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Impurity configurations in metals
Volume 41A, number I
PHYSICS LETTERS
28 August 1972
IMPURITY CONFIGURATIONS IN METALS* C.P. FLYNN Department of Physics and Materials Research Laboratory, University ofIllinois, Urbana, Illinois 61801, USA Received 24 June 1972 The existence of fully ionic and partly ionic impurity configurations in metals is discussed. The continuous range of impurity configurations is indicated and the structural variation with solvent electron gas density clarified for certain impurities.
Most investigations of impurity structure in metals have focussed on cases in which the solvent valence orbitals penetrate the impurity core region to neutralize the defect. However, chemists have for sometime recognized that halides may enter alkali metals as ions neutralized by repelled electrons [1] and strong evidenee for similar ionic effects associated with chalcogens has recently been reported [2, 3]. In this letter we outline the more general nature of ionic effects in impurity structure, and identify the probable extent to which such configurations occur in practical cases, We first study the limit in which the electron gas plays a minor role. Consider an impurity in a dilute monovalent lattice having a large interatomic spacing a. Whether or not a falls beyond the Mott transition the host band states lie approximately at the free atom levels. The energy required to transfer one electron from a neighbouring atom to the impurity is E 1 = I— A e2~a / —
with I the host ionization potential and A the impurity affinity. For I 4.5 eV and A 3.5 eV the transition from an atomic to an ionic impurity configuration lies at e2/a 1 eV, or a 25 a.u. This applied, for example, to the elements I(A = 3.3 eV) and Te(A = 3.6 eV) in K(I = 4.3 eV) but not to Xe for which A —1 eV. The degree to which further ionization steps take place presents more subtle problems. Free doubly charged ions are universally unstable in vacuo and multiple affinities derived from studies of cohesion in salts pertain only to tightly bound orbitals. Provided that a variety of ionicities can, in principle, *
This work was supported in part by the Advanced Research Projects Agency under Contract HC 15-67-C-0221.
be stable in metals the charge state of lowest total energy will, of course, occur in nature. We believe that various ionicities can occur in metals, as in salts [4]. Confusion over this point relates to an oversimplification of one-electron theory. For example, P impurities in Ge clearly introduce holes into all the Ge core bands despite the fact that the core bands lie below a full valence band. Similarly, Tm metal may have only twelfe 4f orbitals occupied below the conduction band. In the same way 0 could, in principle, enter Na with only five 2p orbitals bound below the band bottom, and with two valence electrons repelled to ensure electrical neutrality (although this paramagnetic configuration probably is not the ground state in practice). The important point is that the atomic 0 and the ionic 0= configurations in the metal could conceivably have larger total energies than that of the 0~-metalcomplex. Thus, to assess the degree to which impurity ionization takes place one must in general compare a variety of alternative ionic configurations. With increasing host electron density the valence levels of all impurities must eventually rise into the conduction band first as virtual levels and finally as very broad band states. This happens because the kinetic energy of band states near EF finally overwhelms the impurity potential. The spectrum of impurity configurations begun above and completed by these cases is shown in fig. 1. With increasing electron density an impurity can, for favorable I-A, undergo a transition from the atomic to the first ionic configuration (a and b), and may eventually even complete a full valence shell to attain its fully ionized configuration (c). Whether or not step (c) occurs, the occupied core levels rise into the band with increasing electron density and the virtual levels (d) finally broaden (e) in45
Volume 41.4, number I
PHYSICS LETTERS
28 August 1972
77
r1~
2
~/
~
‘I
~
Te~jSn
~‘:
C
__________________________ _____________________
E Fig. 1. The p wave phase shift for various configurations of Te in metals: (a) atomic; (b) ionic; (c) fully ionic; (d) virtual; (e) broadened band configuration. In (a) and (b) there are respectively two and one p orbitals with i~= 0. The quantity 1 shows the “ionicity” employed for virtual orbitals in fig. 2. EF
to weakly perturbed band states. The degree of ionicity established in steps (b) to (d) depends on details of the host metal and impurity atom structures. Because the preceding arguments are expressed in terms of specific configurations there are two qualifying points to bear in mind. First, the true wave function of the impurity-metal complex must be derived from an interaction among configurations, and is resolved into the simple picture only when the alternative configurations are well spaced in total energy. The results of the interaction cannot be depicted in the one-electron scheme of fig. 1. Second, the distinction between bound and virtual levels loses clarity when electron interactions are superposed on the one-electron scheme [5].No abrupt changes of physical properties are expected in the transition even in oneelectron theory [6], and for interacting electrons the concept of a transition is itself blurred. Fig. 2 provides an assessment of the probable configurations of In, Sn, Sb, Te, I and Xe in monovalent metals as a function of host electron density “e~Xe almost certainly remains atomic while I most probably remains ionic through most of the range shown. It has been observed in detailed susceptibility studies [7] that Sn undergoes a transition from band states to at ~e 1.5 X 1022 A~3,and the same transitions for Te and Sb apparently occur in the range 2.3 x 1022
021
n
Cs____ ~In 1
0223)
/Sn
5Sb \\ ~Te “
\
______
Na
Ag
Xe
023
Fig. 2. The scheme of ionlclty ~e I as(cm a function of host electron density for In, Sn, Sb, Te, I and Xe in monovalent metals. The ionicities of virtual levels (see caption to fig. 1) are indicated by broken lines.
metals [7]. The observed affinities, together with a typical ionization energy, permit rough estimates of ~ for the first ionization. Furthermore, the observed susceptibilities in alkali metals host [2, 3, 71 allow us to make reasonable estimates of the ionicity for ~e in the range of metallic densities. The bound to virtual transition (solid to broken lines in fig. 2) is estimated from the apparently rapid lowering of the energy of Sn orbitals with decreasing 12e in NaK [71,and from theoretical studies [3], but has no sharp significance. Available information makes the broad scheme of the structural properties indicated at least qualitatively reliable. It is not at present clear whether alternative ionicities represent long-lives excited states analogous to core holes in metals. If they exist as discrete excitations these levels should cause tubstantial effects in the impurity excitation spectrum. A search for effects of this nature has been undertaken in our laboratories. References Ill See the review by M.A. Bredig in Molten Salt Chemistry, ed. M. Rlander (lnterscience, New York, 1964) [2] J.A. Rigert and C.P. Flynn, Phys. Rev. Letters 26 (1971) 1177, and to be published. 13] C.P. Flynn and NO. Lipari, Phys. Rev. Letters 27 (1971) 1365, and to be published [4] For a discussion of impurity structure in salts see C.P. Flynn, Point Defects and Diffusion (Oxford 1972) p. 579. [5] N.F. Mott, J. Phys. Rad. 23(1962)594. [6] W. Kohn and C. Majumdar, Phys. Rev. A128 (1965) 1617.
Ill
M.D. Mikolosko, J.A. Rigert and C.P. Flynn, Physics Letters 38.4 (1972) 69.
PHYSICS LETTERS
28 August 1972
IMPURITY CONFIGURATIONS IN METALS* C.P. FLYNN Department of Physics and Materials Research Laboratory, University ofIllinois, Urbana, Illinois 61801, USA Received 24 June 1972 The existence of fully ionic and partly ionic impurity configurations in metals is discussed. The continuous range of impurity configurations is indicated and the structural variation with solvent electron gas density clarified for certain impurities.
Most investigations of impurity structure in metals have focussed on cases in which the solvent valence orbitals penetrate the impurity core region to neutralize the defect. However, chemists have for sometime recognized that halides may enter alkali metals as ions neutralized by repelled electrons [1] and strong evidenee for similar ionic effects associated with chalcogens has recently been reported [2, 3]. In this letter we outline the more general nature of ionic effects in impurity structure, and identify the probable extent to which such configurations occur in practical cases, We first study the limit in which the electron gas plays a minor role. Consider an impurity in a dilute monovalent lattice having a large interatomic spacing a. Whether or not a falls beyond the Mott transition the host band states lie approximately at the free atom levels. The energy required to transfer one electron from a neighbouring atom to the impurity is E 1 = I— A e2~a / —
with I the host ionization potential and A the impurity affinity. For I 4.5 eV and A 3.5 eV the transition from an atomic to an ionic impurity configuration lies at e2/a 1 eV, or a 25 a.u. This applied, for example, to the elements I(A = 3.3 eV) and Te(A = 3.6 eV) in K(I = 4.3 eV) but not to Xe for which A —1 eV. The degree to which further ionization steps take place presents more subtle problems. Free doubly charged ions are universally unstable in vacuo and multiple affinities derived from studies of cohesion in salts pertain only to tightly bound orbitals. Provided that a variety of ionicities can, in principle, *
This work was supported in part by the Advanced Research Projects Agency under Contract HC 15-67-C-0221.
be stable in metals the charge state of lowest total energy will, of course, occur in nature. We believe that various ionicities can occur in metals, as in salts [4]. Confusion over this point relates to an oversimplification of one-electron theory. For example, P impurities in Ge clearly introduce holes into all the Ge core bands despite the fact that the core bands lie below a full valence band. Similarly, Tm metal may have only twelfe 4f orbitals occupied below the conduction band. In the same way 0 could, in principle, enter Na with only five 2p orbitals bound below the band bottom, and with two valence electrons repelled to ensure electrical neutrality (although this paramagnetic configuration probably is not the ground state in practice). The important point is that the atomic 0 and the ionic 0= configurations in the metal could conceivably have larger total energies than that of the 0~-metalcomplex. Thus, to assess the degree to which impurity ionization takes place one must in general compare a variety of alternative ionic configurations. With increasing host electron density the valence levels of all impurities must eventually rise into the conduction band first as virtual levels and finally as very broad band states. This happens because the kinetic energy of band states near EF finally overwhelms the impurity potential. The spectrum of impurity configurations begun above and completed by these cases is shown in fig. 1. With increasing electron density an impurity can, for favorable I-A, undergo a transition from the atomic to the first ionic configuration (a and b), and may eventually even complete a full valence shell to attain its fully ionized configuration (c). Whether or not step (c) occurs, the occupied core levels rise into the band with increasing electron density and the virtual levels (d) finally broaden (e) in45
Volume 41.4, number I
PHYSICS LETTERS
28 August 1972
77
r1~
2
~/
~
‘I
~
Te~jSn
~‘:
C
__________________________ _____________________
E Fig. 1. The p wave phase shift for various configurations of Te in metals: (a) atomic; (b) ionic; (c) fully ionic; (d) virtual; (e) broadened band configuration. In (a) and (b) there are respectively two and one p orbitals with i~= 0. The quantity 1 shows the “ionicity” employed for virtual orbitals in fig. 2. EF
to weakly perturbed band states. The degree of ionicity established in steps (b) to (d) depends on details of the host metal and impurity atom structures. Because the preceding arguments are expressed in terms of specific configurations there are two qualifying points to bear in mind. First, the true wave function of the impurity-metal complex must be derived from an interaction among configurations, and is resolved into the simple picture only when the alternative configurations are well spaced in total energy. The results of the interaction cannot be depicted in the one-electron scheme of fig. 1. Second, the distinction between bound and virtual levels loses clarity when electron interactions are superposed on the one-electron scheme [5].No abrupt changes of physical properties are expected in the transition even in oneelectron theory [6], and for interacting electrons the concept of a transition is itself blurred. Fig. 2 provides an assessment of the probable configurations of In, Sn, Sb, Te, I and Xe in monovalent metals as a function of host electron density “e~Xe almost certainly remains atomic while I most probably remains ionic through most of the range shown. It has been observed in detailed susceptibility studies [7] that Sn undergoes a transition from band states to at ~e 1.5 X 1022 A~3,and the same transitions for Te and Sb apparently occur in the range 2.3 x 1022
021
n
Cs____ ~In 1
0223)
/Sn
5Sb \\ ~Te “
\
______
Na
Ag
Xe
023
Fig. 2. The scheme of ionlclty ~e I as(cm a function of host electron density for In, Sn, Sb, Te, I and Xe in monovalent metals. The ionicities of virtual levels (see caption to fig. 1) are indicated by broken lines.
metals [7]. The observed affinities, together with a typical ionization energy, permit rough estimates of ~ for the first ionization. Furthermore, the observed susceptibilities in alkali metals host [2, 3, 71 allow us to make reasonable estimates of the ionicity for ~e in the range of metallic densities. The bound to virtual transition (solid to broken lines in fig. 2) is estimated from the apparently rapid lowering of the energy of Sn orbitals with decreasing 12e in NaK [71,and from theoretical studies [3], but has no sharp significance. Available information makes the broad scheme of the structural properties indicated at least qualitatively reliable. It is not at present clear whether alternative ionicities represent long-lives excited states analogous to core holes in metals. If they exist as discrete excitations these levels should cause tubstantial effects in the impurity excitation spectrum. A search for effects of this nature has been undertaken in our laboratories. References Ill See the review by M.A. Bredig in Molten Salt Chemistry, ed. M. Rlander (lnterscience, New York, 1964) [2] J.A. Rigert and C.P. Flynn, Phys. Rev. Letters 26 (1971) 1177, and to be published. 13] C.P. Flynn and NO. Lipari, Phys. Rev. Letters 27 (1971) 1365, and to be published [4] For a discussion of impurity structure in salts see C.P. Flynn, Point Defects and Diffusion (Oxford 1972) p. 579. [5] N.F. Mott, J. Phys. Rad. 23(1962)594. [6] W. Kohn and C. Majumdar, Phys. Rev. A128 (1965) 1617.
Ill
M.D. Mikolosko, J.A. Rigert and C.P. Flynn, Physics Letters 38.4 (1972) 69.