Results and Discussions
5.5 Two-Level System
dominate over the three dimensions electron-electron interaction effect. Also the resistance is also insensitivity to the high magnetic fields.
5.5 Two-Level System
It has been reported that scattering of electrons off two-level system can cause a logarithmic increase. R. W. Cochrane and co-workers(75) calculated a mechanism of structure origin, due to scattering by two-level system. They obtain an analytic expression of the following form:
R = R0− Aln(T2+ ∆2), (5.12)
where ∆ is the energy difference between the two atomic tunneling states and A is a constant depending on the number of contributing sites and the strength of the Coulomb interaction. In contrast to the Kondo effect, it has been reported that the resistance due to two-level system scattering is very insensitivity to the external magnetic field. Recent observations in structurally disordered diamagnet ThAsSe show that the electrical resistance displays a logarithmic correction for a wide range in temperature, which is not affected by strong magnetic field up to 16 T.(62; 68) Our observations of temperature dependent resistances which are shown in Fig. 5.14 and Fig. 5.15 are consistent with both experimental and theoretical reports and also support the possibility of two-level system in our system.
Theoretically, A. Zawadowski and co-workers(48) considered the dephasing in metals by two-level system in two-channel Kondo region. In the two-channel Kondo regime, the single-to-single-particle and single-to-many-particle scattering rates are known to respectively decrease and increase with decreasing temper-ature. The single-to-single-particle scattering rates is proportional to T1/2. It is approaching to 0 as temperature is closing to 0. The key point of dephasing is that the single-to-many-particle scattering would cause dephasing, so one can take the single-to-many-particle scattering time as the total dephasing time a low temperature. The theory predicts that there is a broad peak around Kondo temperature. Adding to a power-law decay due to other sources of dephasing
5.5 Two-Level System
(electron-electron interaction or electron-phonon interaction), the total dephas-ing rates would have a broad shoulder around Kondo temperature that is the same as we observe in the experiments.
Y. M. Galperin(87) calculated the dephasing time using a model based on tunneling states of dynamical structure defeats. He predicts that an inelastic scattering time possessing a very weak temperature dependence in a certain perature interval and then crossing over to a slow increase with decreasing tem-perature. Based on the theory, the dephasing rates is proportional to diffusion constant.
Experimentally, Lin and co-workers measured low temperature electron de-phasing time in a series of highly disordered three-dimension AuPd films.(44) For all films, saturation of dephasing times, τϕ0, are observed below about several K.
The saturating temperature is sample dependent and ranges from 0.005 ns to 0.5 ns. Particularly, the τϕ0 is proportional to diffusion constant. On the other hand, Z. Ovadyahu and co-workers(64;65; 66) measured resistances and electron dephasing times in two dimensional In2O3−x and Au-doped In2O3−x. The resis-tances are logarithmical increase as temperature decreasing. The dephasing times are inversely proportional to temperature, except there is a plateau (weak tem-perature dependent) around several K. Particularly, the plateaus range and width depend on the concentration of doped Au. For an un-doped In2O3−x films, no plateau is observed in the reports. It indicates that the appearance of the plateau comes from the breaking of the structural symmetry, a mimic of two-level system.
Until now, we discussed temperature dependent resistance at several high magnetic fields and low temperature electron inelastic scattering time for a series of films with different levels of disorder. The resistances are logarithmic rise from above 10 K down to 30 mK and the behavior is dimensional independent (independent of thickness of films). Particularly, the temperature dependence is insensitivity to magnetic fields up to 15 T. Many theorists predicted a plateau (weak temperature dependent) of inelastic scattering time, that is the same as our results, around Kondo temperature in two-level system. Based on the above discussions, we strongly convince that instead of the spin-flip (Kondo) effect, which is always inferred to the effect of observed saturation of dephasing time, the two-level system dominates the physical behavior of the system.(88)
5.5 Two-Level System
Recently, Imry and co-workers consider a two-level model with loosely bound heavy impurities. In the tunneling model Imry and co-workers take the scatter to reside in a double-minimum potential. The minima are separated by a vector
~b, the tunneling matrix element between the two minima is Ω0, and their energy separation is 2B. The separation 2∆ between the ground state and excited state in the well, respectively, is given by
2∆ = 2 q
Ω20+ B2. (5.13)
The above labelling of the states reflects their spatial symmetry for B = 0. First, they assume that all of the energy splitting of two local minimum potential are uniform and the inelastic scattering is given by
1
τin = 4(αβ)2nsνFσ0
cosh2(∆/(kBT )), (5.14)
where ns isthe concentration of the soft impurities. α and β are the normalized weights in the two wells. αβ = Ω0/(2∆). The combination 2|αβ| is a symmetry parameter, ranging from unity for a symmetric well (B = 0) to zero for a very asymmetric one.
The parameters of the various two-level system within the system, are often distributed. Reasonable distributions are a uniform distribution for B in the range 0 ≤ B ≤ Bmax, and a 1/Ω0 distribution for Ω0, between Ωmin and Ωmax. The latter distribution follows by taking Ω0 to be the exponential of a large negative, uniformly distributed quantity in the corresponding range. One generally expects Ωmax ¿ Bmax. The combined distribution function reads
P (B, Ω0) = 1
Ω0Bmaxln(Ωmax/Ωmin). (5.15) one averages over the distribution of Eq. 5.15, and the inelastic scattering is given by
τin ∝ e−2Ωmin/(kBT ) f or kBT ¿ Ωmin; τin ∝ T f or Ωmin ¿ kBT ¿ Ωmax;
τin∝ const f or Ωmax ¿ kBT. (5.16)
5.5 Two-Level System
Recently, our results catch some theorists’ attentions. B. D´ora and M. Gul´acsi study a nonuniversal contribution to the dephasing rate of conduction electrons due to local vibrational modes.(89) The inelastic scattering rate exhibits strong oscillations at frequencies comparable to the phonon excitation energy, and then saturates to a finite, coupling dependent value. At the extreme strong coupling limit, close to the complete softening of the phonons, the s-matrix vanishes and the inelastic cross section reaches its maximal value. This phonon mediated scattering mechanism is expected to be rather insensitive to the applied magnetic field, in contrast to Kondo-type impurities, and can contribute to the dephasing time in certain alloys containing dynamical defects.
However, the microscopic parameter, the level of symmetry of two wells, ∆, B, and Ω are difficult to know. The more detail discussions of the measured results need further works of theory.
After comparing with all existence theories, it indicates that our system is dominated only by the dynamic structure defeat daphasing. However, the fabri-cating processes and the microstructures of sample strongly affect the behaviors of the inelastic scattering. Only a few works on the effect, many of ideas are still not clear. For example, the dependence of increasing rate and sheet resis-tance and an explicit way to define the levels of disorder, the number of two-level system. At this field, we have to study more to clear the physics in the future.