Đang tải... (xem toàn văn)
Đây là một bài báo khoa học về dây nano silic trong lĩnh vực nghiên cứu công nghệ nano dành cho những người nghiên cứu sâu về vật lý và khoa học vật liệu.Tài liệu có thể dùng tham khảo cho sinh viên các nghành vật lý và công nghệ có đam mê về khoa học
Physica E 32 (2006) 341–345 Nature of the band gap of silicon and germanium nanowires Clive Harris, E.P. O’Reilly à Tyndall National Institute, Lee Maltings, Cork, Ireland Available online 8 February 2006 Abstract Nanowires of both Si and Ge have been predicted to have band gaps that are either direct or indirect depending upon the crystallographic direction along which the nanowire is oriented. We use a sp 3 d 5 s à tight binding model to calculate the band structures for both Ge and Si nanowires oriented along the (1 0 0), (1 1 0) and (1 1 1) directions. We show that the nature of the band gap and the variation of the zone centre band gap with nanowire width depends upon the nanowire stacking direction for both Si and Ge nanowires. We then show, by considering bulk unit cells along the (1 0 0), (1 1 0) and (1 1 1) directions, that it is possible to accurately predict whether a nanowire stacked in the same direction as one of these bulk unit cells has a direct or indirect band structure. r 2006 Elsevier B.V. All rights reserved. PACS: 73.21.Hb; 73.20.At Keywords: Nanowires; Band structure; Silicon; Germanium 1. Introduction A path towards the integration of optically and electrically active material was opened when nanostruc- tured silicon (Si) was found to exhibit strong photolumi- nescence [1]. The integration of optically active Si with electrically active Si would allow both optical and electrical processes to occur on the same chip and would further open the mature Si processing industry to optical devices [2]. From a compatibility point of view, germanium (Ge) is also a promising candidate for direct integration with existing Si technology. Noting this, the possibility of integrating optically active nanostructured Ge [3] with Si technology could also be important. The optical properties of both Si and Ge nanowires have been experimentally studied [4,5]. Differing luminescence properties have been observed for Si nanowires grown along different crystallographic axes [4]. From a theoretical perspective, the band structure of Si nanowires has been more widely studied [6,7] than that of Ge [8]. These theoretical studies showed that the nanowire energy gap could be either direct or indirect, depending upon the crystallographic orientation of the nanowire. We use an sp 3 d 5 s à tight–binding (TB) model here to study the electronic structure of crystalline Ge and Si nanowires, varying in width from $10 ˚ A to over 60 A ˚ .We examine the variation of the direct band gap with width, as well as the nature of the band structure of the nanowires. We confirm previous observations ab out the direct or indirect nature of the nanowires, while also extending these previous observations in Ge [8] to larger nanowire sizes. We explain the nature of the bandgap for both materials, using an argument based upon the bulk band structure of these materials projected onto the crystallographic axes, (1 0 0), (1 1 0), (1 1 1). We confirm that an effective mass approximation (EMA) gives a good description of the conduction band (CB) energy for Si nanowires down to $20 ˚ A in size [9] but show that the CB EMA breaks down for Ge nanowires that are $50 ˚ A. 2. Nanowire electronic structure We present here the electronic structure for both Si and Ge nanowires, oriented along the (1 0 0), (1 1 0) and (1 1 1) directions, calculated using a sp 3 d 5 s à TB Hamiltonian [10]. We add hydrogen (H) atoms to the nanowire surfaces to ARTICLE IN PRESS www.elsevier.com/locate/physe 1386-9477/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2005.12.094 à Corresponding author. E-mail address: eoreilly@tyndall.ie (E.P. O’Reilly). maintain the original tetrahedral bond angles and coordi- nation of the bulk material. We construct all of the nanowires with a square cro ss-section. The TB parameters for Si–Si and Ge–Ge interactions are taken from [11]. The valence band (VB) maximum for bulk Ge and Si is rescaled to be at 0 eV. The Si–H and the Ge–H TB parameters were calculated from fits to SiH 4 and GeH 4 molecules, with the bond lengths taken to be 1.48 A ˚ and 1.52 A ˚ , respectively. Fig. 1 shows the variation with nanowire size of the band gap at the Brilluoin zone centre for nanowires stacked along each of the three crystallographic directions. The band gap increases with decreasing nanowire width, and different direct energy gaps are found for differently oriented nanowires of sim ilar width, in agreement with previous calculations [8,12]. We find for the zone centre energy gap in Ge that E 100 g 4E 111 g 4E 110 g , whereas in Si E 100 g 4E 111 g ’ E 110 g . As the nanowire width D increases, the difference between the Ge (1 0 0) and Ge (1 1 0) energy gap remains almost constant (up to at least 60 A ˚ ). The difference between the Si (1 0 0) and Si (1 1 0) energy gap decreases with D. The band gap for Si nanowires oriented along all three directions is converging towards the G2X bulk band gap. The Ge nanowires converge towards different band gaps: (1 1 0) towards the G 2L gap, whereas (1 0 0) and (1 1 1) nanowires converge towards the G2X value. Figs. 2 and 3 show the band dispersion along the nanowire direction for Si and Ge nanowires with D$20 ˚ A. The Si nanowires maintain a direct band gap, while the Ge nanowires are only direct when stacked along (1 1 0). This is true for all nanowire widths considered here. This is likely to limit the use of ideal Ge nanowires for the efficient production of light. Overall, the effect of the internal stacking of the nanowire is two-fold. It influences the size ARTICLE IN PRESS 1 2 3 4 Energy/eV (100) (110) (111) 0 102030405060 Width/Å Width/Å 0 102030405060 1 2 3 4 Energy/eV (100) (110) (111) (a) (b) Fig. 1. Variation of zone centre band gap with respect to nanowire width for nanowires oriented along (1 0 0), (1 1 0) and (1 1 1) in (a) Si and (b) Ge. Increasing K Increasing K Increasing K 0 1 Energy/eV -1 2 0 1 -1 2 0 1 -1 2 Fig. 2. Band structure of (1 0 0), (1 1 0) and (1 1 1) Si nanowires with D$20 ˚ A. C. Harris, E.P. O’Reilly / Physica E 32 (2006) 341–345342 of the band gap and determines the overall nature of the nanowire band structure. 3. Effective mass approximation The CB e nergy of a nanowire o f cro ss-sectional area D  D can be written using an effective mass approximation as E c ¼ E c0 þ h 2 8D 2 1 m à x þ 1 m à y ! , (1) where m à x and m à y are the effective masses in the x and y directions within the nanowire, with the nanowire axis set along z. Taking the bulk unit cells in Section 4 to represent each nanowire direction, the effective mass values can be readily calculated for each of the indivi dual nanowire stacking directions. From these, the EMA is found to predict the CB energy for Si nanowires down to approximately. 20–30 A ˚ in width as previously seen [9].In the Ge case the EMA is found to break down at much larger nanowire widths. The EMA method fails to predict the CB energy for Ge nanowires under D$50 ˚ A (see Fig. 4). The low value of the transverse effective mass (m à t ) leads the EMA to predict a rapid increase of the CB energy with decreasing nanowire size. However interactions with higher conduction bands push the CB edge down, leading to the breakdown of the EMA. 4. Dependence of nanowire band structure on bulk band structure We show here how the direct or indirect nature of the nanowire band gap can be predicted from the bulk band structure. ARTICLE IN PRESS Increasing K Increasing KIncreasing K 0 -1 1 Energy/eV 2 0 -1 1 2 0 -1 1 2 Fig. 3. Band structure of (1 0 0), (1 1 0) and (1 1 1) Ge nanowires with D$20 ˚ A. Ge (110) Ge (111) Width/Å Energy/eV Si (100) Si (110) Ge (100) Si (111) 5 4 3 2 1 4 3 2 1 4 3 2 1 4 3 2 1 Energy/eV 5 4 3 2 1 Energy/eV 5 4 3 2 1 10 20 30 40 50 60 70 80 90 Width/Å 10 20 30 40 50 60 70 80 90 Fig. 4. Comparison of the zone centre CB energy calculated using the EMA (solid line) and a sp 3 d 5 s à TB model (dotted line). C. Harris, E.P. O’Reilly / Physica E 32 (2006) 341–345 343 The bulk diamond crystal structure can also be represented by unit cells oriented in the (1 0 0), (1 1 0) and (1 1 1) directions. These are constructed by finding the minimum repeat lattice vector a 1 in each direction which is formed from a combination of the primitive lattice basis vectors a 1 , a 2 , a 3 , and then finding two lattice vectors ða 2 ; a 3 Þ perpendicular to a 1 which are also constructed from a 1 , a 2 and a 3 . The bulk unit cell vectors for the three directions of interest here are shown in Table 1, with the unit cell size a set equal to 1. An ideal (1 0 0) nanowire is constructed by repeating layers of (1 0 0)-oriented unit cells. Fig. 5 below shows the reduced Brillouin zone for a (1 0 0) supercell. The bulk states in the plane A provide a complete basis from which to construct the zone centre nanowire states, while the bulk states on plane B provide a complete basis from which to construct the states at the nanowire zone edge. Therefore, if the CB minimum in plane A is at a lower energy than the CB minimum in plane B we can expect the nanowire CB minimum to be at the centre (G point) of the nanowire 1D Brillouin zone. Similarly, if the CB minimum for plane B is lower than the CB minimum for plane A, the nanowire would be indirect in nature. Although only illustrated here for (1 0 0)-oriented nanowires, this argument is also valid for the other two stacking directions which we are considering. Examining the band structure of (1 0 0), (1 1 0) and (1 1 1) bulk unit cells, we can then use the above argument to determine whether the overall energy gap is direct or indirect. Table 2 shows the distribution of the G, X and L states between the A and B planes in the (1 0 0), (1 1 0) and (1 1 1) unit cells. In the (1 0 0) and (1 1 1) bulk uni t cells, the X ðLÞ points are clearly on the A (B) plane. This explains why the Si (1 0 0) and (1 1 1) nanowires are direct in nature, whereas the Ge (1 0 0) and (1 1 1) nanowires have an indirect gap. For the (1 1 0) case, effective mass arguments are required to explain the nature of the energy gap in the associated nanowire. In bulk material the CB minimum for Si lies along the six equivalent X directions, whereas the Ge CB minimum lies at the four L points within the primitive Brilluoin zone. For the (1 1 0) case, it can be seen that the X and L symmetry points lie on both the A and B planes. So it is necessary to assess on which plane the L states are more tightly confined, and hence at a lower energy. States with higher effe ctive masses are more tightly confined. We calculate for bulk Ge a longitudinal mass of m à l ¼ 1 : 6m 0 along G2L near the L point and a much smaller transverse mass m à t ¼ 0 : 1m 0 , similar to typical calculated values [11]. When we calculate the effective mass around the L states on the A plane in Ge, we find that the effective mass varies from 0:1m 0 to 1:6m 0 , whereas the B plane mass varies from 0.1m 0 to 0.25m 0 . We therefore conclude that the L states on the A plane in the (1 1 0) Ge nanowire should be at a lower energy than those on the B plane. Hence, the CB minimum in Ge (1 1 0) nanowires will be at G. Using a similar analysis, we find that the X states on the A plane in a Si (1 1 0) nanowire are more tightly confined than those on the B plane, so that the CB minimum in Si (1 1 0) nanowires will also be at G. 5. Conclusion We have presented the band structure of Ge and Si nanowires oriented along the (1 0 0), (1 1 0) and (1 1 1) crystallographic directions. We find that the zone centre band gap in all three types of Si nanowire converges towards the G2X bulk band gap, whereas the zone centre band gaps for (1 0 0) and (1 1 0) Ge nanowires clearly converge to different bulk band gap values. The (1 0 0) gap converges towards the G2X band gap, while the (1 1 0) nanowire converges towards the narrower G2L band gap. ARTICLE IN PRESS Table 1 Lattice vectors (a 0 s) and reciprocal lattice vectors (b 0 s) for the bulk unit cells (a) (1 0 0), (b) (1 1 0) and (c) (1 1 1) (1 0 0) (1 1 0) (1 1 1) a 1 (1,0,0) (1/2,1/2,0) (1,1,1) a 2 (0,1,0) ð1=2; À1=2; 0Þð1=2; 0; À1=2Þ a 3 (0,0,1) (0,0,1) ð0; 1=2; À1=2Þ b 1 2p/a(1,0,0) 2p/a(1,1,0) 2p/að1=3; 1=3; 1=3Þ b 2 2p/a(0,1,0) 2p/að1; À1; 0Þ 2p/að4=3; À2=3; À2=3Þ b 3 2p/a(0,0,1) 2p/a(0,0,1) 2p/aðÀ2=3; 4=3; À2=3Þ (001) (010) B A (100) Γ X Fig. 5. Representation of the (1 0 0) Brillouin zone for a bulk, with the gamma point marked and the zone edge along (1 0 0) marked as X, two planes A and B are also shown. Table 2 The plane on which the G, X and L primitive unit cell points lie in the 3 cells (1 0 0), (1 1 0) and (1 1 1) Bulk (1 0 0) (1 1 0) (1 1 1) G AAA X AABA L BABB C. Harris, E.P. O’Reilly / Physica E 32 (2006) 341–345344 This can be clearly understood based on the conduction band level ordering in Si and Ge. The full band structures therefore shows Si nanowires to be direct while Ge nanowires are only direct when stacked along the (1 1 0) direction. We conclude that it should be possible to predict the nature of the band gap of a nanowire oriented along any other direction by also investigating the band dispersion in a bulk unit cell oriented along the nanowire direction. Acknowledgments We thank Science Foundation Ireland for funding this work, and Simon Elliott, Justin Holmes and Guillaume Audoit for useful discussions. References [1] L.T. Canham, Appl. Phys. Lett. 57 (1990) 1046. [2] H. Rong, et al., Nature 03273 (2005). [3] M. Zacharias, P.M. Fauchet, Appl. Phys. Lett. 71 (1997) 380. [4] J.D. Holmes, et al., Science 287 (2000) 1471. [5] N.R.B. Coleman, et al., Chem. Phys. Lett. 343 (2001) 1–6. [6] U. Schmid, N.E. Christensen, M. Alouani, M. Cardona, Phys. Rev. B 43 (1991) 14597. [7] M.S. Hybertsen, M. Schluter, Phys. Rev. B 36 (1987) 9683. [8] A.N. Kholod, et al., Phys. Rev. B 70 (2004) 035317. [9] C Y. Yeh, S.B. Zhang, A. Zunger, Phys. Rev. B 50 (1994) 14405. [10] J.M. Jancu, et al., Phys. Rev. B 57 (1998) 6493. [11] T.B. Boykin, G. Klimeck, F. Oyafuso, Phys. Rev. B 69 (2004) 115201. [12] G.D. Sanders, Y C. Chang, Phys. Rev. B 45 (1992) 9202. ARTICLE IN PRESS C. Harris, E.P. O’Reilly / Physica E 32 (2006) 341–345 345 . 1 0) and (1 1 1) directions. We show that the nature of the band gap and the variation of the zone centre band gap with nanowire width depends upon the. the variation of the direct band gap with width, as well as the nature of the band structure of the nanowires. We confirm previous observations ab out the