Microwave transmission spectra in regular and irregular
Ulrich Kuhl · Hans-Jürgen Stöckmann
Hofstadter ButterflyHofstadter calculated the conductance of a twodimensional crystal with a perpendicularly applied magnetic field B [D.R. Hofstadter (1976)]:
|Wave function at
lattice side (n,m):
ψnm = eiakm gn
The gn obey the Harper equation:
(α: number of magnetic flux quanta per unit cell)
- α rational, p/q, => Spectrum splits into q subbands.
- α irrational => Spectrum forms a Cantor set.
- Spectra are self-similar.
- a typically 10-10m => B ≅ 105T needed.
Alternative: superlattices [Schlösser et al. (1996)]
Transfer Matrix AnsatzThe Harper equation can be expressed using a transfer matrix ansatz:
where the transfer matrix Tn is given by
Transfer matrices are multiplicative, :
Only one mode propagates => transfer matrix ansatz:
- time-reversal symmetry => rn=r*n and tn=t*n
- no absorption => |rn|2+|tn| 2=1
kd: phase shift without scatterer
tn, rn: transmission and reflection amplitude
Of course, the transfermatrices Tn are different, but the models have the same properties. Choosing periodic arrangements of tn with different period lengths gives rise to similar patterns as the Hofstadter butterfly.
Single mode propagation in frequency range: 7.5 GHz ≤ ν ≤ 15
=> wavenumber range : 0 ≤ k/(π/d) ≤ 1.8.
The experiment measures the transmission
(i) Every third (a) or every fourth (b)
micrometerscrew is introduced 3mm in the waveguide.
(ii) Micrometerscrew n is introduced 3mm
in the waveguide, if cos(2πnα) > 0.
α = 1/1 (a), 1/2 (b), 1/4 (c), 1/8 (d), 1/16 (e)
Transmission as function of k and α in greyscale printing reproduces the Hofstadter butterfly in every Bloch band:
- Spectra are self-similar.
- Fractal depth of about 3 to 4.
- The low and the high frequency range is not accessible due to damping.
Using only every second micrometerscrew doubles the number of Bloch bands in the accessible frequency range:
- The butterflies in the second and third band are complete.
- Fractal depth is smaller, since only half of the scatterers are used.
Experiment vs. Scattering Theory
- Theory and experiment give the same spectra.
- for micrometerscrew introduced 3 mm.
- Fractal depth is of same order.
(in cooperation with F.M. Izrailev and A.A. Krokhin)
In randomly disordered one-dimensional systems all wavefunctions are localized, therefore no transmission exists. But in systems with correlated disorder localization and mobility edges occur.
The tight binding model
where E is the energy and ε is the site-potential can be
expressed as a quantum model of a linear oscillator with linear
periodic delta-kicks where the amplitude depends on ε.
The inverse localization length is given by where μ is defined by E=2cosμ and . The first term represents pure randomness and the second term gives the correlations.
Choosing appropriate εn, any wanted transmission band structure can be realized [F.M. Izrailev, A.A. Krokhin (1999)].
The theory can be extended to the Kronig-Penney model:
For example, the site potential ε is determined for the distribution of localization lengths where the wavefunctions are delocalized for wavevectors k/(π/d) in the regions 0.3 ... 0.55 and 0.75 ... 0.95 and localized otherwise. The calculated transmission is shown on the right hand side (upper) and the bandstructure is clearly seen. For the complementary distribution of localization lengths the transmission is shown in the lower figure displaying the complementary structure.
ExperimentsThe measured transmission spectra for the two realizations of the distribution of localization lengths are shown above. The depths of the 100 micrometerscrews have been defined by a linear scaling from the site-energies εn from depth 0 to 3 mm.
A similar band structure is measured as it was estimated for the scattering arrangement. The scattering arrangement is shown below.
Using the scattering arrangement for the complementary structure, the transmission bands and gaps are vice versa to the ones shown above.
In comparison a measurement is shown where the depths of the micrometerscrews are random:
Using only 100 scatterers, the calculated band structures can be experimantally realized.
- D.R. Hofstadter, Phys. Rev. B 14, 2239 (1976)
- T. Schlösser et al., Europhys. Lett. 33, 683 (1996)
- F.M. Izrailev, A.A. Krokhin, Phys. Rev. Lett. 82, 4062 (1999)
- U. Kuhl, Dissertation (1998)
- U. Kuhl, H.-J. Stöckmann, Phys. Rev. Lett. 80, 3232 (1998)
- U. Kuhl, F.M. Izrailev, A.A. Krokhin, H.-J. Stöckmann, Appl. Phys. Lett. 77, 633 (2000)
- U. Kuhl, H.-J. Stöckmann, Physica E 9, 384 (2001)