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Dissipative quantum tunneling of defects in a mesoscopic metal
PHYSICA
Physica B 194-196 (1994) 981-982 North-Holland
Dissipative Quantum Tunneling of Defects in a Mesoscopic Metal Norman O. Birge*, K. Chun, G.B. Alers and B. Golding Department of Physics and Astronomy and Center for Fundamental Materials Research, Michigan State University, East Lansing, MI 48824, USA We have studied the dynamics of single bistable defects in sub-micron Bi wires at temperatures 0.1- 2 K. The defect motions can cause large changes in the sample resistance via universal conductance fluctuations. The dynamics are due to the defect particle, coupled to the electron bath, tunneling in a double-well potential with asymmetry s. We clearly observe tunneling rates that increase as the temperature is lowered when kT >> s, as predicted by the theory of dissipative quantum tunneling. Fits of the theory to the data yield a value of the defectelectron bath coupling constant ~x, that is defect-specific.
The quantum-mechanical problem of tunneling in a double-well potential is of great theoretical 1 and experimental interest. 2 The presence of dissipation can have a striking effect on the dynamics of the tunneling particle. In the case of an atom tunneling in a metal, it was first pointed out by Kondo 3 that the tunneling rate can increase with decreasing temperature, due to the non-adiabatic nature of the low-energy electron hole excitations. A striking example of this effect can be observed in the tunneling of a single defect in a metal, studied via the conductance fluctuations of a sub-micron sample. Because the sample is smaller than the phase-breaking length for quantum transport at low temperatures, its conductance is highly sensitive to the positions of the scattering centers. 4 The experiments described here are similar to those reported earlier. 5 We have extended the lower temperature range of the experiments from about 0.5 K to 0.1 K, thus allowing a more detailed quantitative comparison to be made between experiment and theory. The samples studied are thermally-evaporated Bi wires with linewidths of 50100 nm, patterned into 5-terminal devices used in a bridge circuit. At 4.2 K and below, the resistance of a Bi wire often exhibits spontaneous switching between two distinct values, due to the motion of a bistable defect in one of the sample arms. The magnitude of the conductance jump is of order 0.2e2/h at 0.1 K, and is a random function of applied magnetic field6, as expected from theory. 7 The data presented here
were obtained by tuning the magnetic field to obtain the maximum signal, then measuring the transition rates of the defect as a function of temperature. At each temperature, several hundred transitions were recorded and analyzed. Fig. 1 shows the fast and slow transition rates, 7f and 7s, as a function of temperature for a particular defect studied in a magnetic field of 0.14 T. The rates obey the principle of detailed balance over the entire temperature range, i.e. 7s/Tf = e -e/kT, where e/k = 0.23 K is the asymmetry of the double well potential. The data in Fig. 1 exhibit three characteristic temperature regimes. Below 0.3 K, 7f is roughly temperature-independent, while 7s decreases rapidly with decreasing temperature. This is what one would expect from a simple picture
•
1 O0 -
10"1
ii~/l
,
L
'
'
10-1
IIIII
I
I
100
T (K)
Figure 1. Fast and slow transition rates versus temperature for a defect at B = 0.014 T.
0921-4526/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved S S D I 0921-4526(93)E1071-S
982 of spontaneous emission and stimulated absorption for an incoherent two-state system in contact with a thermal reservoir at temperature kT < g. As the temperature is raised above 0.3 K, the rates cross over to a qualitatively different regime. While the ratio of the two rates still obeys detailed balance, the two rates decrease with increasing temperature. This behavior is an essential feature of the dissipative nature of tunneling in metals. Above 1.5 K, the rates increase rapidly, due first to phononassisted tunneling, and eventually to thermal activation over the barrier. We limit our analysis to the temperature range below about 1.5 K. The temperature dependence of the total tunneling rate, 3' = 3'f + 3's, has been calculated in terms of a renormalized tunneling matrix element, Ar, the asymmetry g and the defectelectron bath coupling constant, a: 8
Y
=Ar(27~kT]2~-lcosh(e/2kT)~, . e ,I2 2 ~ bar J ~ ll.C~+l~--~-~)
A fit to the data in Fig. 1 with a = 0.20 is shown as the solid line. The fit shown in Fig. 1 is r61atively insensitive to the precise value of the coupling constant ~. In contrast, Fig. 2 shows data from two defects in different samples over a broad temperature range with kT > e. This is the temperature regime where the rates exhibit the power-law dependence T 2°t'l. (Both data sets have asymmetries ofe/k ~ 100 mK as determined by plots of the ratio of the rates versus 1/T.) Fig. 2 shows the total rate obtained from the power spectrum of the raw data. The data sets in Fig. 2 both exhibit power laws, with values of or of 0.32 and 0.20, respectively. Our results demonstrate the unique ability of this type of experiment to determine the microscopic parameters of individual defects. Such information may be invaluable in studies of disordered systems, where microscopic parameters are characterized by broad distributions. * Supported in part by NSF grant DMR-9023458. REFERENCES 1.
2.
3. 4.
5. 6.
10o~ , 104
, 1O0 T (K)
Figure 2. Total rate, y, for two defects in different samples. Solid lines are fits with a = 0.30 and 0.20 for upper and lower traces, respectively.
7. 8.
"Quantum Tunneling in Condensed Matter", ed. Yu. Kagan and A.J. Leggett, Elsevier, New York 1993. "Quantum Aspects of Molecular Motion in Solids", ed. A. Heidemann et al., Springer, New York 1987. J. Kondo, Physica 124B, 25; 125B, 279; and 126B, 377 (1984). B.L. Artshuler and B.Z. Spivak, Pis'ma Zh. Eksp. Teor. 42, 363 (1985); S. Feng, P.A. Lee and A.D. Stone, Phys. Rev. Lett. 56, 1960 (1986). B. Golding, N.M. Zimmerman and S.N. Coppersmith, Phys. Rev. Lett. 68, 998 (1992). N.M. Zimmerman, B. Golding and W.H. Haemmerle, Phys. Rev. Lett. 67, 1322 (1991). P.A. Lee, A.D. Stone and H. Fukuyama, Phys. Rev. B 35, 1039 (1987) and refs. therein. H. Grabert and U. Weiss, Phys. Rev. Lett. 54, 1605 (1985); M.P.A. Fisher and A.T. Dorsey, Phys. Rev. Lett. 54, 1609 (1985); see also A.J. Leggett et al., Rev. Mod. Phys. 59, 1 (1987).
Physica B 194-196 (1994) 981-982 North-Holland
Dissipative Quantum Tunneling of Defects in a Mesoscopic Metal Norman O. Birge*, K. Chun, G.B. Alers and B. Golding Department of Physics and Astronomy and Center for Fundamental Materials Research, Michigan State University, East Lansing, MI 48824, USA We have studied the dynamics of single bistable defects in sub-micron Bi wires at temperatures 0.1- 2 K. The defect motions can cause large changes in the sample resistance via universal conductance fluctuations. The dynamics are due to the defect particle, coupled to the electron bath, tunneling in a double-well potential with asymmetry s. We clearly observe tunneling rates that increase as the temperature is lowered when kT >> s, as predicted by the theory of dissipative quantum tunneling. Fits of the theory to the data yield a value of the defectelectron bath coupling constant ~x, that is defect-specific.
The quantum-mechanical problem of tunneling in a double-well potential is of great theoretical 1 and experimental interest. 2 The presence of dissipation can have a striking effect on the dynamics of the tunneling particle. In the case of an atom tunneling in a metal, it was first pointed out by Kondo 3 that the tunneling rate can increase with decreasing temperature, due to the non-adiabatic nature of the low-energy electron hole excitations. A striking example of this effect can be observed in the tunneling of a single defect in a metal, studied via the conductance fluctuations of a sub-micron sample. Because the sample is smaller than the phase-breaking length for quantum transport at low temperatures, its conductance is highly sensitive to the positions of the scattering centers. 4 The experiments described here are similar to those reported earlier. 5 We have extended the lower temperature range of the experiments from about 0.5 K to 0.1 K, thus allowing a more detailed quantitative comparison to be made between experiment and theory. The samples studied are thermally-evaporated Bi wires with linewidths of 50100 nm, patterned into 5-terminal devices used in a bridge circuit. At 4.2 K and below, the resistance of a Bi wire often exhibits spontaneous switching between two distinct values, due to the motion of a bistable defect in one of the sample arms. The magnitude of the conductance jump is of order 0.2e2/h at 0.1 K, and is a random function of applied magnetic field6, as expected from theory. 7 The data presented here
were obtained by tuning the magnetic field to obtain the maximum signal, then measuring the transition rates of the defect as a function of temperature. At each temperature, several hundred transitions were recorded and analyzed. Fig. 1 shows the fast and slow transition rates, 7f and 7s, as a function of temperature for a particular defect studied in a magnetic field of 0.14 T. The rates obey the principle of detailed balance over the entire temperature range, i.e. 7s/Tf = e -e/kT, where e/k = 0.23 K is the asymmetry of the double well potential. The data in Fig. 1 exhibit three characteristic temperature regimes. Below 0.3 K, 7f is roughly temperature-independent, while 7s decreases rapidly with decreasing temperature. This is what one would expect from a simple picture
•
1 O0 -
10"1
ii~/l
,
L
'
'
10-1
IIIII
I
I
100
T (K)
Figure 1. Fast and slow transition rates versus temperature for a defect at B = 0.014 T.
0921-4526/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved S S D I 0921-4526(93)E1071-S
982 of spontaneous emission and stimulated absorption for an incoherent two-state system in contact with a thermal reservoir at temperature kT < g. As the temperature is raised above 0.3 K, the rates cross over to a qualitatively different regime. While the ratio of the two rates still obeys detailed balance, the two rates decrease with increasing temperature. This behavior is an essential feature of the dissipative nature of tunneling in metals. Above 1.5 K, the rates increase rapidly, due first to phononassisted tunneling, and eventually to thermal activation over the barrier. We limit our analysis to the temperature range below about 1.5 K. The temperature dependence of the total tunneling rate, 3' = 3'f + 3's, has been calculated in terms of a renormalized tunneling matrix element, Ar, the asymmetry g and the defectelectron bath coupling constant, a: 8
Y
=Ar(27~kT]2~-lcosh(e/2kT)~, . e ,I2 2 ~ bar J ~ ll.C~+l~--~-~)
A fit to the data in Fig. 1 with a = 0.20 is shown as the solid line. The fit shown in Fig. 1 is r61atively insensitive to the precise value of the coupling constant ~. In contrast, Fig. 2 shows data from two defects in different samples over a broad temperature range with kT > e. This is the temperature regime where the rates exhibit the power-law dependence T 2°t'l. (Both data sets have asymmetries ofe/k ~ 100 mK as determined by plots of the ratio of the rates versus 1/T.) Fig. 2 shows the total rate obtained from the power spectrum of the raw data. The data sets in Fig. 2 both exhibit power laws, with values of or of 0.32 and 0.20, respectively. Our results demonstrate the unique ability of this type of experiment to determine the microscopic parameters of individual defects. Such information may be invaluable in studies of disordered systems, where microscopic parameters are characterized by broad distributions. * Supported in part by NSF grant DMR-9023458. REFERENCES 1.
2.
3. 4.
5. 6.
10o~ , 104
, 1O0 T (K)
Figure 2. Total rate, y, for two defects in different samples. Solid lines are fits with a = 0.30 and 0.20 for upper and lower traces, respectively.
7. 8.
"Quantum Tunneling in Condensed Matter", ed. Yu. Kagan and A.J. Leggett, Elsevier, New York 1993. "Quantum Aspects of Molecular Motion in Solids", ed. A. Heidemann et al., Springer, New York 1987. J. Kondo, Physica 124B, 25; 125B, 279; and 126B, 377 (1984). B.L. Artshuler and B.Z. Spivak, Pis'ma Zh. Eksp. Teor. 42, 363 (1985); S. Feng, P.A. Lee and A.D. Stone, Phys. Rev. Lett. 56, 1960 (1986). B. Golding, N.M. Zimmerman and S.N. Coppersmith, Phys. Rev. Lett. 68, 998 (1992). N.M. Zimmerman, B. Golding and W.H. Haemmerle, Phys. Rev. Lett. 67, 1322 (1991). P.A. Lee, A.D. Stone and H. Fukuyama, Phys. Rev. B 35, 1039 (1987) and refs. therein. H. Grabert and U. Weiss, Phys. Rev. Lett. 54, 1605 (1985); M.P.A. Fisher and A.T. Dorsey, Phys. Rev. Lett. 54, 1609 (1985); see also A.J. Leggett et al., Rev. Mod. Phys. 59, 1 (1987).