For a growth orientation between trigonal-bisectrix plane, the anisotropy degree of the L(1)-point Dirac cone is calculated and plotted in Figure 4, where we define the anisotropy degree as the ratio of the maximum and the minimum values of the v(k) for the massive L(1)-point Dirac fermions. The anisotropy degree of the L(1)-point Dirac fermions can be varied within a wide range from ∼1 to ∼15 according to Figure 4 by growing the film along different crystalline directions. On the other hand, the

trigonal-bisectrix crystalline plane is the mirror plane for the reflection symmetry of the crystal lattice,51 so the L(2) point and the L(3) point are always in mirror symmetry to each other, and hence the bottom of the L(2)- and L(3)-point conduction band edges are degenerate in energy. The difference in energy between the L(1)-point conduction band edge and the L(2)point (L(3)-point) conduction band edge as a function of film growth orientation is also plotted in Figure 4, as a guidance for experiments, which shows how the L(1) point becomes different from the L(2) and L(3) points in energy. The 100 nm thick films and the 50 nm thick films are illustrated as examples. The technology for experimental implementations of the Bi1−xSbx thin films we described above is foreseeable. Single Bi and Bi1−xSbx thin films grown normal to the trigonal axis,26,51 as well as polycrystal Bi thin films grown with preference to a low symmetry direction have been synthesized,52,53 and so have single crystal Bi and Bi1−xSbx nanowires been grown along various crystalline directions.54−63 The cryogenic measurement of transport, optical, and magnetical properties of bulk Bi and Bi1−xSbx have also been developed to a very high level of sophistication over decades of efforts.6,14,18,21,33,34,38 For a single crystalline Bi1−xSbx thin film sample, an anisotropic single-Dirac-cone should be observed. For a mosaic single crystalline Bi1−xSbx thin film sample grown along a lowsymmetry direction, but with disorder in the in-plane direction of grains, the in-plane anisotropy would be sacrificed depending on the degree of the disorder. When the in-plane disorder achieves a total randomness, an isotropic single-Dirac-cone should be observed, which might be interesting for researches related to the phase factor of Dirac fermions. In conclusion, we have developed an iterative-two-dimensional-two-band model to describe a general 2D nonparabolic anisotropic dispersion relation. On the basis of this methodology, we have made a prediction that anisotropic single-Diraccones can be constructed in Bi1−xSbx thin films. Some critical cases of L(1)-point single-Dirac-cones are illustrated as examples. Novel physical phenomena associated with massless and massive Dirac fermions that have been previously reported in other materials systems could hopefully also be observed in Bi1−xSbx thin films. Because the Bi1−xSbx thin film system has special features as we discussed above, we can also expect to observe new physical phenomena that have never been observed in other systems.

■

AUTHOR INFORMATION

Corresponding Author

Figure 4. Anisotropy degree of the L(1)-point single-Dirac-cone vs film growth orientation within the trigonal-bisectrix crystalline plane (blue). The differences in energy between the L(1)-point conduction band edge and the L(2) (L(3))-point conduction band edge ΔE, for 100 nm thick films (green) and 50 nm (red) thick films as a function of film growth orientation are also shown. θ is the angle between the growth orientation and the trigonal axis in the trigonal-bisectrix plane. θ = 0 stands for the trigonal growth, and θ = π/2 stands for the bisectrix growth. When the film growth orientation is in the trigonalbisectrix plane, the L(2) point and the L(3) point are in reflection symmetry and hence degenerate in energy. The green curve and the red curve show how the L(1) point becomes different from the L(2) and L(3) points, when the film growth orientation changes from the trigonal direction to the bisectrix direction within the trigonal-bisectrix plane.

*S.T.: e-mail, [email protected]; telephone, 1-617-253-6860; fax, 1-617-253-6827. M.S.D.: e-mail: [email protected]; telephone, 1-617-253-6864; fax: 1-617-253-6827. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS

We thank Dr. David Hsieh, Dr. Elena Rogacheva, and Dr. Iwao Matsuda for valuable discussions. We acknowledge the support from AFOSR MURI Grant number FA9550-10-1-0533, subaward 60028687. The views expressed are not endorsed by the sponsor. 2025

dx.doi.org/10.1021/nl300064d | Nano Lett. 2012, 12, 2021−2026

Nano Letters

■

Letter

(41) Freeman, W. L.; Gettys, W. E. Phys. Rev. B 1978, 17 (2), 529− 534. (42) Subotowicz, M.; Jałochowski, M.; Mikołajczak, B.; Mikołajczak, P. Phys. Status Solidi (A) 1973, 17 (1), 79−87. (43) Akhtar, S.; Khawaja, E. Phys. Status Solidi (A) 1985, 87 (1), 335−340. (44) Das, V. D.; Soundararajan, N. Phys. Rev. B 1987, 35 (12), 5990. (45) Fuchs, K. The conductivity of thin metallic films according to the electron theory of metals; Cambridge University Press: Cambridge, U.K., 1938; p 100. (46) Sondheimer, E. Adv. Phys. 1952, 1 (1), 1−42. (47) Mallik, R. C.; Das, V. D. Vacuum 2005, 77 (3), 275−285. (48) Mallik, R. C.; Das, V. D. J. Appl. Phys. 2005, 98, 023710. (49) Völklein, F.; Kessler, E. Thin Solid Films 1987, 155 (2), 197− 208. (50) Komnik, Y. F.; Bukhshtab, E.; Nikitin, Y. V. Thin Solid Films 1978, 52 (3), 361−364. (51) Hirahara, T.; Sakamoto, Y.; Saisyu, Y.; Miyazaki, H.; Kimura, S.; Okuda, T.; Matsuda, I.; Murakami, S.; Hasegawa, S. Phys. Rev. B 2010, 81 (16), 165422. (52) Vereecken, P.; Sun, L.; Searson, P. C.; Tanase, M.; Reich, D.; Chien, C. J. Appl. Phys. 2000, 88, 6529. (53) Dauscher, A.; Boffoué, M.; Lenoir, B.; Martin-Lopez, R.; Scherrer, H. Appl. Surf. Sci. 1999, 138, 188−194. (54) Lin, Y. M.; Cronin, S. B.; Ying, J. Y.; Dresselhaus, M.; Heremans, J. P. Appl. Phys. Lett. 2000, 76, 3944. (55) Li, L.; Zhang, Y.; Li, G.; Zhang, L. Chem. Phys. Lett. 2003, 378 (3−4), 244−249. (56) Heremans, J.; Thrush, C. M.; Lin, Y.-M.; Cronin, S.; Zhang, Z.; Dresselhaus, M. S.; Mansfield, J. F. Phys. Rev. B 2000, 61 (4), 2921− 2930. (57) Li, L.; Zhang, Y.; Li, G.; Wang, X.; Zhang, L. Mater. Lett. 2005, 59 (10), 1223−1226. (58) Wang, J.; Wang, X.; Peng, Q.; Li, Y. Inorg. Chem. 2004, 43 (23), 7552−7556. (59) Jin, C. G.; Jiang, G. W.; Liu, W. F.; Cai, W. L.; Yao, L. Z.; Yao, Z.; Li, X. G. J. Mater. Chem. 2003, 13 (7), 1743−1746. (60) Cheng, Y. T.; Weiner, A. M.; Wong, C. A.; Balogh, M. P.; Lukitsch, M. J. Appl. Phys. Lett. 2002, 81, 3248. (61) Shim, W.; Ham, J.; Kim, J.; Lee, W. Appl. Phys. Lett. 2009, 95, 232107. (62) Xu, Y.; Ren, Z.; Ren, W.; Cao, G.; Deng, K.; Zhong, Y. Nanotechnology 2008, 19, 115602. (63) Zhang, Z.; Sun, X.; Dresselhaus, M. S.; Ying, J. Y.; Heremans, J. Phys. Rev. B 2000, 61 (7), 4850−4861.

REFERENCES

(1) Novoselov, K. S.; Geim, A. K.; Morozov, S. V.; Jiang, D.; Katsnelson, M. I.; Grigorieva, I. V.; Dubonos, S. V.; Firsov, A. A. Nature 2005, 438 (7065), 197−200. (2) Zhang, Y.; Tan, Y.-W.; Stormer, H. L.; Kim, P. Nature 2005, 438 (7065), 201−204. (3) Katsnelson, M. I.; Novoselov, K. S.; Geim, A. K. Nat. Phys. 2006, 2 (9), 620−625. (4) Pathak, S.; Shenoy, V. B.; Baskaran, G. Phys. Rev. B 2010, 81 (8), 085431. (5) Gusynin, V.; Sharapov, S.; Carbotte, J. Phys. Rev. Lett. 2006, 96 (25), 256802. (6) Bolotin, K.; Sikes, K.; Jiang, Z.; Klima, M.; Fudenberg, G.; Hone, J.; Kim, P.; Stormer, H. Solid State Commun. 2008, 146 (9−10), 351− 355. (7) Moore, J. E. Nature 2010, 464 (7286), 194−198. (8) Moore, J. Nat. Phys. 2009, 5 (6), 378−380. (9) Fu, L.; Kane, C. L. Phys. Rev. Lett. 2008, 100 (9), 96407. (10) Qi, X. L.; Li, R.; Zang, J.; Zhang, S. C. Science 2009, 323 (5918), 1184. (11) Linder, J.; Tanaka, Y.; Yokoyama, T.; Sudbø, A.; Nagaosa, N. Phys. Rev. Lett. 2010, 104 (6), 67001. (12) Shen, R.; Shao, L. B.; Wang, B.; Xing, D. Y. Phys. Rev. B 2010, 81 (4), 041410. (13) Park, C.; Yang, L.; Son, Y.; Cohen, M.; Louie, S. Nat. Phys. 2008, 4 (3), 213−217. (14) Jain, A. Phys. Rev. 1959, 114 (6), 1518−1528. (15) Tanuma, S. J. Phys. Soc. Jpn. 1961, 16, 2349. (16) Golin, S. Phys. Rev. 1968, 176 (3), 830. (17) Chu, H.; Kao, Y. Phys. Rev. B 1970, 1 (6), 2377−2384. (18) Vecchi, M. P.; Dresselhaus, M. S. Phys. Rev. B 1974, 10 (2), 771. (19) Hiruma, K.; Kido, G.; Miura, N. Solid State Commun. 1981, 38 (9), 859−863. (20) Rogacheva, E. J. Phys. Chem. Solids 2008, 69 (2−3), 580−584. (21) Vecchi, M.; Pereira, J.; Dresselhaus, M. Phys. Rev. B 1976, 14 (2), 298. (22) Chao, P.; Chu, H.; Kao, Y. Phys. Rev. B 1974, 9 (10), 4030. (23) Hsieh, D.; Qian, D.; Wray, L.; Xia, Y.; Hor, Y. S.; Cava, R. J.; Hasan, M. Z. Nature 2008, 452 (7190), 970−974. (24) van Hulst, J. A.; Jaeger, H. M.; Radelaar, S. Phys. Rev. B 1995, 52 (8), 5953−5961. (25) Partin, D.; Heremans, J.; Morelli, D.; Thrush, C.; Olk, C.; Perry, T. Phys. Rev. B 1988, 38 (6), 3818. (26) Rogacheva, E.; Orlova, D. Y. S.; Dresselhaus, M. S.; Tang, S. Size Effect in Bi-Sb Solid Solutions Thin Films; Materials Research Society Symposium Proceeding; 2011. (27) Buot, F. Phys. Rev. A 1973, 8 (3), 1570. (28) Lax, B.; Mavroides, J. G. Solid State Physics; Academic Press: New York, 1960; Vol. 11. (29) Callaway, J. Quantum Theory of the Solid State; Academic Press: San Diego, 1991. (30) Lin, Y.; Rabin, O.; Cronin, S.; Ying, J.; Dresselhaus, M. Appl. Phys. Lett. 2002, 81, 2403. (31) Rabin, O.; Lin, Y.; Dresselhaus, M. Appl. Phys. Lett. 2001, 79, 81. (32) Dinger, R.; Lawson, A. Phys. Rev. B 1973, 7 (12), 5215−5227. (33) Isaacson, R.; Williams, G. Phys. Rev. 1969, 185 (2), 682−688. (34) Datars, W.; Vanderkooy, J. IBM J. Res. Dev. 1964, 8 (3), 247− 252. (35) Brandt, N.; Ponomarev, Y.; Chudinov, S. J. Low Temp. Phys. 1972, 8 (5), 369−420. (36) There are three bisectrix axes associated with the three L points, respectively. In this Letter, we refer to the bisectrix axis associated with the L(1) point. (37) The intrinsic Fermi level is shown in Figure 3c. (38) Hartman, R. Phys. Rev. 1969, 181 (3), 1070. (39) Sandomirskii, V. Sov. J. Exp. Theor. Phys. 1967, 25, 101. (40) Asahi, H.; Humoto, T.; Kawazu, A. Phys. Rev. B 1974, 9 (8), 3347. 2026

dx.doi.org/10.1021/nl300064d | Nano Lett. 2012, 12, 2021−2026