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Optical absorption in virtual bound state alloys
Volume 40A, number 3
PHYSICS LETTERS
OPTICAL ABSORPTION
IN VIRTUAL
17 July 1972
BOUND STATE ALLOYS
D. BEAGLEHOLE Victoria University of Wellington, New Zealand Received 30 May 1972 We present simple models for the absorption of radiation by conduction electron relaxation and by impurity and host electron excitation. The former can be interpreted in terms of a frequency dependent relaxation time.
Transitional impurities in simple host metals form virtual bound states whose density of states in Friedel's [1 ] and Anderson's [2] models are Lorentzian. The energy of the center of the distribution E d adjust itself self-consistently with respect to the Fermi energy to give the appropriate impurity d-electron occupancy. Absorption of radiation in these alloys takes place by two processes, by relaxtion of the host conduction electrons scattered by the impurity states, and by excitation of impurity electrons to empty conduction states and excitation of host electrons to empty impurity states above the Fermi level. These processes have been described theoretically by Caroli [3] in the Friedel model and by KjOllerstr6m [4] in the Anderson model. A simple picture presented here gives results equivalent to Caroli's, and perhaps provides a more intuitive interpretation o f the experimental resuits. Conduction electron absorption: frequency dependent relaxation time. In the Drude model the conduction electrons are taken to have a single relaxation time r; the frequency dependent conductivity is then given by o(60) = o0/(1 + iwr); o 0 is the dc conductivity, o 0 = ne2r/m. For impurity scattering l / r is proportional to the impurity concentration x, and for the dc conductivity to the density of impurity states at the Fermi level. We assume that 1/r as a function of frequency is proportional to the average of the impurity states over a frequency range +-w about the Fermi level. Taking the impurity density of states as Lorentzian nd(E ) = 10 A/Tr {(E - Ed)2 + A 2} 1/ff60) becomes 1/ff60) = {1/r(0)h60 } {arctan (E F - E d + h60)/A -- arctan (E F - E d - h60)/A}. In the region 60r >> 1 corresponding to Caroli's calculation, this expression for r(60) gives just his o(w). Excitation from and to impurity states. Excitation from an atomic d state (with 1 electron) of energy E d into a free electron continuum [5] gives a contribution to the real part of o(60) 01(60 ) = (e2/lOmn60)~M2k where h60 = ~/2k2 [2m - Ea and k must be greater than k~. M is the grad matrix element between d and continuum states M= f ffd Vr ffk d3r" Assuming that M 2 is constant over energies of the order of E F - E d this expression may be generalised to the case of a Lorentzian distribution of d states o1(6o ) = (xe2M2/31r2mw)k F {arctan(EF _ E d ) / A _ arctan(E F - E d - h60)/A} . If we include also excitation of conduction electron into empty impurity states o 1(60) becomes 209
Volume 40A, number 3
PHYSICS LETTERS
17 July 1972
O1(O0) = (xe 2 M2/37r2mco)k F {arctan(E F - E d +'fiw)/A -- arctan(E F - E d -- ll6o)/A} whose energy dependence is again just that of Caroli's expression. Comparison of absorption processes. The line shapes for these two absorption processes are different, The average over the impurity density of states which enters 1/r(~o) has its maximum value when h~o reaches to the maxim u m of the density of states, and so this process has its largest contribution when hw = E F - E d. On the other hand, the absorption due to impurity absorption continues to grow until all impurity states are participating, i.e. until hw spans the full impurity distribution. Caroli and KjOllerstr6m have estimated that conduction electron relaxation gives the larger absorption. This has been confirmed in recent experiments on dilute CuNi alloys [6, 7].
References [1] [2] [3] [4] [5] [6] [7]
J. Friedel, Suppl. Nuovo Cimento 7 (1958) 287. P.W. Anderson, Phys. Rev. 124 (1961) 41. B. Caroli, Phys. Kondens Materie 1 (1963) 346. B. Kj611erstr6m,Phil. Mag. 19 (1969) 1207. D. Beaglehole, Proc. Phys. Soc. 87 (1966)461. D. Beaglehole and C. Kunz, Ver. Deutschen Physikalischen Gesellschaft 9 (1971) 656. H.D. Drew and R.E. Doezema, to be published.
210
PHYSICS LETTERS
OPTICAL ABSORPTION
IN VIRTUAL
17 July 1972
BOUND STATE ALLOYS
D. BEAGLEHOLE Victoria University of Wellington, New Zealand Received 30 May 1972 We present simple models for the absorption of radiation by conduction electron relaxation and by impurity and host electron excitation. The former can be interpreted in terms of a frequency dependent relaxation time.
Transitional impurities in simple host metals form virtual bound states whose density of states in Friedel's [1 ] and Anderson's [2] models are Lorentzian. The energy of the center of the distribution E d adjust itself self-consistently with respect to the Fermi energy to give the appropriate impurity d-electron occupancy. Absorption of radiation in these alloys takes place by two processes, by relaxtion of the host conduction electrons scattered by the impurity states, and by excitation of impurity electrons to empty conduction states and excitation of host electrons to empty impurity states above the Fermi level. These processes have been described theoretically by Caroli [3] in the Friedel model and by KjOllerstr6m [4] in the Anderson model. A simple picture presented here gives results equivalent to Caroli's, and perhaps provides a more intuitive interpretation o f the experimental resuits. Conduction electron absorption: frequency dependent relaxation time. In the Drude model the conduction electrons are taken to have a single relaxation time r; the frequency dependent conductivity is then given by o(60) = o0/(1 + iwr); o 0 is the dc conductivity, o 0 = ne2r/m. For impurity scattering l / r is proportional to the impurity concentration x, and for the dc conductivity to the density of impurity states at the Fermi level. We assume that 1/r as a function of frequency is proportional to the average of the impurity states over a frequency range +-w about the Fermi level. Taking the impurity density of states as Lorentzian nd(E ) = 10 A/Tr {(E - Ed)2 + A 2} 1/ff60) becomes 1/ff60) = {1/r(0)h60 } {arctan (E F - E d + h60)/A -- arctan (E F - E d - h60)/A}. In the region 60r >> 1 corresponding to Caroli's calculation, this expression for r(60) gives just his o(w). Excitation from and to impurity states. Excitation from an atomic d state (with 1 electron) of energy E d into a free electron continuum [5] gives a contribution to the real part of o(60) 01(60 ) = (e2/lOmn60)~M2k where h60 = ~/2k2 [2m - Ea and k must be greater than k~. M is the grad matrix element between d and continuum states M= f ffd Vr ffk d3r" Assuming that M 2 is constant over energies of the order of E F - E d this expression may be generalised to the case of a Lorentzian distribution of d states o1(6o ) = (xe2M2/31r2mw)k F {arctan(EF _ E d ) / A _ arctan(E F - E d - h60)/A} . If we include also excitation of conduction electron into empty impurity states o 1(60) becomes 209
Volume 40A, number 3
PHYSICS LETTERS
17 July 1972
O1(O0) = (xe 2 M2/37r2mco)k F {arctan(E F - E d +'fiw)/A -- arctan(E F - E d -- ll6o)/A} whose energy dependence is again just that of Caroli's expression. Comparison of absorption processes. The line shapes for these two absorption processes are different, The average over the impurity density of states which enters 1/r(~o) has its maximum value when h~o reaches to the maxim u m of the density of states, and so this process has its largest contribution when hw = E F - E d. On the other hand, the absorption due to impurity absorption continues to grow until all impurity states are participating, i.e. until hw spans the full impurity distribution. Caroli and KjOllerstr6m have estimated that conduction electron relaxation gives the larger absorption. This has been confirmed in recent experiments on dilute CuNi alloys [6, 7].
References [1] [2] [3] [4] [5] [6] [7]
J. Friedel, Suppl. Nuovo Cimento 7 (1958) 287. P.W. Anderson, Phys. Rev. 124 (1961) 41. B. Caroli, Phys. Kondens Materie 1 (1963) 346. B. Kj611erstr6m,Phil. Mag. 19 (1969) 1207. D. Beaglehole, Proc. Phys. Soc. 87 (1966)461. D. Beaglehole and C. Kunz, Ver. Deutschen Physikalischen Gesellschaft 9 (1971) 656. H.D. Drew and R.E. Doezema, to be published.
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