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Resonance tunneling through a single interstitial Mn impurity in a GaAs quantum well
ARTICLE IN PRESS
Physica E 40 (2008) 1665–1667 www.elsevier.com/locate/physe
Resonance tunneling through a single interstitial Mn impurity in a GaAs quantum well P. Dahan Ruppin Academic Center, School of Engineering, Emek-Hefer 40250, Israel Available online 13 October 2007
Abstract The electronic structure of the impurity d-levels of interstitial Mni in a GaAs quantum well (QW) is found to create deeply bound donor states located within the band gap. This behavior results from lowering the symmetry of the band states in 2DEG, which, in turn, lifts symmetry bans on the hybridization matrix elements. An impurity-assisted tunneling current is enabled by discrete bound donor states with energies located below the first conduction QW subband. r 2007 Elsevier B.V. All rights reserved. PACS: 71.55.i; 73.21.b; 85.30.Mn Keywords: Magnetic impurity; Two dimensional electron gas; Resonance tunneling
1. Introduction In bulk GaAs, the interstitial site, Mni , is known to act as a double donor [1,2], as is expected in saturated bonds in which the 4s electron transfers directly to the d-shell: A0 ð4s2 3dn Þ ! A0 ð3dnþ2 Þ [3]. The (0=þ) and (þ=2þ) transitions are calculated to lie within the GaAs conduction band and it is shown that Mni at the tetrahedral, T d , interstitial site is coordinated by As and exhibits a single charge state Mn2þ for all values of the Fermi energy [2]. In this paper, we will show the different electronic structures of Mni in a GaAs quantum well (QW), compared with bulk GaAs. The differences are related to the changes in the d-level properties that occur when the T d symmetry in QW is lowered to a tetragonal, D2d symmetry. As a result, a resonance tunneling current through a deeply bound donor state of interstitial Mn is expected. 2. Model
e E iG ¼ eð6Þ G þ MðE iG Þ,
The calculation for the Mni in a GaAs QW begins with the impurity Hamiltonian, H e ¼ H 0 þ V d ðr R0 Þ þ U 0 ðr R0 Þ,
where H 0 is the Hamiltonian of an electron that is confined in the z direction by the potential V ðzÞ; U 0 ðr R0 Þ is the lattice crystal field potential acting on the impurity at site R0 , and V d ðrÞ is the impurity potential. According to the resonance scattering model for transition metal impurities [4–6], Hamiltonian (1) is solved using the basis fe jl;j ; jdGm g, which includes: (i) the atomic d orbital, jdGm , which retains its 3D character because its radius, rd , is small when compared with the width of the well, V ðzÞ; and (ii) the wave e l;j , orthogonalized to jdGm . functions of the QW states, j Here, l akk and j are the quantum numbers that describe the finite electron motion in the xy-plane of the conduction and valence bands, a, and in the confining potential V ðzÞ, respectively. It follows from Hamiltonian (1) that the energies of the dð6Þ shell electron, E iG ðþ=2þÞ, are then determined by the following equation:
(1)
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(2)
1 6 5 e where, eð6Þ G ¼ Eðd Þ Eðd Þ, M g ðEÞ ¼ VG0 Q ðEÞV is the self-energy part, where V is the hybridization matrix element, Q ¼ 1 UG0 ; G0 ¼ ð1 E H0 Þ1 , U is the scattering potential and g ¼ e; t2 is the irreducible representation of the crystalline point group T d .
ARTICLE IN PRESS P. Dahan / Physica E 40 (2008) 1665–1667
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The electronic structure of Mni in GaAs is calculated and it is shown that the ð0=þÞ and ðþ=2þÞ transitions lie within the conduction band with the down-spin t2 # levels [2]. This is the typical behavior of 3d impurities in zincblend crystals in which the states in the lower part of the conduction band do not influence the behavior of localized states [7,8] since their hybridization with the impurity dstate is symmetry-forbidden. As a result, the hybridization matrix elements V Gm;c ðkÞ ¼ hdGmjU 0 jckik for small k. In a QW, the situation of the same level is quite different: The envelope function, wj ðzÞ, within the integral lifts the selection rules for the hybridization integrals V Gm;cj ðkk Þ ¼ hdGmjU 0 jlji, since this axially symmetric function contains all spherical harmonics, Y l0 . It is instructive to review the symmetries of the states, as follows: Accounting the electron spin of the conduction band states in the T d group, the 1 G1 conduction band becomes 2 G6 . The resulting bound d-electron states with crystal field splitting transform according to irreducible representations G3 and G5 . However, when accounting for the spin–orbit interaction in the T d group (see, e.g., Ref. [7]), G3 becomes G8 and G5 states split into G8 þ G7 states. After lowering the symmetry, the newly split representations 2 G6 þ2 G7 of the tetragonal D2d replace the 4 G8 of T d . As a result, the wave function of the sixth electron in the d shell of the charged MnðþÞ ion, which transforms according to representation i 2 G6 , hybridizes strongly with conduction band states so that the matrix elements, V G6 ;cj , are constant at kk ! 0. The nonzero value of V G6 ;cj ðkk ¼ 0Þ in a QW leads to a e g ðE iG Þ. Then, retain the logarithmic singularity of M 6 conduction band contribution, Eq. (2) takes the form jV G6 ;c j2 E iG6 þ D1 1 ð6Þ ln (3) E iG6 eG Q ðEÞ ¼ 0, E iG6 D1 2D1 where 2D1 is the first subband width of the 2D constant density of states. Unlike the bulk case [2], solutions to our Eq. (3) can be both of the deep levels in the forbidden band, E ie and E it2 , and the so-called crystal field resonances in the conduction band (Fig. 1). To estimate the energy position of E ie and EiΓ − dΓ Eie
dt2 de
Eit2
M(EiΓ)
Fig. 1. Graphical solution of Eq. (3) for the t2 and e levels (the dashed line represents the bottom and top edges of the first conduction QW subband). The atomic levels, g , are shifted below the first conduction subband, creating deeply bound donor states at E it2 ¼ 160 meV and E ie ¼ 120 meV.
E it2 , we start with the prime energy ed within the conduction band, with crystal field splitting DCF ¼ 0:4 eV, V G6 ;c ¼ 0:8 eV, scattering potential U ¼ 0:5 eV and D1 ¼ 1 eV. Hence the two lowest solutions are located below the first conduction subband edge, at E it2 ¼ 160 meV and E ie ¼ 120 meV (Fig. 1). Dimension lowering also significantly influences the renormalized crystal field splitting, DCF ¼ E it2 E ie , which is quenched so as to become DCF ¼ 40 meV. We choose these reasonable values to fit the donor binding energy and crystal field splitting of a similar electronic structure that was first revealed for a donor Mni in a GaMnAs=GaAs p2i2n diode using resonant magneto-tunneling spectroscopy [9]. In bulk semiconductors, the main contribution to the tail part of the wave function is usually an antibonding combination from the heavy-hole band, which has relatively little weight compared with that of the atomic d function. The constant value of V G6 ;c ðkk ¼ 0Þ in the QW case results in the divergence of the function MðE iG6 Þ and e 0 ðE iG ! E c1 Þ ! 1 in a singularity of its derivative M cG6 6 near the conduction subband edge. As a result, the din a QW can be electron wave function of the MnðþÞ i represented as a bonding combination: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi e0 e 0 cs , c ¼ 1= 1 þ M þ M cð6Þ (4) dG6 cG cG iG 6
6
6
which is significantly more extended and has a different symmetry than that of the bulk case. This wave function, Eq. (4), transforms according to the irreducible representation G6 . Using the Bardeen approach [10], the impurity-assisted tunneling current is IG /
X n;G
e0 M 2 cG 0 j Fn;G ðkk Þj re ðE iG ÞdðE n E iG Þ, e 1þM cG
where Fn;G ðkk Þ is the Fourier transform of the impurity wave function cð6Þ iG6 , re ðE iG Þ is the occupancy of a state in the emitter with energy E iG , and E n is the energy of the nth emitter state. Only the tail part of Eq. (4), which is an extended function, is considered here since its contribution to the overlap integrals with the 2D emitter plane wave function, jekk i, is dominant. Since the donor bound states energies, E iG , are located below the first 2D subband, the electron energy in the emitter becomes aligned with E iG at a reverse-bias voltage between the emitter and the impurity within the QW. Due to the relatively small crystal field splitting, the radial part of the tail function, Eq. (4), is quite similar for both G6 ðt2 Þ and G6 ðeÞ states. Their contribution is, however, significantly different since the value of e 0 ðE iG Þ is a very sensitive function with respect to the M energy position near the QW conduction subband. Since e 0 ðE iG Þ4M e 0 ðE iG Þ, and the E iG6 ðt2 Þ lies below E iG6 ðeÞ, M e t2 difference between them is significant since both states are very close to the divergence point. The probability density of the wave function is, therefore, different for the two
ARTICLE IN PRESS P. Dahan / Physica E 40 (2008) 1665–1667
donor states, and the relative contribution of the core and of the tail parts can be measured using magneto-tunneling spectroscopy. It is therefore expected that the contribution to the tunneling current amplitude of the tail part of the E iG6 ðeÞ state be greater than that of the E iG6 ðt2 Þ state. In conclusion, this paper shows that the electronic structure of the d-levels of an interstitial manganese impurity, Mni , in a GaAs QW is different from that of the bulk case. The differences are related to the properties of the d-level and to the crystalline symmetry, whereby the tetrahedral, T d symmetry in QW is lowered to a tetragonal, D2d symmetry. As a result, the symmetry bans on the hybridization matrix elements are lifted, leading to a strong hybridization of the d-states with the states at the bottom of the conduction QW subband. Hence, the Mni introduces its localized donor states E ie and E it2 into the energy gap with rather loosely binding energy and significant quenching crystal field splitting. The localized donor state is described by a bonding superposition with higher lying QW conduction subband states, which form a dominant extended Bloch tail of the s-function. This electronic structure supports an impurity-assisted tunneling current enabled by deeply bound donor states with energy located below the first conduction QW subband. A similar
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electronic structure was revealed for the first time for a donor Mni in a GaMnAs=GaAs p2i2n diode using resonant magneto-tunneling spectroscopy [9]. Our theory may explain these observations as well.
References [1] S.C. Erwin, A.G. Petukhov, Phys. Rev. Lett. 89 (2002) 227201; J. Blinovski, P. Kacman, Phys. Rev. B 67 (2003) 121204; F. Maca, J. Masek, Phys. Rev. B 65 (2002) 235209. [2] P. Mahadevan, A. Zunger, Phys. Rev. B 68 (2003) 075202. [3] G.W. Ludwig, H.M. Woodbury, in: D. Turnbull, F. Seitz (Eds.), Solid State Physics, vol. 13, Academic Press, New York, 1962, pp. 223–286. [4] F.D. Haldan, P.W. Andersonl, Phys. Rev. B 13 (1976) 2553. [5] V.N. Fleurov, K.A. Kikoinl, J. Phys. C: Solid State Phys. 9 (1976) 1673. [6] G. Picoli, A. Chomette, M. Lannoo, Phys. Rev. B 30 (1984) 7138. [7] K.A. Kikoin, V.N. Fleurov, Transition Metal Impurities in Semiconductors, Singapore, World Scientific, 1994. [8] A. Zunger, in: F. Seitz, D. Turnbull, (Eds.), Solid State Physics, vol. 39, Academic Press, New York, 1986, p. 275. [9] O. Makarovsky, et al., Presented in HMF-17 Wurzburg, Germany, 2006. [10] T.M. Fromhold, F.W. Sheard, L. Eaves, Acta Phys. Polon. A 82 (1992) 737.
Physica E 40 (2008) 1665–1667 www.elsevier.com/locate/physe
Resonance tunneling through a single interstitial Mn impurity in a GaAs quantum well P. Dahan Ruppin Academic Center, School of Engineering, Emek-Hefer 40250, Israel Available online 13 October 2007
Abstract The electronic structure of the impurity d-levels of interstitial Mni in a GaAs quantum well (QW) is found to create deeply bound donor states located within the band gap. This behavior results from lowering the symmetry of the band states in 2DEG, which, in turn, lifts symmetry bans on the hybridization matrix elements. An impurity-assisted tunneling current is enabled by discrete bound donor states with energies located below the first conduction QW subband. r 2007 Elsevier B.V. All rights reserved. PACS: 71.55.i; 73.21.b; 85.30.Mn Keywords: Magnetic impurity; Two dimensional electron gas; Resonance tunneling
1. Introduction In bulk GaAs, the interstitial site, Mni , is known to act as a double donor [1,2], as is expected in saturated bonds in which the 4s electron transfers directly to the d-shell: A0 ð4s2 3dn Þ ! A0 ð3dnþ2 Þ [3]. The (0=þ) and (þ=2þ) transitions are calculated to lie within the GaAs conduction band and it is shown that Mni at the tetrahedral, T d , interstitial site is coordinated by As and exhibits a single charge state Mn2þ for all values of the Fermi energy [2]. In this paper, we will show the different electronic structures of Mni in a GaAs quantum well (QW), compared with bulk GaAs. The differences are related to the changes in the d-level properties that occur when the T d symmetry in QW is lowered to a tetragonal, D2d symmetry. As a result, a resonance tunneling current through a deeply bound donor state of interstitial Mn is expected. 2. Model
e E iG ¼ eð6Þ G þ MðE iG Þ,
The calculation for the Mni in a GaAs QW begins with the impurity Hamiltonian, H e ¼ H 0 þ V d ðr R0 Þ þ U 0 ðr R0 Þ,
where H 0 is the Hamiltonian of an electron that is confined in the z direction by the potential V ðzÞ; U 0 ðr R0 Þ is the lattice crystal field potential acting on the impurity at site R0 , and V d ðrÞ is the impurity potential. According to the resonance scattering model for transition metal impurities [4–6], Hamiltonian (1) is solved using the basis fe jl;j ; jdGm g, which includes: (i) the atomic d orbital, jdGm , which retains its 3D character because its radius, rd , is small when compared with the width of the well, V ðzÞ; and (ii) the wave e l;j , orthogonalized to jdGm . functions of the QW states, j Here, l akk and j are the quantum numbers that describe the finite electron motion in the xy-plane of the conduction and valence bands, a, and in the confining potential V ðzÞ, respectively. It follows from Hamiltonian (1) that the energies of the dð6Þ shell electron, E iG ðþ=2þÞ, are then determined by the following equation:
(1)
Tel.: +972 9 8981377; fax: +972 9 8981302.
E-mail address: [email protected] 1386-9477/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2007.10.023
(2)
1 6 5 e where, eð6Þ G ¼ Eðd Þ Eðd Þ, M g ðEÞ ¼ VG0 Q ðEÞV is the self-energy part, where V is the hybridization matrix element, Q ¼ 1 UG0 ; G0 ¼ ð1 E H0 Þ1 , U is the scattering potential and g ¼ e; t2 is the irreducible representation of the crystalline point group T d .
ARTICLE IN PRESS P. Dahan / Physica E 40 (2008) 1665–1667
1666
The electronic structure of Mni in GaAs is calculated and it is shown that the ð0=þÞ and ðþ=2þÞ transitions lie within the conduction band with the down-spin t2 # levels [2]. This is the typical behavior of 3d impurities in zincblend crystals in which the states in the lower part of the conduction band do not influence the behavior of localized states [7,8] since their hybridization with the impurity dstate is symmetry-forbidden. As a result, the hybridization matrix elements V Gm;c ðkÞ ¼ hdGmjU 0 jckik for small k. In a QW, the situation of the same level is quite different: The envelope function, wj ðzÞ, within the integral lifts the selection rules for the hybridization integrals V Gm;cj ðkk Þ ¼ hdGmjU 0 jlji, since this axially symmetric function contains all spherical harmonics, Y l0 . It is instructive to review the symmetries of the states, as follows: Accounting the electron spin of the conduction band states in the T d group, the 1 G1 conduction band becomes 2 G6 . The resulting bound d-electron states with crystal field splitting transform according to irreducible representations G3 and G5 . However, when accounting for the spin–orbit interaction in the T d group (see, e.g., Ref. [7]), G3 becomes G8 and G5 states split into G8 þ G7 states. After lowering the symmetry, the newly split representations 2 G6 þ2 G7 of the tetragonal D2d replace the 4 G8 of T d . As a result, the wave function of the sixth electron in the d shell of the charged MnðþÞ ion, which transforms according to representation i 2 G6 , hybridizes strongly with conduction band states so that the matrix elements, V G6 ;cj , are constant at kk ! 0. The nonzero value of V G6 ;cj ðkk ¼ 0Þ in a QW leads to a e g ðE iG Þ. Then, retain the logarithmic singularity of M 6 conduction band contribution, Eq. (2) takes the form jV G6 ;c j2 E iG6 þ D1 1 ð6Þ ln (3) E iG6 eG Q ðEÞ ¼ 0, E iG6 D1 2D1 where 2D1 is the first subband width of the 2D constant density of states. Unlike the bulk case [2], solutions to our Eq. (3) can be both of the deep levels in the forbidden band, E ie and E it2 , and the so-called crystal field resonances in the conduction band (Fig. 1). To estimate the energy position of E ie and EiΓ − dΓ Eie
dt2 de
Eit2
M(EiΓ)
Fig. 1. Graphical solution of Eq. (3) for the t2 and e levels (the dashed line represents the bottom and top edges of the first conduction QW subband). The atomic levels, g , are shifted below the first conduction subband, creating deeply bound donor states at E it2 ¼ 160 meV and E ie ¼ 120 meV.
E it2 , we start with the prime energy ed within the conduction band, with crystal field splitting DCF ¼ 0:4 eV, V G6 ;c ¼ 0:8 eV, scattering potential U ¼ 0:5 eV and D1 ¼ 1 eV. Hence the two lowest solutions are located below the first conduction subband edge, at E it2 ¼ 160 meV and E ie ¼ 120 meV (Fig. 1). Dimension lowering also significantly influences the renormalized crystal field splitting, DCF ¼ E it2 E ie , which is quenched so as to become DCF ¼ 40 meV. We choose these reasonable values to fit the donor binding energy and crystal field splitting of a similar electronic structure that was first revealed for a donor Mni in a GaMnAs=GaAs p2i2n diode using resonant magneto-tunneling spectroscopy [9]. In bulk semiconductors, the main contribution to the tail part of the wave function is usually an antibonding combination from the heavy-hole band, which has relatively little weight compared with that of the atomic d function. The constant value of V G6 ;c ðkk ¼ 0Þ in the QW case results in the divergence of the function MðE iG6 Þ and e 0 ðE iG ! E c1 Þ ! 1 in a singularity of its derivative M cG6 6 near the conduction subband edge. As a result, the din a QW can be electron wave function of the MnðþÞ i represented as a bonding combination: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi e0 e 0 cs , c ¼ 1= 1 þ M þ M cð6Þ (4) dG6 cG cG iG 6
6
6
which is significantly more extended and has a different symmetry than that of the bulk case. This wave function, Eq. (4), transforms according to the irreducible representation G6 . Using the Bardeen approach [10], the impurity-assisted tunneling current is IG /
X n;G
e0 M 2 cG 0 j Fn;G ðkk Þj re ðE iG ÞdðE n E iG Þ, e 1þM cG
where Fn;G ðkk Þ is the Fourier transform of the impurity wave function cð6Þ iG6 , re ðE iG Þ is the occupancy of a state in the emitter with energy E iG , and E n is the energy of the nth emitter state. Only the tail part of Eq. (4), which is an extended function, is considered here since its contribution to the overlap integrals with the 2D emitter plane wave function, jekk i, is dominant. Since the donor bound states energies, E iG , are located below the first 2D subband, the electron energy in the emitter becomes aligned with E iG at a reverse-bias voltage between the emitter and the impurity within the QW. Due to the relatively small crystal field splitting, the radial part of the tail function, Eq. (4), is quite similar for both G6 ðt2 Þ and G6 ðeÞ states. Their contribution is, however, significantly different since the value of e 0 ðE iG Þ is a very sensitive function with respect to the M energy position near the QW conduction subband. Since e 0 ðE iG Þ4M e 0 ðE iG Þ, and the E iG6 ðt2 Þ lies below E iG6 ðeÞ, M e t2 difference between them is significant since both states are very close to the divergence point. The probability density of the wave function is, therefore, different for the two
ARTICLE IN PRESS P. Dahan / Physica E 40 (2008) 1665–1667
donor states, and the relative contribution of the core and of the tail parts can be measured using magneto-tunneling spectroscopy. It is therefore expected that the contribution to the tunneling current amplitude of the tail part of the E iG6 ðeÞ state be greater than that of the E iG6 ðt2 Þ state. In conclusion, this paper shows that the electronic structure of the d-levels of an interstitial manganese impurity, Mni , in a GaAs QW is different from that of the bulk case. The differences are related to the properties of the d-level and to the crystalline symmetry, whereby the tetrahedral, T d symmetry in QW is lowered to a tetragonal, D2d symmetry. As a result, the symmetry bans on the hybridization matrix elements are lifted, leading to a strong hybridization of the d-states with the states at the bottom of the conduction QW subband. Hence, the Mni introduces its localized donor states E ie and E it2 into the energy gap with rather loosely binding energy and significant quenching crystal field splitting. The localized donor state is described by a bonding superposition with higher lying QW conduction subband states, which form a dominant extended Bloch tail of the s-function. This electronic structure supports an impurity-assisted tunneling current enabled by deeply bound donor states with energy located below the first conduction QW subband. A similar
1667
electronic structure was revealed for the first time for a donor Mni in a GaMnAs=GaAs p2i2n diode using resonant magneto-tunneling spectroscopy [9]. Our theory may explain these observations as well.
References [1] S.C. Erwin, A.G. Petukhov, Phys. Rev. Lett. 89 (2002) 227201; J. Blinovski, P. Kacman, Phys. Rev. B 67 (2003) 121204; F. Maca, J. Masek, Phys. Rev. B 65 (2002) 235209. [2] P. Mahadevan, A. Zunger, Phys. Rev. B 68 (2003) 075202. [3] G.W. Ludwig, H.M. Woodbury, in: D. Turnbull, F. Seitz (Eds.), Solid State Physics, vol. 13, Academic Press, New York, 1962, pp. 223–286. [4] F.D. Haldan, P.W. Andersonl, Phys. Rev. B 13 (1976) 2553. [5] V.N. Fleurov, K.A. Kikoinl, J. Phys. C: Solid State Phys. 9 (1976) 1673. [6] G. Picoli, A. Chomette, M. Lannoo, Phys. Rev. B 30 (1984) 7138. [7] K.A. Kikoin, V.N. Fleurov, Transition Metal Impurities in Semiconductors, Singapore, World Scientific, 1994. [8] A. Zunger, in: F. Seitz, D. Turnbull, (Eds.), Solid State Physics, vol. 39, Academic Press, New York, 1986, p. 275. [9] O. Makarovsky, et al., Presented in HMF-17 Wurzburg, Germany, 2006. [10] T.M. Fromhold, F.W. Sheard, L. Eaves, Acta Phys. Polon. A 82 (1992) 737.