Aharonov-Bohm conductance of a disordered single-channel quantum ring
We study the effect of weak disorder on tunneling conductance of a single-channel quantum ring threaded by magnetic flux. We assume that temperature is higher than the level spacing in the ring and smaller than the Fermi energy. In the absence of disorder, the conductance shows sharp dips (antiresonances) as a function of magnetic flux. We discuss different types of disorder and find that the short-range disorder broadens antiresonances, while the long-range one leads to arising of additional resonant dips. We demonstrate that the resonant dips have essentially non-Lorentzian shape. The results are generalized to account for the spin-orbit interaction which leads to splitting of the disorder-broadened resonant dips, and consequently to coexisting of two types of oscillations (both having the form of sharp dips): Aharonov-Bohm oscillations with magnetic flux and Aharonov-Casher oscillations with the strength of the spin-orbit coupling. We also discuss the effect of the Zeeman coupling.
The Aharonov-Bohm (AB) effect bohmD () is one of the beautiful manifestations of the wave nature of electrons. The key physical issue — the sensitivity of the phase of an electronic wavefunction to a magnetic flux — enables the design of quantum AB interferometers aronov87D (); AB1 (); AB2 (); AB3 (); AB4 (); AB5 (); AB6 (); AB7 (); AB8 (); AB9 (); Preden1 (); Preden2 (); Bykov1 (); Bykov2 (); Zhang1 (); Ofek (); Weisz (); Yacoby5 (); Chang () that can be tuned by an external magnetic field. Such interferometers occupy a worthy place in the quantum interferometry based on low-dimensional electronic nanosystems. A single-channel ballistic ring tunnel-coupled to the leads and threaded by the magnetic flux is the simplest realization of the AB interferometer (see Fig. 1). The interference of clockwise and counterclockwise electron trajectories manifests itself in the oscillations of the ring conductance with the period (here is the magnetic flux measured in the units of the flux quantum ). bohmD (); aronov87D ()
At low temperature and weak tunneling coupling, AB conductance exhibits narrow resonant peaks both in clean and disordered single-channel rings buttD () (see also Refs. Moskalets1, ; Li, ; Mao, ; Feldman, ; Kokoreva, for discussion of disordered case). The peak arises each time when one of the field-dependent energy levels in the ring crosses the Fermi energy Hence, the positions of the AB resonances depend on buttD () (AB resonances are also affected by the Coulomb blockade kinD (); Grifoni ()). Based on this physical picture one could expect the suppression of the resonance structure at where is the level spacing in the ring. Remarkably, this naive expectation is incorrect and the interference effects are not entirely suppressed by the thermal averaging. Specifically, for the conductance of the noninteracting ring with weak tunnel coupling to the contacts exhibits sharp narrow dips (antiresonances) at where is an arbitrary integer number (see Fig. 2). jaglaD (); dmitrievD () It was also shown that the electron-electron interaction leads to arising of a fine structure of the antiresonances: each antiresonance splits into a series of narrow dips which correspond to blocking of the tunneling current by the persistent one dmitrievD () (in contrast to the Coulomb blockade this effect is robust to increasing of temperature).
Additional physics comes into play in the presence of the spin-orbit (SO) interaction. In particular, the rotation of the electron spin in the built-in SO magnetic field results in a spin phase shift between clockwise and counterclockwise waves. This phase is additional with respect to AB phase and exists even at zero external magnetic field () so that zero-field conductance exhibits the Aharonov-Casher (AC) effect: ACD (); StoneD () periodic oscillations with the strength of the SO coupling. The AC oscillations were the focus of intensive theoretical Stone1D (); history1D (); history6D (); history2D (); history3D (); history4D (); history5D (); citro2D (); pletD (); citro1D (); kovD (); RomeoD (); lobosD (); plet1D (); moldovD (); AharonyD (); MichettiD () research and their signatures were observed experimentally.exp1D (); exp2D () Recently, we demonstrated that these oscillations are also not suppressed by thermal averaging, my () in a full analogy with the AB ones. Specifically, at SO interaction splits AB antiresonances into pairs of symmetrical (with respect to ) antiresonances. We also showed that the Zeeman interaction leads to appearance of two additional negative peaks on each period my ().
What, to the best of our knowledge, has not been discussed in the literature is the effect of disorder on the tunneling conductance through a single-channel ring at relatively high temperatures, .Moskalets () The aim of the current study is to fill this gap.
In this paper, we study the tunneling transport of non-interacting electrons through a disordered single-channel quantum ring of length threaded by a magnetic flux We assume that is much smaller than the Fermi energy but large compared to (throughout the paper we linearize electron spectrum near thus neglecting small variation of the electron velocity within the temperature band). The tunneling coupling characterized by tunneling probability is assumed to be weak, which implies that the ring is almost closed.
We discuss different types of disorder and find that the short-range disorder broadens antiresonances at while the long-range one leads to arising of additional antiresonances at We also find that the resonant dips have essentially non-Lorentzian shape. The results are generalized to account for the spin-orbit interaction which leads to splitting of the disorder-broadened resonant dips, and consequently to coexisting of two types of oscillations (both having the form of sharp periodic dips): Aharonov-Bohm oscillations with magnetic flux and Aharonov-Casher oscillations with the strength of the spin-orbit coupling. Additional disorder-broadened resonant dips arise in the presence of the Zeeman coupling.
I Clean ring
We start with discussion of the high-temperature conductance of the clean ring following Refs. dmitrievD, ; my, . This section aims to introduce basic notions and clarify our approach to the problem. Later this approach will be generalized to describe the effect of disorder.
The conductance is given by the Landauer formula:
is the thermal average of the transmission coefficient and is the Fermi-Dirac function (here we take into account double spin degeneracy).
We consider symmetrical setup (see Fig. 1) with identical point contacts described by the scattering matrix,
whose elements buttD ()
represent amplitudes of scattering from three incoming channels to three outgoing ones (see Fig. 3). Here is a real parameter characterizing the strength of the tunneling coupling to the contact: weak coupling corresponds to while an open contact is described by
The transmission amplitude can be calculated by summation of the amplitudes of all the trajectories connecting contact and contact including the trajectories with backscatterings by contacts (the processes and on Fig. 3). Let us denote by the number of times the electron passes the contact without exiting the ring (). The trajectories with a given consist of the odd number of semicircles and thus have the same length . The sum of the amplitudes of such trajectories can be written as where is the electron wavenumber. Hence, the transmission amplitude is written as
Next, we separate contributions of trajectories ending with lower and upper semicircle thus writing Introducing vector with two components, and one may easily derive the following recurrence relations:
where the the matrix is given by
The element [multiplied by ] is the sum of the amplitudes of the trajectories starting at the contact and making a single return to the same contact (indices and specify, respectively, the final and initial directions of motion: ). The components of the vector
yield contributions of shortest counterclockwise and clockwise trajectories, respectively.
Using Eq. (5), we express transmission coefficient in terms of
The terms with in Eq. (10) vanish after thermal averaging in the discussed case so that the expression for the averaged transmission coefficient becomes
The dependence is shown on Fig. 2. The physical explanation of the dip (antiresonance) at is quite simple (here and below we consider the interval ). Let us demonstrate that at the contribution of any trajectory is exactly canceled by contribution of the trajectory mirrored with respect to the line connecting and . Indeed, the sum of the amplitudes of these two trajectories is proportional to , where is a difference between the number of clockwise and counterclockwise revolutions, At this sum turns to zero for any Thus, the antiresonance is due to the destructive interference of mirrored paths.
For weak tunneling coupling, the antiresonance is well approximated by the Lorentz-shape dip:
It is worth noting, that in the vicinity of antiresonance one can neglect the backscattering on the contacts. Indeed, at the off-diagonal elements of which are proportional to multiplied by small factors which implies that backscattering is effectively suppressed by a factor Physically, the effective suppression of backscattering is explained by destructive interference of two processes. In the first process an electron is reflected by a contact (say, contact ) and returns to this contact after one revolution around the ring. The amplitude of such a process is (the sign is prescribed by direction of the propagation) where the amplitude appeared because the contact was passed without reflection. In the second process the electron passes the contact without reflection and then is reflected by contact and returns to The corresponding amplitude is given by Evidently, for the amplitudes of these processes exactly cancel each other.
It is worth noting that backscattering is important in vicinity of integer values of flux. dmitrievD () In particular, putting in Eq. (I) we come to incorrect conclusion that there are resonant peaks at in the evident contradiction with Eq. (12).
The approach discussed above allows one to find transmission coefficient for arbitrary and However, it is technically cumbersome and lacks physical transparency. Below we derive the main result of this section, Eq. (13), by using an alternative method. This method is valid only in the vicinity of where backscattering by contacts can be neglected. However, it has a number of advantages compared to the first one: it is more illustrative physically, and much more easily generalized to account for disorder.
The key idea is that for and the tunneling amplitude through the ring may be presented as a sum of the transition amplitudes through intermediate states corresponding to quasistationary levels of almost closed ring. The appropriate analytical expression is derived in Appendix B and reads
is the Green function of the closed ring, describing the transition from the contact to contact is the energy of the tunneling electron, and are the electron energies in the closed ring, corresponding to wave functions The quantities
are the amplitudes of transition through the corresponding quasistationary states (see Fig. 4). Here is the tunneling rate given by
In the closed clean ring there are two types of the electron states, corresponding to counterclockwise and clockwise propagation, labeled below by indices and respectively. Wave functions and energies of these states read
where constant arises in course of linearization of the spectrum near the Fermi energy [here obeys the following equation: ]. As seen, for each level is double-degenerate, Finite lifts the degeneracy of these levels:
so that for any and transmission coefficient turns to zero even before energy averaging. The minus sign in the r.h.s. of Eq. (20) appeared due to the property
This is the property that leads to destructive interference and formation of the dip in the tunneling conductance. Physically, this is an alternative way to describe a compensation of mirrored paths discussed above.
The double sum in this equation contains “classical” terms proportional to , as well as the interference ones, (with ). The main contribution to the integral in Eq. (22) comes from the vicinities of poles (with the size on the order of ) of the amplitudes and . It is easy to see from Eqs. (16) and (22) that interference terms are comparable with classical ones only if For , energies and are close to each other, (see Fig. 4), and differ from the energies of other levels by a much larger distance ( or larger). The interference contributions to Eq. (22) containing products of the amplitudes from different pairs, can be neglected compared to the “classical” terms and to the interference terms, containing products of the amplitudes from the same pair. Therefore, we can rewrite Eq. (22) as a sum over pairs of close levels
Hence, the paths over which electron passes through the ring can be split into pairs of interfering paths, corresponding to quasi-degenerate intermediate states and (see Fig. 4).
Now we demonstrate that contributions of different pairs in Eq. (23) in fact differ by the thermal factor only, so that the problem can be reduced to analysis of the transition through a single pair. First we notice, that the energy dependence of the thermal factor is smooth and in the th term of the sum one may replace with Next we change the integration variable in this term: Then, dependence on remains only in the factors This dependence disappears after calculation of modulus squared in Eq. (23). Finally, we calculate sum over (here we use inequality ), and arrive to the following equation
which expresses in terms of transition through a single pair of close levels with energies The minus sign in front of the second fraction in Eq. (24) appeared due to the property (21). One may separate in Eq. (24) ”classical” contribution, (integral form the sum of the squared amplitudes), from the interference one, Performing integration, we find and Summing these terms we restore Eq. (13). We notice that, as expected, interference term gives significant contribution only in the region of small
Ii The ring with impurities
In the above calculations we considered the case of the clean ring. Now we discuss the effect of disorder on the high-temperature conductance of the ring.
ii.1 Long-range disorder
One of the realizations of the disorder is a weak smooth random potential with the correlation length much exceeding the electron Fermi wavelength. In this case, backscattering by disorder is exponentially suppressed, so that the potential only leads to the additional phase shift between the right and left-moving electron waves propagating from contact to contact along upper and lower shoulder of interferometer, respectively (with zero winding number). We denote the disorder-induced phase difference between these two waves as . Such an interferometer is evidently equivalent to the clean one having two arms with the lengths and where The conductance of the latter interferometer was calculated in Ref. dmitrievD, . From Eq. (A3) of Ref. dmitrievD, we find
This equation is valid provided that As seen, for there are two dips in the conductance (at and at ), the widths and the depths of the dips being oscillating functions of [in particular, ]. Hence, long-range disorder leads to appearance of the additional antiresonance in the conductance at and modifies the antiresonance near .
ii.2 Short-range disorder
ii.2.1 Calculation of the transmission coefficient
Another realization of disorder is the potential created by weak short-range impurities, randomly distributed along the ring with the concentration Let us characterize the strength of disorder by the scattering rate in the infinite wire calculated by the golden rule. For short-range potential, transport and quantum scattering rates coincide and are given by where is the reflection amplitude for a single impurity (). Substituting in this equation (here is the number of impurities in the ring) we get
We restrict ourselves to discussion of the ballistic case,local () , or, equivalently,
We will see that the main effect of the short-range potential is the broadening of the antiresonances. One could expect that scattering by disorder leads to essential increase of the resonance width, when becomes shorter than lifetime of the electron in the ring, which implies Another expectation is that in the regime when electron experiences many scatterings during the lifetime and therefore acquires random phase, the interference is suppressed, and, consequently, the depth of the dip essentially decreases. However, we will show that the scattering on the impurities comes into play at much smaller disorder strength when so that for the dip is essentially broadened. Also, in contrast to the naive expectation, its depth remains on the order of
Now we generalize the method, introduced in the first part of the previous section. To do this we should modify the matrix , taking into account the scattering on the impurities. This matrix becomes complicated, since it includes the amplitudes of all the trajectories with scatterings on both contacts and impurities, after which an electron returns to contact However, in the case and the matrix can be simplified.
As a first step we expand matrix in a Taylor series up to the first order with respect to denoting .
For the clean ring () we have
We neglected off-diagonal elements of since, as we explained above, backscattering on the contacts is effectively suppressed at Now we write the expansion for dirty ring in a way that reproduces Eq. (28) for
Here both and depend on The matrix should vanish at so that at small Calculations using Eq. (63) show that one can neglect the term in Eq. (29) (as well as terms on the order of and higher). Physically, this implies neglect of the processes involving both scattering by impurities and forward scattering by contacts during one revolution around the ring.
Let us now discuss the properties of the matrix According to the definition of the matrix (see previous section) the matrix relates the amplitudes and of the incoming and outcoming waves, respectively, at the point (see Fig. 5)
Hence, the matrix is the matrix describing a complex scatterer consisting of impurities, located at points between and Having in mind that this matrix should be unitary and taking into account the time reversal symmetry we write this matrix in the most general form:
Here is the small forward scattering phase for a complex scatterer consisting of impurities. This phase is added to the geometrical phase and, therefore, drops out after thermal averaging. The off-diagonal element, is, up to a phase factor, the reflection amplitude from a complex of impurities. One can expand with respect to In the lowest order in we obtain
This expression takes into account only one backscattering on impurities during a revolution around the ring, and is valid in the case .
As we see, the matrix is now dependent on , so that Eq.(11) does not generally follow from Eq.(10). However, if we assume that impurities are randomly distributed along the ring (some special non-random distributions will be discussed at the end of Sec. II.2) the terms with in Eq.(10) do not survive the averaging over . Therefore we can use Eq.(11) while performing averaging over :
Since the probability to scatter on an impurity before the first visit of the point is small, we can neglect the scattering terms in the vector entering Eq. (33) and use Eq. (9). The sum in Eq. (33) can be calculated with the use of Eq. (63) which simplifies after expansion of the numerator and denominator with respect to and Calculations yield the following expression for the transmission coefficient:
This equation is valid provided that and The relation between and can be arbitrary.
The approach, discussed above, can be also used to calculate the transmission coefficient for arbitrary In particular, one can show that at there appears a small dip with the amplitude on the order of (to obtain this result one should take into account backscattering by contacts). In the following discussion we focus only on the antiresonance at
Let us first consider a ring with a single impurity. As seen from Eq. (34), the transmission coefficient is given by the Lorentz-shape antiresonance:
We see that the transmission coefficient at is no longer equal to zero and the antiresonance broadens so that its width becomes . We also find that the depth of the dip changes from to with increasing In other words, in contrast to the antiresonance width, its depth remains the same order of magnitude.
In order to perform the averaging over in the case of many impurities, we notice, that for random impurity distribution the averaging over is equivalent to averaging over . For two impurities the average is easily calculated. The result reads
For the case , we rewrite Eq. (34) using the identities , and get the following expression:
where is the modified Bessel function of the second kind.
Assuming now that the number of impurities is large, we get and after simple calculation obtain
where This dependence is plotted in Fig. 6. We see that similar to the case of a single impurity, the transmission coefficient at is no longer equal to zero and the antiresonance broadens. It is also notable that for the dip has a non-Lorentzian shape.
Let us discuss two limiting cases. For the minimal value of conductance is given by and the width of the antiresonances increases from to where
The relative contribution of the disorder to the resonance width, is enhanced by a factor in comparison with naive expectation In the opposite limiting case, we get so that the depth of the antiresonance is two times smaller compared with the case of clean ring, while the width is given by
ii.2.2 Two-level approximation
Next we discuss the obtained results in terms of transition amplitudes through intermediate quasi-stationary states of almost closed ring. We recall that in the vicinity of the flux the energy levels of the clean ring can be split into pairs of close levels corresponding to clockwise and counterclockwise propagation of the electron inside the ring. Energy distance between levels in the pairs equals to (see Fig. 4, left panel). For the case when impurity potential [here is the potential of the -th impurity] is sufficiently weak (see corresponding criterion below) it can be simply accounted for by perturbation theory for two close levels.
Calculation yields for energies and wave functions of potential-disturbed states:
is the matrix element of the impurity potential expressed in terms of reflection amplitude . For , the inequality which ensures validity of the two-level perturbation theory is given by . For randomly distributed impurities, the amplitude of the potential is estimated as . Hence, the perturbation theory applies for .
Let us first discuss the dependence of the conductance on qualitatively. First we note that due to the repulsion between levels in the impurity potential the minimal distance between levels in any pair is given by (this distance corresponds to ). For similar to the case of a clean ring, there exist a dip in the transmission coefficient which arises due to the interference between transition amplitudes though pairs of close levels (see Fig. 4), while the ”classical” term is featureless at
In contrast, in the opposite limiting case, the energy distance between levels in any pair becomes much larger than , and, consequently, contribution of the interference terms to the transmission coefficient is small compared to “classical” ones. This, however, does not lead to disappearance of the dip in the transmission coefficient. It turns out that in the dirty ring the “classical” terms acquire sharp dependence on and decrease by a factor within a narrow region
Indeed, as seen from Eqs. (44)-(45), for the wave functions in the th pair are simply given by clockwise- and counterclockwise-moving waves and The “classical” contribution to the transmission coefficient from each of these levels, say level is proportional to In the opposite case, disorder potential strongly mixes clockwise- and counterclockwise-propagating waves. Consider, for simplicity, the case From Eqs. (44)-(45) we see that and where Averaging over random phase and taking into account Eq. (21) we obtain twice smaller value This implies existence of a dip of a width in the transmission coefficient: decreases by a factor within this width.
The transition from the interference to “classical” picture of formation of the dip can be illustrated by an example of the ring with a single impurity. The conductance of such a ring is given by Eq. (35). The “classical” and interference contributions to this expression read
As seen, the interference contribution leads to formation of the dip at and can be neglected for
The rigorous calculations of the conductance can be performed in a way analogous to the case of the clean ring. The transmission coefficient is expressed via a Green function of the closed ring and can be approximately presented as a sum over pairs of interfering paths through intermediate states and Equations (16) and (23) still hold with the following replacement: and In contrast to the case of the clean ring, the summands in Eq (23) are now different not only due to the thermal factor but also because of the strong dependence of on However, for random distribution of impurities the summation over within the temperature window is equivalent to averaging over the impurity positions. After such averaging the dependence on remains only in the thermal factor and the problem is again reduced to the case of transition through a single pair of close levels. The transmission coefficient may be written as a sum of classical and interference terms
ii.2.3 Special impurity distributions
Above we assumed that impurities are randomly distributed along the ring. If this is not the case, the results might be quite different. The reason is that there exist some impurity distributions for which the summation over within the temperature band is not equivalent to the averaging over the impurity positions, so that calculation should be performed more carefully. One may check that the method discussed in Sec. II.2.2 reproduces Eq. (34) with the replacement and Let us now give some examples of special impurity distributions for which Eqs. (36)-(39) are invalid. If the impurities are distributed symmetrically with respect to the line connecting the contacts, then and, consequently, On the language of trajectories this can be explained by cancelation of the contributions of the mirrored paths just as in the case of the clean ring. The width of the dip is on the order of . If impurities are distributed symmetrically with respect to the line perpendicular to and crossing the center of the ring, then and the amplitude of the dip in goes to zero with increasing for Finally, if impurities are distributed symmetrically with respect to the ring center, one gets and dependence becomes the same as in the clean ring.
Iii The ring with impurities and spin-orbit interaction
In this section we discuss the effect of impurities on the conductance of a ring with spin-orbit and Zeeman interactions. The case of a clean ring was studied in detail in Ref. my, .
The Hamiltonian of a clean ring with SO-interaction induced by axially-symmetric built-in field is given by
is the kinetic energy,
is the Zeeman term ( is the Zeeman splitting energy in the external magnetic field parallel to the axis) and describes the SO coupling:
Here is the angle between effective SO-induced magnetic field and the axis, is the dimensionless parameter characterizing the strength of SO interaction, stands for the anticommutator.
The problem is studied in the quasiclassical case () in which the effect of the SO interaction is described by the rotation of the electron spin in effective magnetic field which varies along the electron trajectory. The stationary wavefunctions and energies in this case read:
Here we introduced the notations
As seen from Eq.(54) the degeneracy of the levels occurs for the following eight values of magnetic flux: and where However, at four of these values, and the resonances are absent because of the backscattering on the contacts (see appendix B in Ref. my, ). At four other points, the backscattering is negligible, and there appear the antiresonances with the width The amplitudes of the antiresonances at and are and respectively, where
At the degeneracy occurs between the states and at between and at between and