3.4 The Loschmidt echo and spin relaxation
3.4.4 Loschmidt echo in the Monte Carlo simulations
Our proposal to separate the dephasing and relaxation processes in the above described problem of spin relaxation, essentially mimics the magnetic resonance approach. In this approach, the Loschmidt echo can be induced by i) letting the
evolution evolution p/2 pulse
NMR signal
after t echo
"recovery time"
after p
pulse after t
echo
"evolution time"
t=0 M
0
Rotating frame of reference
after p/2
pulse
p pulse
0.0 0.5 1.0 1.5 2.0
-1.0 -0.5 0.0 0.5 1.0
time (in units of t echo
)
Figure 3.18: Schematics of the dephasing process in NMR experiments and the method of spin-echo. The figure assumes a right handed precession direction withωL. The spin magnetization lies in the (x0, y0) plane after a π/2 pulse when dephasing due to a spread in the Larmor frequencies starts: spins in blue and red precess faster or slower than ωL, respectively. After an evolution time of τecho, a π pulse is applied which rotates the spins around an axis perpendicular to z0. Clearly, the blue and red spins are now behind or before the average spin direction and as a result these will be aligned coherently after another τecho time, when the spin echo occurs. The lower panel depicts the corresponding NMR signal.
0 1 2 3 0.0
0.5 1.0
InvertedNMRSpinEcho(arb.u.)
time (in units of T 2
) echo envelope
exp(- t/T 2
)
~T 2
*
-50 0 50
0.0 0.5 1.0
NMR signal
1/T 2
*
FT-NMRsignal
NMR frequency (in units of 1/ T 2
) spin-packet
1/T 2
Figure 3.19: Schematics of theT2 measurement. NMR spin echo experiments are performed with varying time delay between theπ/2and π pulses. The individual echoes have a linewidth of2T2∗ but the resulting spin echo envelope followse−t/T2. The corresponding FT NMR signal reflects this behavior: it contains a broad signal whose width is 1/T2∗ and it consists of individual spin-packets whose width is 1/T2.
Figure 3.20: Example of Loschmidt echo simulations with the method explained in the text. tflip indicates the time point of SOC reversal. All simulations were run withΓ = 0.01L. The Fourier transformed spin decay is also shown (lower panel).
Note that the sidelobes, which are present in the original dephasing problem (black curve), are absent for the echo envelope (red curve).
system evolve under the simultaneous action of dephasing and spin relaxation for a given time, t, ii) inverting the built-in SOC fields: Ω(k)→ −Ω(k) and iii) detecting the amplitude of Loschmidt echo maximum at2t. The spin-relaxation time can be determined from the envelope of the consecutive Loschmidt echoes.
The result is shown in Fig. 3.20. Technical details of a highly optimized algorithm are given in A.2. Both the original spin decay and the echo envelope can be Fourier transformed and the result is shown in Fig. 3.20. The Fourier transformed spin decay signal contains the sidelobes beside a Lorentzian centered at ω = 0.
The central Lorentzian describes the spin relaxation for longer times, whereas the sidelobes describe the rapid dephasing. Interestingly, the sidelobes are missing for the Fourier transformed Loschmidt echo envelope and only a central line, which carries all spectral weight, is preserved. This is a clear indication that the Loschmidt echo concept can successfully separate spin relaxation from dephasing.
I highlight a conceptual difference between the NMR spin echo and our numerical approach: in NMR, the spins lie in a plane and the spin direction was reversed by mirroring, which yielded essentially that the consecutive action of the same Hamiltonian let the spins evolve toward the echo. Our approach is more similar to the original concept of the Loschmidt echo, i.e. it is the SOC Hamiltonian which is reversed rather than the spin direction.
I finally note that this reversal of the internal SOC field could be realized experimentally in a system which is dominated by a Bychkov-Rashba type SOC due to an external electric field. This is well approximated in graphene, where the dominance of the Bychkov-Rashba type SOC over the intrinsic contribution was predicted [62]. For such a system, it is expected that the electric field and the resulting SOC fields have a distribution in the device. Then, a sudden reversal of the external electric field would lead to a Loschmidt echo. Certainly, this scenario could be challenged by several experimental factors including e.g. finite electric field switching field.
Summary
This thesis focused on the description of spin relaxation in metals and semicon-ductors. Prior to the thesis there had been several open questions including how well the Elliott-Yafet theory can be validated in metals experimentally, whether the complete phase diagram of spin-relaxation can be constructed in zincblende semiconductors, how one can separate the effects of spin dephasing from spin-relaxation, and whether the two major theories of spin-spin-relaxation, Elliott-Yafet and D’yakonov-Perel’, could be unified and brought to a common formalism. I managed to find the answers to these open questions and namely, I presented the accurate method for the experimental verification of the Elliott-Yafet theory in elemental metals. I developed a novel, Monte Carlo simulation based approach, which allows to study spin-relaxation in semiconductors without inversion symmetry with an arbitrary distribution of the spin-orbit coupling and for arbitrary values of the momentum relaxation. I also presented an intuitive way to unify the above two relaxation theories, which also allowed for the application of the Monte Carlo approach in materials with inversion symmetry, i.e. on a range of materials for which the original method was not thought to be applicable. A method, based on the concept of the Loschmidt echo was presented, which allows the separation of the spin dephasing from spin relaxation.
75
Thesis points
1. I pointed out two shortcomings of the phenomenological Beuneu-Monod relation, which was developed to explain the spin-relaxation time in elemental metals by correlating the experimental electron spin resonance line-width with the so-called spin-orbit admixture coefficients and the momentum-relax-ation theory. Namely that i) the momentum-relaxmomentum-relax-ation involves the Debye temperature and the electron-phonon coupling whose variation among the elemental metals was neglected, ii) the Elliott-Yafet theory involves matrix elements of the spin-orbit coupling (SOC), which are however not identical to the SOC induced energy splitting of the atomic levels, even though the two have similar magnitudes. I obtained refined values for the empirical spin-orbit admixture parameters for the alkali metals by considering the proper description of the momentum relaxation theory. [T1]
2. I developed a stochastic model for the calculation of the dynamic spin-susceptibility for materials without inversion symmetry for an arbitrary distribution of the spin-orbit coupling and for an arbitrary value of the quasi-particle scattering. The calculation yields numerically the spin-relaxation time. I validated the model by comparing its predictions to analytic cal-culations in the clean (no momentum scattering) and dirty limits (large momentum scattering) [T2].
3. Using the stochastic model, I studied the full phase space of spin relaxation as a function of SOC strength, its distribution, and the magnitude of the momentum relaxation rate. This allowed the identification of two novel spin-relaxation regimes; where spin relaxation is strongly non-exponential and when the spin relaxation equals the momentum relaxation. I also found a compelling analogy between the spin-relaxation theory and the NMR motional narrowing. Using the stochastic model, I calculated the dynamic spin-susceptibility for a variety of SOC distributions [T2].
77
4. I developed an intuitive model which allowed the unification of the Elliott-Yafet and the D’yakonov-Perel’ spin-relaxation mechanisms. I showed that the respective Hamiltonians of the two theories can be transformed to each other. I showed that the so-called generalized Elliott-Yafet theory, which was developed for the case of large momentum scattering, can be straightforwardly obtained using the D’yakonov-Perel’ approach [T3].
5. I showed that the intuitive unification of the EY and DP theories not only provides an insight to the intimate relationship between the two theories but allows to numerically obtain spin-relaxation times using the stochastic approach, which was developed for the DP case. I presented that this can be successfully applied to calculate the experimentally determined spin-relaxation time in MgB2, which shows a significant momentum scattering rate, it thus cannot be handled with the conventional Elliott-Yafet model [T3].
6. I developed an intuitive numerical tool which allows the separation of de-phasing and spin-relaxation processes. The method essentially mimics the concept of the Loschmidt echo, i.e. it introduces a time reversal for the built-in magnetic fields and keeps track of the resultbuilt-ing ensemble magnetization decay. I showed that the envelope of the Loschmidt echoes recovers the true spin-relaxation processes, which are otherwise unobservable due to strong dephasing [T4].
The publications related to the thesis points are as follows:
[T1] L. Szolnoki, A. Kiss, L. Forró, and F. Simon: Empirical Monod-Beuneu relation of spin relaxation revisited for elemental metals, Physical Review B
89, 115113 (2014).
[T2] L. Szolnoki, A. Kiss, B. Dóra, and F. Simon: Spin-relaxation time in materi-als with broken inversion symmetry and large spin-orbit coupling, Scientific
Reports 7, 9949 (2017).
[T3] L. Szolnoki, B. Dóra, A. Kiss, J. Fabian, and F. Simon: Intuitive approach to the unified theory of spin relaxation, Physical Review B 96, 245123 (2017).
[T4] L. Szolnoki, et al. manuscript in preparation
Other published works which are related to the thesis but are not included in the thesis points:
[5] G. Fábián, B. Dóra, Á. Antal, L. Szolnoki, L. Korecz, A. Rockenbauer, N.
M. Nemes, L. Forró, and F. Simon: Testing the Elliott-Yafet spin-relaxation mechanism in KC8: A model system of biased graphene, Physical Review B
85, 235405 (2012).
[6] A. Kiss, L. Szolnoki, and F. Simon: The Elliott-Yafet theory of spin relaxation generalized for large spin-orbit coupling, Scientific Reports 6, 22706 (2016).
[7] B. G. Márkus, L. Szolnoki, D. Iván, B. Dóra, P. Szirmai, B. Náfrádi, L.
Forró, and F. Simon: Anisotropic Elliott-Yafet theory and application to KC8 potassium intercalated graphite, Physica Status Solidi B 253, 2293 (2016)
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Appendix
A.1 The user interface of the simulation software
The source code of the simulation software was published as the electronic Sup-plementary Materials for Ref. [9] and is also available at Github1. There is also a more recent, cleaned up but incomplete software2 that also includes an advanced spin-echo measurement implementation. Its user interface differs from the first implementation.
The software was developed on Debian3, but it should be possible to compile on any POSIX compliant operating system (on Windows it should compile under Cygwin or WSL). The software depends on the Armadillo4 and Boost5 external libraries.
The software has a command line interface which allows scripting large number of simulations with varying simulation parameters. The --help option produces all the other available command line options.
$ bin/main --help Allowed options:
-h [ --help ] produce help message
-v [ --version ] print version number
--autocorr set autocorr measurement
--spins arg (=10) set number of spins --duration arg (=300) set simulation duration --timestep arg (=1) set timestep
--omega arg (=0.20000000000000001) set the absolute value of Larmor
1https://github.com/leni536/DP_random_walk
2https://github.com/leni536/DP_random_walk2
3https://www.debian.org/
4http://arma.sourceforge.net/
5https://www.boost.org/
87
precession
--delta_omega arg (=0) set the width of omega distribution --seed arg (=rand) set the seed for the random generator -o [ --output ] arg (=-) output file path
-m [ --model ] arg (=naiv) name of the model
--meas arg (=prep) name of measurement method -b [ --B_meas ] arg (=0) measurement field
--tmin arg (=0) starting time, B_meas turns in at t=0 Since all simulation parameters have a default value, when run without argu-ments, the program outputs the result of a sample simulation to the standard output . The output contains a header which describes all simulation parameters so the simulation can be reproduced. After the header section it outputs the spin component of interest in the function of time. The time interval and sampling interval is the same as specified in the command line.
$ bin/main | head -20
# Djakonov-Perel simulation
# t=0 Sz=1, no magnetic field
# version: commit_17cc1d6e5275608347fa311f7f7e64735470c311
# spins: 10
# duration: 300
# timestep: 1
# omega: 0.2
# seed: 2494990189
# model: naiv
# meas: prep
# B_meas: 0
# tmin: 0
# autocorr: false
# t, Sz 0, 1
1, 0.990826 2, 0.97055 3, 0.946542 4, 0.925218 5, 0.910056