- Email: [email protected]
GaAs nanowire superlattices
ARTICLE IN PRESS
Physica E 37 (2007) 204–207 www.elsevier.com/locate/physe
Electronic properties of InAs/GaAs nanowire superlattices Y.M. Niquet De´partement de Recherche Fondamentale sur la Matie`re Condense´e, SP2M/L_Sim, CEA Grenoble, 38054 Grenoble Cedex 9, France Available online 28 August 2006
Abstract The electronic properties of strained InAs/GaAs nanowire superlattices are computed using a semi-empirical sp3 d5 s tight-binding model, taking strains, piezoelectric fields and image charge effects into account. Strain relaxation appears to be efficient in nanowire heterostructures, but is highly inhomogeneous in thin InAs layers. It digs a well in the conduction band that traps the electrons at the surface of the nanowires. This likely decreases the oscillator strength and might ease the capture of the electrons by nearby surface defects. r 2006 Elsevier B.V. All rights reserved. PACS: 73.21.Hb; 78.67.Lt Keywords: Nanowires; Strains; Electronic structure
1. Introduction Semi-conductor nanowires grown from metal catalysts [1] are promising building blocks for one-dimensional physics and nanoelectronics. In particular, the composition of the nanowires can be modulated along their growth axis, which allows the synthesis of nanowire heterostructures and superlattices [2,3]. Various devices such as resonant tunneling diodes, single-electron transistors or memories have already been realized using, for example, InAs/InP nanowire heterostructures [4–6]. The design of quantum well superlattices is limited by the lattice mismatch between the different materials. Small enough nanowire superlattices can, however, relax strains by deforming their surface. The epitaxy of thick lattice mismatched layers has actually been demonstrated in nanowire heterostructures. The effects of strain relaxation on the electronic properties of the nanowires are, nonetheless, little known. Previous work on etched nanowires suggest that the carriers might be localized (in compressed layers) near the surface where strong relaxation takes place [7,8]. This, of course, is not desirable for most experiments and applications. Tel.: +33 4 38 78 43 22; fax: +33 4 38 78 51 97.
E-mail address: [email protected]. 1386-9477/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2006.07.011
In this work, we compute the strain distribution and tight-binding electronic structure of InAs/GaAs nanowire superlattices [9,10]. We show that strain relaxation is indeed efficient, but highly inhomogeneous in thin InAs layers. It indeed digs a well at the surface of the nanowires, that traps the electrons. We introduce the theory and models in Section 2, then discuss the results in Section 3.
2. Theory In this section, we first describe the nanowire structures, then the physics included in the calculation: strains, piezoelectric fields, and image charge effects. We consider cylindrical, ½1 1 1-oriented InAs/GaAs nanowire superlattices (NWSLs) with radius R ¼ 8 nm and period L ’ 48 nm. The thickness of the InAs layers is 4ptp16 nm. The dangling bonds at the surface of the nanowires are saturated with hydrogen atoms. We let ¯ and z ¼ ½1 1 1. x ¼ ½1 1¯ 0, y ¼ ½1 1 2, The lattice mismatch between InAs and GaAs is k ¼ 6:69%. The relaxed atomic positions and actual period L of the superlattice are computed using Keating’s valence force field (VFF) model [11]. The VFF energy depends on the nearest neighbors bond lengths and bond angles through ‘‘bond stretching’’ and ‘‘bond bending’’
ARTICLE IN PRESS Y.M. Niquet / Physica E 37 (2007) 204–207
elastic constants. The latter are fitted to the bulk modulus and Poisson ratio n111 of the materials [12]. Strains displace the anions with respect to the cations, which gives rise to a piezoelectric polarization and potential. The former is proportional to the shear strains in the cubic axis set fx0 ¼ ½1 0 0; y0 ¼ ½0 1 0; z0 ¼ ½0 0 1g: ~ ¼ 2e14 y0 z0 ; x0 z0 ; x0 y0 , P (1) where e14 ¼ 0:045 C=m2 in InAs and e14 ¼ 0:160 C=m2 in ~ is first converted into the equivalent bound GaAs [12]. P ~ Poisson’s equation is then ~P. charge density rp ð~ rÞ ¼ r solved with a finite differences method to find the piezoelectric potential V p ð~ rÞ [13,14]. The one-particle electronic structure of the nanowires must be corrected for image charge effects. An electron added to the NWSL indeed repels nearby valence electrons onto the surface [15]. The interaction between the additional electron and these (negative) surface image charges decreases the affinity and rises the conduction band states. Conversely, the hole added upon ionization of the NWSL gets surrounded by a cloud of valence electrons, leaving positive image charges at the surface. The repulsion between the hole and its image charges also increases the ionization energy and lowers the valence band states. The image charges (also known as self-energy) corrections therefore result in a substantial opening of the one-particle bandgap of the nanowires. We have discussed the underlying physics and theory in Ref. [16]. In a semi-classical approach, the effect of the image charges can be modeled by a local potential SðrÞ, that repels the electrons and holes from the surface, thus enhancing lateral confinement. The one-particle states of the NWSLs are last computed with a first nearest neighbors, two-center orthogonal sp3 d5 s tight-binding (TB) model [17]. We use Boykin’s parametrization taking strains and spin–orbit coupling into account [18]. We look for a few eigenpairs around the bandgap using a Jacobi–Davidson algorithm as discussed in Ref. [16].
205
3. Results In this section, we first discuss strain relaxation, then the piezoelectric potential in the NWSLs; we last analyze their electronic properties. 3.1. Strain relaxation The hydrostatic strain dO=O ¼ xx þ yy þ zz is plotted in Fig. 1 for InAs layers with thickness t ¼ 4 and 16 nm. dO=O is the (first order) variation of the volume of the unit cell with reference to the unstrained material. It would be as large as dO=O ¼ 9:56% in a (hypothetic [19]) InAs/ GaAs quantum well superlattice grown on GaAs. In small enough NWSLs, the InAs layers can relax strains by protruding outwards. Strain relaxation first takes place at the surface of the nanowires, then extends deeper and deeper inside with increasing t. The periphery of the InAs layers is even overrelaxed (dO=O40) to help relieving the core. The expansion of the InAs layers goes with a build up of tensile strains in GaAs, that slowly relax away from the interfaces. The residual strain in the InAs layers can be characterized by the total variation DV =V of their volume with reference to the free-standing case (the average dO=O). It is only 3:84% for t ¼ 4 nm, and drops below 1% for t ¼ 16 nm, showing that the InAs layers are almost completely relaxed. Strain relaxation is very inhomogeneous in the thinnest InAs layers. It mostly takes place in a shell of width t at the surface, leaving the core heavily compressed. Compressive strains, however, shift the conduction band states upwards, while tensile strains shift them downwards. The effects of strains on the conduction band can indeed be approximated by a local potential [20]: DE c ð~ rÞ ¼ a c
dO ð~ rÞ, O
(2)
Fig. 1. The hydrostatic strain dO=O in InAs layers with thickness t ¼ 4 or 16 nm. The dots are the atoms in the ðyzÞ plane of the plot. The vertical, dashdotted lines delimit the InAs layers. The spacing between isolevel curves is 1%, the white one being dO=O ¼ 0.
ARTICLE IN PRESS Y.M. Niquet / Physica E 37 (2007) 204–207
206
where ac ¼ 6:3 eV in the present TB model. The strain profile of Fig. 1 therefore digs a ’ 400 meV deep well at the surface of the nanowire. We will discuss its effect on the electronic structure in paragraph 3.3. 3.2. Piezoelectric potential The piezoelectric potential is plotted in Fig. 2 for an InAs layer with thickness t ¼ 4 nm. In an InAs/GaAs quantum well superlattice, the piezoelectric polarization pffiffiffi ~ ¼ ð2e14 = 3Þðzz k Þ~ would be P z in InAs (and zero in GaAs). The bound charge density rp ð~ rÞ would therefore consist of two uniform sheets of charge located at the interfaces. The piezoelectric field would be E z ¼ 3:7 107 V=m in InAs, resulting in a ’ 150 meV difference of potential across the layer. The piezoelectric field pattern in the NWSL still resembles the one of a (finite) parallel plate capacitor. Indeed, the bound charge density rp ð~ rÞ is mainly distributed at the interfaces. There is, however, a longrange tail of charge on each side in GaAs, that enhances the piezoelectric field in the barriers. The difference of potential across the InAs layer is moreover reduced to ’ 50 meV by strain relaxation. The average piezoelectric field
in the InAs layer further decreases for larger t, being significant only around the interfaces. 3.3. Electronic structure The lowest-lying electron (E1) and hole (H1) wavefunctions are plotted for t ¼ 4 nm in Fig. 3. Their energies (with respect to the valence band edge of bulk GaAs) are EðH1Þ ¼ 0:238 eV and EðE1Þ ¼ 1:141 eV. As discussed in Section 3.1, strain relaxation digs a well in the conduction band at the surface of the nanowire. This well, though partially filled by the image charge potential, traps the electron. The hole remains, on the contrary, well localized around the nanowire axis. The valence band potentials, whose dependence on strains is more intricate [20], are actually much flatter across the nanowire. The hole, being heavier than the electron, is, however, very sensitive to the piezoelectric field (Fig. 2), that confines the H1 wavefunction to the right. Trapping the electron so close to the surface is not very desirable in most experiments and applications. Nearby surface defects might indeed capture the electron more easily. Moreover, the electron and hole wavefunctions show little overlap, which likely decreases the oscillator strength of the H1 ! E1 transition. These consequences of strain relaxation have been anticipated before in etched nanowires [7,8]. The well in the conduction band should, however, become shallow at larger t, as strain relaxation gains the whole InAs layer. The ‘‘hole’’ in the center of the E1 wavefunction actually closes with increasing t and disappears around t ¼ 10 nm (see Fig. 4). The one-particle bandgap energy decreases from E g ¼ 0:903 eV for t ¼ 4 nm to E g ¼ 0:552 eV for t ¼ 16 nm. The conduction band indeed rapidly shifts downward upon strain relaxation. Preliminary calculations show that the electron–hole interaction does not change the whole picture, and decreases the bandgap energy by about 0.1 eV. 4. Conclusion
Fig. 2. The piezoelectric potential V p ð~ rÞ in an InAs layer with thickness t ¼ 4 nm. The spacing between isolevel curves is 10 mV, the white one being V p ð~ rÞ ¼ 0.
We have computed the electronic structure of strained InAs/GaAs nanowire superlattices using a semi-empirical
Fig. 3. The highest hole (H1) and lowest electron (E1) wavefunctions in an InAs layer with thickness t ¼ 4 nm.
ARTICLE IN PRESS Y.M. Niquet / Physica E 37 (2007) 204–207
207
References
Fig. 4. The lowest electron (E1) wavefunction in an InAs layer with thickness t ¼ 16 nm.
sp3 d5 s tight-binding model. We have shown that strain relaxation, although efficient, can dig a well in the conduction band of the thinnest InAs layers, that traps the electrons at the surface of the nanowires. This is not desirable for most applications. This well, however, disappears when the thickness t of the InAs layers is comparable with the radius R of the nanowires. InAs/InP heterostructures should also exhibit such a surface well. It will, however, be more shallow (k ¼ 3:13%) and might trap the electrons only at larger R’s. We stress that strain relaxation would raise an additional barrier (instead of a well) at the surface of layers under tensile strain. The growth of a shell around the nanowire might also homogenize the strain distribution and prevent the formation of a surface well. Acknowledgments This work was supported by the French ‘‘Action Concerte´e Incitative’’ (ACI) ‘‘TransNanofils’’ and by the European Integrated Project (IP) NODE (EU Contract Nr 015783 NODE).
[1] A.M. Morales, C.M. Lieber, Science 279 (1998) 208. [2] Y. Wu, R. Fan, P. Yang, Nanoletters 2 (2002) 83. [3] M.S. Gudiksen, L.J. Lauhon, J. Wang, D.C. Smith, C.M. Lieber, Nature 415 (2002) 617. [4] M.T. Bjo¨rk, B.J. Ohlsson, C. Thelander, A.I. Persson, K. Deppert, L.R. Wallenberg, L. Samuelson, Appl. Phys. Lett. 81 (2002) 4458. [5] C. Thelander, T. Martenson, M.T. Bjo¨rk, B.J. Ohlsson, M.W. Larsson, L.R. Wallenberg, L. Samuelson, Appl. Phys. Lett. 83 (2003) 2052. [6] C. Thelander, H.A. Nilsson, L.E. Jensen, L. Samuelson, Nanoletters 5 (2005) 635. [7] H. Straub, G. Brunthaler, W. Faschinger, G. Bauer, C. Vieu, J. Cryst. Growth 159 (1996) 451. [8] Y.M. Niquet, C. Priester, C. Gourgon, H. Mariette, Phys. Rev. B 57 (1998) 14850. [9] B.J. Ohlsson, M.T. Bjo¨rk, A.I. Persson, C. Thelander, L.R. Wallenberg, M.H. Magnusson, K. Deppert, L. Samuelson, Physica E 13 (2002) 1126. [10] N. Panev, A.I. Persson, N. Sko¨ld, L. Samuelson, Appl. Phys. Lett. 83 (2003) 2238. [11] P.N. Keating, Phys. Rev. 145 (1966) 637. [12] O. Madelung, M. Schulz, H. Weiss, Landolt–Bo¨rnstein Numerical Data and Functional Relationships in Science and Technology, New Series, Group III, vol. 17a, Springer, New York, 1982. [13] S. Salon, M.V.K. Chari, Numerical Methods in Electromagnetism, Academic Press, San Diego, 1999. [14] We used a static dielectric constant k0 ¼ 15 inside the nanowire to compute the piezoelectric potential, and a high-frequency dielectric constant k1 ¼ 12 to compute the image charge potential SðrÞ. [15] C. Delerue, G. Allan, M. Lannoo, Phys. Rev. Lett. 90 (2003) 076803. [16] Y.M. Niquet, A. Lherbier, N.H. Quang, M.V. Ferna´ndez-Serra, X. Blase, C. Delerue, Phys. Rev. B 73 (2006) 165319. [17] J.C. Slater, G.F. Koster, Phys. Rev. 94 (1954) 1498. [18] T.B. Boykin, G. Klimeck, R.C. Bowen, F. Oyafuso, Phys. Rev. B 66 (2002) 125207. [19] The critical thickness for 2D growth is actually very small. We would like to stress that these calculations, although showing strong elastic relaxation, cannot rule out the formation of dislocations or the Stranski–Krastanov like growth of InAs pyramids in the NWSLs. [20] G.L. Bir, G.E. Pikus, Symmetry and Strain-Induced Effects in Semiconductors, Wiley, New York, 1974.
Physica E 37 (2007) 204–207 www.elsevier.com/locate/physe
Electronic properties of InAs/GaAs nanowire superlattices Y.M. Niquet De´partement de Recherche Fondamentale sur la Matie`re Condense´e, SP2M/L_Sim, CEA Grenoble, 38054 Grenoble Cedex 9, France Available online 28 August 2006
Abstract The electronic properties of strained InAs/GaAs nanowire superlattices are computed using a semi-empirical sp3 d5 s tight-binding model, taking strains, piezoelectric fields and image charge effects into account. Strain relaxation appears to be efficient in nanowire heterostructures, but is highly inhomogeneous in thin InAs layers. It digs a well in the conduction band that traps the electrons at the surface of the nanowires. This likely decreases the oscillator strength and might ease the capture of the electrons by nearby surface defects. r 2006 Elsevier B.V. All rights reserved. PACS: 73.21.Hb; 78.67.Lt Keywords: Nanowires; Strains; Electronic structure
1. Introduction Semi-conductor nanowires grown from metal catalysts [1] are promising building blocks for one-dimensional physics and nanoelectronics. In particular, the composition of the nanowires can be modulated along their growth axis, which allows the synthesis of nanowire heterostructures and superlattices [2,3]. Various devices such as resonant tunneling diodes, single-electron transistors or memories have already been realized using, for example, InAs/InP nanowire heterostructures [4–6]. The design of quantum well superlattices is limited by the lattice mismatch between the different materials. Small enough nanowire superlattices can, however, relax strains by deforming their surface. The epitaxy of thick lattice mismatched layers has actually been demonstrated in nanowire heterostructures. The effects of strain relaxation on the electronic properties of the nanowires are, nonetheless, little known. Previous work on etched nanowires suggest that the carriers might be localized (in compressed layers) near the surface where strong relaxation takes place [7,8]. This, of course, is not desirable for most experiments and applications. Tel.: +33 4 38 78 43 22; fax: +33 4 38 78 51 97.
E-mail address: [email protected]. 1386-9477/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2006.07.011
In this work, we compute the strain distribution and tight-binding electronic structure of InAs/GaAs nanowire superlattices [9,10]. We show that strain relaxation is indeed efficient, but highly inhomogeneous in thin InAs layers. It indeed digs a well at the surface of the nanowires, that traps the electrons. We introduce the theory and models in Section 2, then discuss the results in Section 3.
2. Theory In this section, we first describe the nanowire structures, then the physics included in the calculation: strains, piezoelectric fields, and image charge effects. We consider cylindrical, ½1 1 1-oriented InAs/GaAs nanowire superlattices (NWSLs) with radius R ¼ 8 nm and period L ’ 48 nm. The thickness of the InAs layers is 4ptp16 nm. The dangling bonds at the surface of the nanowires are saturated with hydrogen atoms. We let ¯ and z ¼ ½1 1 1. x ¼ ½1 1¯ 0, y ¼ ½1 1 2, The lattice mismatch between InAs and GaAs is k ¼ 6:69%. The relaxed atomic positions and actual period L of the superlattice are computed using Keating’s valence force field (VFF) model [11]. The VFF energy depends on the nearest neighbors bond lengths and bond angles through ‘‘bond stretching’’ and ‘‘bond bending’’
ARTICLE IN PRESS Y.M. Niquet / Physica E 37 (2007) 204–207
elastic constants. The latter are fitted to the bulk modulus and Poisson ratio n111 of the materials [12]. Strains displace the anions with respect to the cations, which gives rise to a piezoelectric polarization and potential. The former is proportional to the shear strains in the cubic axis set fx0 ¼ ½1 0 0; y0 ¼ ½0 1 0; z0 ¼ ½0 0 1g: ~ ¼ 2e14 y0 z0 ; x0 z0 ; x0 y0 , P (1) where e14 ¼ 0:045 C=m2 in InAs and e14 ¼ 0:160 C=m2 in ~ is first converted into the equivalent bound GaAs [12]. P ~ Poisson’s equation is then ~P. charge density rp ð~ rÞ ¼ r solved with a finite differences method to find the piezoelectric potential V p ð~ rÞ [13,14]. The one-particle electronic structure of the nanowires must be corrected for image charge effects. An electron added to the NWSL indeed repels nearby valence electrons onto the surface [15]. The interaction between the additional electron and these (negative) surface image charges decreases the affinity and rises the conduction band states. Conversely, the hole added upon ionization of the NWSL gets surrounded by a cloud of valence electrons, leaving positive image charges at the surface. The repulsion between the hole and its image charges also increases the ionization energy and lowers the valence band states. The image charges (also known as self-energy) corrections therefore result in a substantial opening of the one-particle bandgap of the nanowires. We have discussed the underlying physics and theory in Ref. [16]. In a semi-classical approach, the effect of the image charges can be modeled by a local potential SðrÞ, that repels the electrons and holes from the surface, thus enhancing lateral confinement. The one-particle states of the NWSLs are last computed with a first nearest neighbors, two-center orthogonal sp3 d5 s tight-binding (TB) model [17]. We use Boykin’s parametrization taking strains and spin–orbit coupling into account [18]. We look for a few eigenpairs around the bandgap using a Jacobi–Davidson algorithm as discussed in Ref. [16].
205
3. Results In this section, we first discuss strain relaxation, then the piezoelectric potential in the NWSLs; we last analyze their electronic properties. 3.1. Strain relaxation The hydrostatic strain dO=O ¼ xx þ yy þ zz is plotted in Fig. 1 for InAs layers with thickness t ¼ 4 and 16 nm. dO=O is the (first order) variation of the volume of the unit cell with reference to the unstrained material. It would be as large as dO=O ¼ 9:56% in a (hypothetic [19]) InAs/ GaAs quantum well superlattice grown on GaAs. In small enough NWSLs, the InAs layers can relax strains by protruding outwards. Strain relaxation first takes place at the surface of the nanowires, then extends deeper and deeper inside with increasing t. The periphery of the InAs layers is even overrelaxed (dO=O40) to help relieving the core. The expansion of the InAs layers goes with a build up of tensile strains in GaAs, that slowly relax away from the interfaces. The residual strain in the InAs layers can be characterized by the total variation DV =V of their volume with reference to the free-standing case (the average dO=O). It is only 3:84% for t ¼ 4 nm, and drops below 1% for t ¼ 16 nm, showing that the InAs layers are almost completely relaxed. Strain relaxation is very inhomogeneous in the thinnest InAs layers. It mostly takes place in a shell of width t at the surface, leaving the core heavily compressed. Compressive strains, however, shift the conduction band states upwards, while tensile strains shift them downwards. The effects of strains on the conduction band can indeed be approximated by a local potential [20]: DE c ð~ rÞ ¼ a c
dO ð~ rÞ, O
(2)
Fig. 1. The hydrostatic strain dO=O in InAs layers with thickness t ¼ 4 or 16 nm. The dots are the atoms in the ðyzÞ plane of the plot. The vertical, dashdotted lines delimit the InAs layers. The spacing between isolevel curves is 1%, the white one being dO=O ¼ 0.
ARTICLE IN PRESS Y.M. Niquet / Physica E 37 (2007) 204–207
206
where ac ¼ 6:3 eV in the present TB model. The strain profile of Fig. 1 therefore digs a ’ 400 meV deep well at the surface of the nanowire. We will discuss its effect on the electronic structure in paragraph 3.3. 3.2. Piezoelectric potential The piezoelectric potential is plotted in Fig. 2 for an InAs layer with thickness t ¼ 4 nm. In an InAs/GaAs quantum well superlattice, the piezoelectric polarization pffiffiffi ~ ¼ ð2e14 = 3Þðzz k Þ~ would be P z in InAs (and zero in GaAs). The bound charge density rp ð~ rÞ would therefore consist of two uniform sheets of charge located at the interfaces. The piezoelectric field would be E z ¼ 3:7 107 V=m in InAs, resulting in a ’ 150 meV difference of potential across the layer. The piezoelectric field pattern in the NWSL still resembles the one of a (finite) parallel plate capacitor. Indeed, the bound charge density rp ð~ rÞ is mainly distributed at the interfaces. There is, however, a longrange tail of charge on each side in GaAs, that enhances the piezoelectric field in the barriers. The difference of potential across the InAs layer is moreover reduced to ’ 50 meV by strain relaxation. The average piezoelectric field
in the InAs layer further decreases for larger t, being significant only around the interfaces. 3.3. Electronic structure The lowest-lying electron (E1) and hole (H1) wavefunctions are plotted for t ¼ 4 nm in Fig. 3. Their energies (with respect to the valence band edge of bulk GaAs) are EðH1Þ ¼ 0:238 eV and EðE1Þ ¼ 1:141 eV. As discussed in Section 3.1, strain relaxation digs a well in the conduction band at the surface of the nanowire. This well, though partially filled by the image charge potential, traps the electron. The hole remains, on the contrary, well localized around the nanowire axis. The valence band potentials, whose dependence on strains is more intricate [20], are actually much flatter across the nanowire. The hole, being heavier than the electron, is, however, very sensitive to the piezoelectric field (Fig. 2), that confines the H1 wavefunction to the right. Trapping the electron so close to the surface is not very desirable in most experiments and applications. Nearby surface defects might indeed capture the electron more easily. Moreover, the electron and hole wavefunctions show little overlap, which likely decreases the oscillator strength of the H1 ! E1 transition. These consequences of strain relaxation have been anticipated before in etched nanowires [7,8]. The well in the conduction band should, however, become shallow at larger t, as strain relaxation gains the whole InAs layer. The ‘‘hole’’ in the center of the E1 wavefunction actually closes with increasing t and disappears around t ¼ 10 nm (see Fig. 4). The one-particle bandgap energy decreases from E g ¼ 0:903 eV for t ¼ 4 nm to E g ¼ 0:552 eV for t ¼ 16 nm. The conduction band indeed rapidly shifts downward upon strain relaxation. Preliminary calculations show that the electron–hole interaction does not change the whole picture, and decreases the bandgap energy by about 0.1 eV. 4. Conclusion
Fig. 2. The piezoelectric potential V p ð~ rÞ in an InAs layer with thickness t ¼ 4 nm. The spacing between isolevel curves is 10 mV, the white one being V p ð~ rÞ ¼ 0.
We have computed the electronic structure of strained InAs/GaAs nanowire superlattices using a semi-empirical
Fig. 3. The highest hole (H1) and lowest electron (E1) wavefunctions in an InAs layer with thickness t ¼ 4 nm.
ARTICLE IN PRESS Y.M. Niquet / Physica E 37 (2007) 204–207
207
References
Fig. 4. The lowest electron (E1) wavefunction in an InAs layer with thickness t ¼ 16 nm.
sp3 d5 s tight-binding model. We have shown that strain relaxation, although efficient, can dig a well in the conduction band of the thinnest InAs layers, that traps the electrons at the surface of the nanowires. This is not desirable for most applications. This well, however, disappears when the thickness t of the InAs layers is comparable with the radius R of the nanowires. InAs/InP heterostructures should also exhibit such a surface well. It will, however, be more shallow (k ¼ 3:13%) and might trap the electrons only at larger R’s. We stress that strain relaxation would raise an additional barrier (instead of a well) at the surface of layers under tensile strain. The growth of a shell around the nanowire might also homogenize the strain distribution and prevent the formation of a surface well. Acknowledgments This work was supported by the French ‘‘Action Concerte´e Incitative’’ (ACI) ‘‘TransNanofils’’ and by the European Integrated Project (IP) NODE (EU Contract Nr 015783 NODE).
[1] A.M. Morales, C.M. Lieber, Science 279 (1998) 208. [2] Y. Wu, R. Fan, P. Yang, Nanoletters 2 (2002) 83. [3] M.S. Gudiksen, L.J. Lauhon, J. Wang, D.C. Smith, C.M. Lieber, Nature 415 (2002) 617. [4] M.T. Bjo¨rk, B.J. Ohlsson, C. Thelander, A.I. Persson, K. Deppert, L.R. Wallenberg, L. Samuelson, Appl. Phys. Lett. 81 (2002) 4458. [5] C. Thelander, T. Martenson, M.T. Bjo¨rk, B.J. Ohlsson, M.W. Larsson, L.R. Wallenberg, L. Samuelson, Appl. Phys. Lett. 83 (2003) 2052. [6] C. Thelander, H.A. Nilsson, L.E. Jensen, L. Samuelson, Nanoletters 5 (2005) 635. [7] H. Straub, G. Brunthaler, W. Faschinger, G. Bauer, C. Vieu, J. Cryst. Growth 159 (1996) 451. [8] Y.M. Niquet, C. Priester, C. Gourgon, H. Mariette, Phys. Rev. B 57 (1998) 14850. [9] B.J. Ohlsson, M.T. Bjo¨rk, A.I. Persson, C. Thelander, L.R. Wallenberg, M.H. Magnusson, K. Deppert, L. Samuelson, Physica E 13 (2002) 1126. [10] N. Panev, A.I. Persson, N. Sko¨ld, L. Samuelson, Appl. Phys. Lett. 83 (2003) 2238. [11] P.N. Keating, Phys. Rev. 145 (1966) 637. [12] O. Madelung, M. Schulz, H. Weiss, Landolt–Bo¨rnstein Numerical Data and Functional Relationships in Science and Technology, New Series, Group III, vol. 17a, Springer, New York, 1982. [13] S. Salon, M.V.K. Chari, Numerical Methods in Electromagnetism, Academic Press, San Diego, 1999. [14] We used a static dielectric constant k0 ¼ 15 inside the nanowire to compute the piezoelectric potential, and a high-frequency dielectric constant k1 ¼ 12 to compute the image charge potential SðrÞ. [15] C. Delerue, G. Allan, M. Lannoo, Phys. Rev. Lett. 90 (2003) 076803. [16] Y.M. Niquet, A. Lherbier, N.H. Quang, M.V. Ferna´ndez-Serra, X. Blase, C. Delerue, Phys. Rev. B 73 (2006) 165319. [17] J.C. Slater, G.F. Koster, Phys. Rev. 94 (1954) 1498. [18] T.B. Boykin, G. Klimeck, R.C. Bowen, F. Oyafuso, Phys. Rev. B 66 (2002) 125207. [19] The critical thickness for 2D growth is actually very small. We would like to stress that these calculations, although showing strong elastic relaxation, cannot rule out the formation of dislocations or the Stranski–Krastanov like growth of InAs pyramids in the NWSLs. [20] G.L. Bir, G.E. Pikus, Symmetry and Strain-Induced Effects in Semiconductors, Wiley, New York, 1974.