Open access peer-reviewed chapter

A Quantum Trajectory Interpretation of Magnetic Resistance in Quantum Dots

By Ciann-Dong Yang and Shih-Ming Huang

Submitted: October 6th 2017Reviewed: January 25th 2018Published: April 4th 2018

DOI: 10.5772/intechopen.74409

Downloaded: 364

Abstract

For a complete description of the electronic motion in a quantum dot, we need a method that can describe not only the trajectory behavior of the electron but also its probabilistic wave behavior. Quantum Hamilton mechanics, which possesses the desired ability of manifesting the wave-particle duality of electrons moving in a quantum dot, is introduced in this chapter to recover the quantum-mechanical meanings of the classical terms such as backscattering and commensurability and to give a quantum-mechanical interpretation of the observed oscillation in the magneto-resistance curve. Solutions of quantum Hamilton equations reveal the existence of electronic standing waves in a quantum dot, whose occurrence is found to be accompanied by a jump in the electronic resistance. The comparison with the experimental data shows that the predicted locations of the resistance jump match closely with the peaks of the measured magneto-resistance.

Keywords

  • quantum dots
  • quantum Hamilton mechanics
  • standing waves
  • quantum trajectory
  • magneto-resistance

1. Introduction

As the size of electronic devices is narrowed down to the nanoscale, quantum effects become so prominent that classical mechanics is no longer able to provide an accurate description for electrons moving in nanostructures. However, due to the lack of the sense of trajectory in quantum mechanics, classical or semi-classical mechanics so far has been the sole tool in determining ballistic orbits in quantum dots. Classical orbits satisfying commensurability conditions of geometrical resonances were derived in the literature to determine the magneto-transport behavior of periodic quantum systems. It was reported that the observed regular peaks in the magneto-resistance corresponded to backscattering of commensurate orbits [1], and the critical magnetic fields determined from the backscattering orbits showed an excellent agreement with the observed peak positions in the magneto-resistance curves [2]. A recent study showed that the ballistic motion of electrons within quantum dots can be controlled by an externally applied magnetic field so that the resulting conductance images resemble the classical transmitted and backscattered trajectories [3].

The use of an anisotropic harmonic function, instead of an abrupt hard potential, to describe the confining potential in a quantum dot was shown to be helpful to improve the accuracy of predicting magneto-resistance peaks based on backscattering orbits [4]. Nowadays, the confinement potential forming an electron billiard can be practically patterned to almost arbitrary profile, through which ballistic orbits with chaotic dynamics can be generated to characterize magneto transport [5]. However, the chaotic behavior and its change with magnetic field could not be described in the usual quantum-mechanical picture due to the lack of a trajectory interpretation. Regarding this aspect, the classical description becomes a valued tool for detailed understanding of the transition from low to high magnetic fields in quantum dot arrays [6]. On the other hand, quantum mechanical model for electron billiards was known as quantum billiards [7], in which moving point particles are replaced by waves. Quantum billiards are most convenient for illustrating the phenomenon of Fano interference [8] and its interplay with Aharonov-Bohm interference [9], which otherwise cannot be described by classical methods.

From the existing researches, we have an observation that the ballistic motion in electron billiards was solely described by classical mechanics, while the wave motion in quantum billiards could only be described by quantum mechanics. The aim of this chapter is to give a unified treatment of electron billiards and quantum billiards. We point out that quantum Hamilton mechanics [10, 11] can describe both ballistic motion and wave motion of electrons in a quantum dot to provide us with a quantum commensurability condition to determine backscattering orbits as well as with the wave behavior to characterize the magneto-resistance in a quantum dot.

Quantum Hamilton mechanics is a dynamical realization of quantum mechanics in the complex space [12], under which each quantum operator is realized as a complex function and each wavefunction is represented by a set of complex-valued Hamilton equations of motion. With quantum Hamilton mechanics, we can recover the quantum-mechanical meanings of the classical commensurability condition by showing that there are integral numbers of oscillation in the radial direction, as an electron undergoes a complete angular oscillation around a quantum dot. When the radial and angular dynamics are commensurable, the shape of electronic quantum orbits is found to be stationary like a standing wave. Furthermore, the wave number N, distributed on the circumference of the quantum dot, can be controlled by the applied magnetic field. It will be shown that the classical backscattering orbits discovered in the literature resemble the shape of the quantum standing waves derived here with their wave numbers equal to the numbers of electron’s bounces within the quantum dot.

The electronic standing-wave motions considered in this chapter will reveal that a jump of the magneto-resistance in quantum dots is accompanied by a phenomenon of magnetic stagnation, which is a quantum effect that an electron is stagnated or trapped within a quantum dot by an applied magnetic field in such a way that the electron’s cyclotron angular velocity is exactly counterbalanced by its quantum angular velocity. We point out that magnetic stagnation is a degenerate case from the electronic standing-wave motion as the wave number Napproaches to infinity. The magnetic fields yielding the phenomenon of magnetic stagnation can be determined by the quantum Hamilton equations derived here. Knowing these critical magnetic fields allows us to control the magneto-resistance precisely.

In the following sections, we first introduce quantum Hamilton mechanics and apply it to derive Hamilton equations, which are then used in Section 2 to describe the electronic quantum motions in a quantum dot. By solving the Hamilton equations of motion, Section 3 demonstrates electronic standing-wave motions in various quantum states and characterizes the magnetic field leading to the phenomenon of magnetic stagnation. In Section 4, we show that the magnetic stagnation is the main cause to the resistance oscillation of quantum dots in low magnetic field by comparing the theoretical predictions obtained from Section 3 with the experimental results of the magneto-resistance curve [4, 13].

2. Quantum Hamilton dynamics in a 2D quantum dot

To probe the quantum to classical transition, which involves both classical and quantum features, quantum dots are the most natural systems [14]. Analyzing such systems, we need an approach that can provide both classical and quantum descriptions. Quantum Hamilton mechanics is one of the candidates satisfying this requirement. This chapter will apply quantum Hamilton mechanics to an open quantum dot with circular shape, which is connected to reservoirs with strong coupling. The electronic transport through an open quantum dot can be realized by nano-fabrication techniques as a two-dimensional electron gases system (2DES) at an AlGaAs/GaAs heterostructure, as depicted in Figure 1.

Figure 1.

Schematic illustration of a semiconductor heterostructure with a circular dot between the two tunnel barriers.

Under the framework of quantum Hamilton mechanics [10, 12], the equivalent mathematical model of a quantum dot is described as an electron moving in an electromagnetic field with scalar potential Vand vector potential A. The related Hamiltonian operator Ĥcan be realized as the following complex Hamiltonian function,

Htqp=12mp+ceAp+ceA+Vtq+2imp.E1

We adopt polar coordinates q=rθand momentum p=prpθin the above equation to describe the electronic quantum motion in a 2D circular quantum dot. The resulting complex Hamiltonian Eq. (1) becomes

H=12mpr+ecAr2+1r2pθ+ecAθ2+Vrθ+2im1rpr+prr+1r2pθθ,E2

where m=0.067meis the electron’s effective mass in AlGaAs/GaAs heterostructure. The scalar potential Vrθacts as a confinement potential in the dot, and is modeled by the parabolic function V=kr2/2=mω02r2/2to simulate a soft-wall potential. The vector potential Ais determined from the applied magnetic field Bvia the relation B=×A. Here, we consider a constant Balong the zdirection, which amounts to Ar=0and Aθ=Br2/2. Substituting the above assignments of Vand Ainto the complex Hamiltonian Eq. (2), we obtain

H=12mpr2+1r2pθ2+ωLpθ+12mω2r2+2im1rpr+prr+1r2pθθE3

where ωL=eB/2mcis the Larmor frequency and ω=ω02+ωL2is the composite frequency. The quantum Hamilton-Jacobi equation associated with the Hamiltonian Hdefined in Eq. (1) reads

St+Htqppi=S/qi=0,E4

where Sis the quantum action function to be determined. By making use of the substitution

pr=Sr,pθ=Sθ,E5

the quantum Hamilton-Jacobi Eq. (4) associated with the Hamiltonian in Eq. (3) turns out to be

St+12mSr2+1r2Sθ2+ωLSθ+12mω2r2i2m1rSr+2Sr2+1r22Sθ2=0.E6

The recognition of the complex Hamiltonian Hin Eqs. (1) and (3) as a complex realization of the Hamiltonian operator Ĥis confirmed by the fact that the quantum Hamilton-Jacobi equation in Eqs. (4) and (6) yields the same Schrodinger equation as constructed from Ĥ. This can be seen by applying the following transformation

S=ilnΨ,E7

to Eq. (6) to produce the expected Schrodinger equation:

iΨt=22m2Ψr2+1rΨr+1r22Ψθ2iωLΨθ+12mω2r2Ψ.E8

Due to the time-independent nature of the applied potentials Aand V, the wavefunction Ψin Eq. (8) assumes the following form of solution,

Ψtrθ=eiEt/ψrθ,E9

where ψrθsatisfies the time-independent Schrodinger equation

Ĥψ22m2r2+1rr+1r22θ2iωLθ+12mω2r2ψ=.E10

On the other hand, Eq. (3) can be rewritten by using the substitutions Eqs. (5) and (7) as

Ĥψ=,E11

where Hand Ĥare defined, respectively, by Eqs. (3) and (10). This is a direct proof of the fact that the complex Hamiltonian His a functional realization of the Hamiltonian operator Ĥin a complex space. Indeed, it can be shown [10] that every quantum operator Âcan be realized as a complex function Avia the relation Âψ=. The combination of Eqs. (10) and (11) reveals the energy conservation law H=E, which is a natural result of Hamilton mechanics by noting that the Hamiltonian Hin Eq. (3) does not contain time texplicitly and must be a motion constant equal to the system’s total energy E.

Upon performing the differentiations pr/rand pθ/θinvolved in Eq. (3), we have to specify in advance the action function Sor equivalently the wavefunction ψvia the relation Eq. (7). This requirement makes the complex Hamiltonian Hstate-dependent. For a given quantum state described by ψ, the complex Hamiltonian Hcan be expressed explicitly as:

H=12mpr2+1r2pθ2+ωLpθ+12mω2r2+2im1rpr+i2lnψr2+1r2i2lnψθ2.E12

Apart from deriving the Schrodinger equation, the above complex Hamiltonian also gives electronic quantum motions in the state ψin terms of the Hamilton equations of motion,

drdt=Hpr=1mpr+2im1r=imlnψr+2im1r,E13
dt=Hpθ=1mr2pθ+ωL=imr2lnψθ+ωL.E14

The appearance of the imaginary number i=1on the right-hand side of the above equations indicates that the quantum trajectory rtθthas to be defined in the complex space as rtθt=rRt+irItθRt+Itto guarantee the solvability of Eqs. (13) and (14). It is just the coupling connection between the real and imaginary parts that gives rise to the quantum phenomena, as we have observed in the real world, such as wave-particle duality [15], tunneling [16], and Heisenberg uncertainty principle [17]. For a given 1D wavefunction Ψtxexpressed in Cartesian coordinates, the complex Hamiltonian Eq. (1) has a simple form:

Htxp=12mp2+Vtx+2impx=12mp2+Vtx22m2lnΨtxx2.E15

The Hamilton equation for xturns out to be

ẋ=Hp=pm=1mSx=imΨx,xCE16

which can be conceived of as a complex-valued version of Bohmian mechanics [18, 19]. The complex quantum trajectory method based on Eq. (16) has been recently developed into a potential computational tool to analyze wave-packet interference [20] and wave-packet scattering [21].

The wavefunction ψhas to be solved in advance from the Schrodinger Eq. (10), before we determine the electron’s quantum trajectory rtθtfrom Eqs. (13) and (14). In terms of the dimensionless radial distance ρ=/1/2r, the eigenvalues En,land the related eigenfunction ψn,lcan be solved analytically as [22].

En,l=2n+l+1ω+lωL,E17
ψn,lρθ=Rn,lρΘlθ=Cn,leρ2/2ρlLnlρ2eilθ,E18

where n=0,1,2,is the radial quantum number, l=0,±1,±2,is the angular quantum number, and Cn,lis a normalization factor. The electronic motion in the eigenstate ψn,lnow can be established by integrating Eqs. (13) and (14) with ψgiven by Eq. (18):

=1idlnRn,lρ+12ρfn,lρ,E19
=lρ2+ωLω.E20

where the dimensionless time is expressed by τ=. Eq. (20) indicates that the angular dynamics θτis influenced by the magnetic field Bvia the relation ωL=eB/2mcand reveals the existence of a critical Bsuch that the Larmor angular velocity ωL/ωcounterbalances the quantum angular velocity l/ρ2to yield /=0. The stagnation magnetic field denotes the critical Bthat stagnates the electron with zero angular displacement within a quantum dot. The occurrence of magnetic stagnation retards the electronic transport and causes a jump in resistance. In the following sections, we will characterize the stagnation magnetic field from the equations of motion Eqs. (19) and (20) and verify the consistency between this theoretical prediction and the experimental measurement of resistance.

3. Standing waves and critical magnetic field

The conductivity of a quantum dot depends on how electrons move under the confinement potential within the quantum dot. Eqs. (19) and (20) provides us with all the required information to describe the underlying electronic quantum motion. The radial motion ρτdescribed by Eq. (19) and the angular motion θτdescribed by Eq. (20) are, individually, periodic time functions, whose periods, Tρand Tθ, can be computed by using the residue theorem. In case that the radial and angular motions are not commensurable, i.e., Tρ/TθQ, the overall motion is not periodic and the electron’s orbit precesses continuously around the periphery of the quantum dot, as shown in Figure 2a. By way of this precession orbit, an electron can pass through the quantum dot from the entrance to the exit and contribute to the conductance.

Figure 2.

Four types of electronic quantum trajectory in a quantum dot. (a) a magnetic field (Bc=0.4T) not satisfying the commensurability condition Eq. (21) yields precessing trajectories. (b) A critical magnetic field Bc=0.204T yields standing-wave like quantum trajectory, whose five wavelengths on the circumference do not contact the exit of the quantum dot. (c) A magnetic field Bc=0.26T yields similar standing-wave trajectory as that in part (b) but with six wavelengths which contact both the entrance and exit of the quantum. (d) At Bc=0.65T, a stagnation magnetic field, the electron is stagnated within an isolated region of θ.

On the other hand, if Tρ/Tθis a rational number, the shape of the electron’s orbit is stationary like a standing wave, as shown in Figure 2b. Except that the orientation of the standing wave happens to align with the direction from the entrance to the exit, as shown in Figure 2c, passage through the quantum dot is prohibited, when a standing-wave motion emerges. A standing-wave (non-precessing) orbit has to satisfy the commensurability condition,

Tθ=NTρE21

where Nis a positive integer. This condition ensures that when the electron undergoes a complete oscillation in the θdirection, there are integral numbers of oscillation in the rdirection. Once electronic standing waves emerge in a quantum dot, the electron after a complete θrevolution will return to the entrance to the quantum dot and consequently contribute to the resistance of the quantum dot.

As shown in Figure 2d, the standing-wave motion degenerates into a confined motion such that the electron is trapped into a closed trajectory, in the extreme case N. When the electron is trapped or stagnated, it is in no way to pass through the quantum dot and causes a remarkable increase in resistance. The special magnetic field corresponding to Nplays the major role in the magneto-resistance and is to be derived below.

The pattern and the orientation of the standing waves can be controlled by the applied magnetic field Bvia the relation Eq. (20), which indicates that the angular motion depends on the parameter,

ωLω=ωLω02+ωL2=eB/2mcω02+e2B2/2mc2,E22

which, in turn, is solely determined by the magnetic field B. We are able to control the resistance of the quantum dot by varying the magnitude of Bto satisfy the commensurability condition Eq. (21). Our next issue is to characterize the critical magnetic field Bcthat satisfies the commensurability condition Eq. (21). The period Tρin Eq. (21) can be evaluated by the radial dynamics Eq. (19). The radial motion ρtis a periodic time function whose trace on the complex ρplane forms a closed path cρ, along which the period Tρcan be computed as a contour integral as

Tρ==cρfn,lρ.E23

According to the residue theorem, the contour integral in Eq. (23) is equal to 2πitimes the sum of the residues of 1/fn,lρevaluated at its poles within the contour cρ, i.e.,

Tρ=cρfn,lρ=2πikRk,E24

where Rkis the residue of 1/fn,lρevaluated at its kthpole. Let Ωkbe the set containing all of contours which enclose the same poles of 1/fn,lρ. According to the residue theorem, the integrals along the contours belonging to the same set Ωkhave the same contour integral, denoted by Tρk. If the number of different ways of pole encirclement is M, we can define Mcontour sets, Ω1, Ω2, …, ΩM, with each contour set corresponding to one particular way of pole encirclement. Along all the possible contours, the period Tρdefined by Eq. (23) can only have Mdiscrete values, Tρ1, Tρ2, …, TρM, defined by

Tρ=cρfn,lρ=Tρk,cρΩk,k=1,2,,M.E25

The sequence Tρ1Tρ2TρMthen constitutes a set of quantization levels for the period Tρin the quantum state ψn,lρθ.

(A) Standing Wave with l=0

In case of l=0, the radial dynamics and azimuth dynamics are decoupled according to Eqs. (19) and (20). A look on the ground state nl=00is helpful to understand some common features in the states with l=0. The related wavefunction is given by Eq. (18) as R0,0ρ=eρ2/2and Θ0θ=1. Substituting this wavefunction into Eqs. (19) and (20) yields the equations of motion for the ground-state electron,

=i2ρ212ρ,=ωLω.E26

It appears that that the ground-state electron rotates with a constant angular velocity ωL/ωaround its equilibrium radial position ρeq=2/2. Therefore, the azimuth period Tθis simply 2π/ωL/ω, and the radial period Tρcan be computed from Eqs. (24) and (26) as

Tρ=1icρ2ρ2ρ21=π,E27

where cρis any contour enclosing the pole ρeq=2/2on the complex plane of ρ.

The commensurability condition Eq. (21) with the calculated Tρand Tθfor the ground state turns out to be

ωLω=2N,N=3,4,5,,E28

where we note ωL/ω<1from its definition in Eq. (22). The critical magnetic field Bcnow can be solved from Eq. (28) as

Bc=B0N2/41,N=3,4,5,,E29

where B0is the magnetic field whose Larmor frequency ωLis equal to the natural frequency ω0of the harmonic oscillator, i.e., B0=2mc/eω0. The relation expressed by Eq. (29) characterizes all the magnetic fields that force the electron to behave like a standing wave in the ground state of a quantum dot.

Regarding excited states, there are multiple periods in the radial motion ρτas indicated by Eq. (25). Taking first excited state nl=10as an illustrating example, the quantum dynamics is described by

=i2ρ411ρ2+62ρρ22=ωLω,E30

which has four equilibrium points at

ρeq=±11±732E31

According to different encirclements of equilibrium points, four sets of complex trajectories ρτcan be identified as shown in Figure 3a, where Ω1and Ω2denote the sets of all trajectories enclosing only one equilibrium point, Ω3denotes the set enclosing two equilibrium points, and Ω4denotes the set enclosing all the four equilibrium points.

Figure 3.

(a) Four sets of complex trajectories ρτ are identified according to different encirclements of equilibrium points in the state nl=10. (b) Typical time response of an increasing Reθ corresponds to the trajectory shown in Figure 2b. (c) Typical time response of an oscillatory Reθ corresponds to the trajectory shown in Figure 2d.

Corresponding to the four different ways of encirclement, the four quantization levels of Tρcan be computed from Eq. (24) as

Tρ=2π73±373292,π,2π.E32

The commensurability condition for the occurrence of standing wave in the four contour sets now can be derived from Eq. (21) as

ωLω=73+37316N,N7,ρτΩ17337316N,N3,ρτΩ22N,N3,ρτΩ31N,N2,ρτΩ4E33

The related critical magnetic field Bccan be determined by substituting Eq. (33) into Eq. (22). Comparing Eq. (28) with Eq. (33), we can see that the critical Bc, which raises standing waves in the ground state, also raises standing waves in the first excited state. The peaks of the magneto-resistance just concentrate on the dominant critical magnetic field that concurrently produces standing waves in different states.

(B) Standing Wave with l>0.

In the case of l>0, the cyclotron angular velocity ωL/ωand the quantum angular velocity l/ρ2are in the same direction so as to give an intensified resultant θ̇=l/ρ2+ωL/ω. The coupling between the azimuth motion θτand the radial dynamics ρτmakes the evaluation of Tθmore difficult; however, because ρτis a periodic function, we can evaluate θ̇in Eq. (20) by simply replacing l/ρ2with its average value l/ρ2ave, if only the period of θτis concerned,

Tθ=2πθ̇ave=2πl/ρ2ave+ωL/ω=NTρ.E34

The time average l/ρ2aveis computed over one period of ρτand can be converted into a contour integral along the contour cρtraced by ρτon the complex plane:

lρ2ave=lTρ0Tρρ2τ=lTρcρρ2fn,lρ.E35

Substituting the above l/ρ2aveinto Eq. (34), we obtain the critical value of ωL/ωas

ωLω=2πNTρlρ2ave.E36

Due to the constraint 0ωL/ω1, the admissible integer Nlies in the interval

2π/Tρ1+l/ρ2ave<N2π/Tρl/ρ2ave,E37

where Tρand l/ρ2aveare given by Eqs. (24) and (35), respectively. The admissible range of Nis narrowed by increasing angular quantum number l, as can be seen from inequality Eq. (37). There is a maximum allowable lbeyond which inequality Eq. (37) has no integer solution and standing-wave motion within the quantum dot disappears. To compare with the quantum state nl=10considered previously, let us study the state nl=11whose quantum motion is described by

=i2ρ411ρ2+62ρρ22,=1ρ2+ωLω.E38

The period Tρis the same as that derived in Eq. (32), and the period Tθcan be computed by Eq. (34) with l/ρ2aveevaluated by the contour integral Eq. (35) as

1ρ2ave=l11+73/12,ρτΩ1l1173/12,ρτΩ22l/3,ρτΩ3Ω4E39

Using Tρand l/ρ2avein Eqs. (36) and (37), the critical value of ωL/ωin the state nl=11becomes

ωLω73+37316N11+7312,N=3,ρτΩ17337316N117312,3N14,ρτΩ22/N2/3,N=2,3,ρτΩ31/N2/3,N=1,ρτΩ4E40

The comparison between Eqs. (33) and (40) leads to the observation that the number of the allowed integer Ndecreases dramatically when lincreases from 0 to 1. Since the total different number of Naccounts for the number of different ways by which standing wave can be formed, the possibility for the occurrence of standing-wave motion and thus the electronic resistance decreases with increasing angular quantum number l. The main reason is that the increment of the angular velocity θ̇=l/ρ2+ωL/ωwith large laccelerates the electron’s angular motion around the quantum dot and thus improves the conductance of the quantum dot.

(C) Standing Waves with l<0

In this case, the cyclotron angular velocity ωL/ωand the quantum angular velocity l/ρ2are in opposite directions so as to give a weakened resultant θ̇=l/ρ2+ωL/ω. The resultant angular velocity θ̇avemay be positive, negative or zero, depending on the magnitude of l/ρ2ave, which can be classified into three categories: (1) l/ρ2ave0, (2) 1<l/ρ2ave<0, and (3) l/ρ2ave1, as listed in Table 1.

Range of lCritical ωL/ωRange of integer N
l/ρ2ave0ωLω=2πNTρlρ2ave2π/Tρ1+l/ρ2ave<N2π/Tρl/ρ2ave
1<l/ρ2ave<0ωLω=2πNTρlρ2aveN>2π/Tρ1+l/ρ2ave
ωLω=lρ2aveN
ωLω=2πNTρlρ2aveN2π/Tρl/ρ2ave
l/ρ2ave1ωLω=2πNTρlρ2ave2π/Tρl/ρ2aveN<2π/Tρ1+l/ρ2ave

Table 1.

The relation between critical Larmor frequency and angular quantum number l.

The cases of l=0and l>0considered previously belong to category (1) with θ̇ave>0, while the case of l<0belongs to categories (2) and (3). Taking into account the motion with θ̇ave<0, Eq. (36) now becomes

ωLω=±2πNTρlρ2ave,E41

where the admissible integer Nfor the three categories is summarized in Table 1.

The critical magnetic field Bcgiven by Eq. (41) with θ̇ave>0and θ̇ave<0produces standing-wave motions oscillating, respectively, counterclockwise and clockwise around the quantum dot, as shown in Figure 2b and c. For an angular quantum number with 1<l/ρ2ave<0in category (2), there exists a special Larmor angular velocity ωL/ωsuch that it counterbalances the quantum angular velocity l/ρ2aveto yield

θ̇ave=l/ρ2ave+ωL/ω=0.E42

The critical Bcsatisfying θ̇ave=0produces isolated standing waves that form closed trajectories as shown in Figure 2d. This critical Bcnullifies the electron’s net angular displacement and is called stagnation magnetic field. Because a passage through the quantum dot requires a net angular displacement equal to π, an electron with θ̇ave=0is unable to pass the quantum dot and makes no contribution to the conductivity.

In a case study of l<0, we consider the state of nl=11, whose quantum equations of motion read

=i2ρ411ρ2+62ρρ22,=1ρ2+ωLω.E43

The radial trajectories ρτare the same as those depicted in Figure 3a. Along different sets of radial trajectory, different modes of standing-wave motion are excited. According to the value of l/ρ2ave=1/ρ2avegiven by Eq. (39), it is found that the trajectory set Ω1belongs to category (3), while the sets Ω2, Ω3and Ω4belong to category (2), as tabulated in Table 2.

SetFrequency rangeCritical frequencyInteger N
Ω10ωLω<1ωLω=73+37316N+11+73124N9
Ω2117312ωLω<1ωLω=7337316N+117312N4
ωLω=117312ωLω=117312N
0ωLω<117312ωLω=7337316N+117312N15
Ω3ωLω>23ωLω=2N+23N7
ωLω=23ωLω=23N
0ωLω<23ωLω=2N+23N3
Ω4ωLω>23ωLω=1N+23N4
ωLω=23ωLω=23N
0ωLω<23ωLω=1N+23N2

Table 2.

Distribution of the critical frequencies in the state nl=11.

Typical standing waves in Ω1, Ω2and Ω3are shown in Figure 4 for N=5,7and 9. We can see that the geometrical meaning of the integer Ndefined in Eq. (21) is just the number of electronic waves distributed on the circumference of the quantum dot. Due to θ̇ave<0in Ω1trajectory set, as indicated in Table 2, the mean rotation direction of the electron in Ω1is clockwise. Because θ̇avemerely denotes the mean angular velocity, locally we may have θ̇>0during some short periods in which the electron rotates in an opposite direction as shown in Figure 3b.

Figure 4.

Typical standing-wave motions in the state nl=1−1 with wave number N=5,7 and 9. The trajectory sets Ω1, Ω2, and Ω3 refer to the three sets of radial trajectory defined in Figure 3a.

In the state ψ, we have two stagnation frequencies at ωL/ω=1173/12and ωL/ω=2/3. In the presence of magnetic stagnation, Larmor angular velocity ωL/ωis counterbalanced by the quantum angular velocity l/ρ2avesuch that the electron’s net angular displacement Δθaveis zero and the electron is stagnated within the quantum dot. The instantaneous dynamics of ρτand θτare solved from Eq. (43) at the stagnation frequency ωL/ω=2/3and the results are shown in Figure 3c. As expected, the net change of θτis zero after a period of oscillation. The projection of the computed complex trajectory on the real xyplane is a closed path as illustrated in Figure 2d. This closed path produced by magnetic stagnation isolates the electron from the exit of the quantum dot and is the main cause of electronic resistance.

Apart from the consequence of θ̇ave=0, the effect of magnetic stagnation is also reflected in the wave number N. From Eq. (21), the relation between θ̇aveand Ncan be expressed by

N=2π/Tρθave=2π/Tρl/ρ2ave+ωL/ωE44

There are infinitely many wavelengths distributed on the circumference of the quantum dot, as θ̇aveapproaches to zero. The variation of the wave number Nwith respect to the critical Larmor frequency ωL/ωfor the quantum state nl=11is demonstrated in Figure 5a. A prominent change of Nappears in the vicinity of the two stagnation frequencies ωL/ω=1173/12and 2/3, at which the wave number Napproaches to infinity. These two stagnation frequencies coincide with the locations of the resistance peaks by comparing with the experimental results as shown in Figure 5b.

Figure 5.

(a) The variation of wave number N with respect to the Larmor frequency ωL/ω in the quantum state nl=1−1. (b) The two stagnation frequencies, ωL/ω=11−73/12 and 2/3, coincide with the two peaks of the experimental curve of resistance.

4. Experimental verification

This section will compare the above theoretical predictions with the existing experimental data [4, 13] to confirm the fact that the effect of magnetic stagnation is the main cause to the resistance oscillation of quantum dots in low magnetic field. The experiment was performed in an AlGaAs/GaAs heterostructure with a carrier concentration ne=2.5×1011cm2. Resistance was measured at temperature T=1.4Kusing a sensitive lock-in amplifier at currents of typically 1 nA and a frequency of 12 Hz. The resulting resistance measurement in the range of low magnetic field B1.3is depicted in Figure 5b showing a strong peak located around B=0.22Tand three weak peaks at B=0.65T, 0.97T, and 1.21T.

Thus far, our analysis on quantum trajectory focuses on some specific states. In order to know the influence of the applied magnetic field on the resistance, we have to consider all the possible quantum states occupied in the device. At temperature T=1.4K, where the resistance is measured, the possible states to be occupied can be estimated by the Fermi-Dirac distribution,

fE=11+eEn,lEF/kBT,E45

where En,lis the energy level given by Eq. (17), and EF=8.5meV is the Fermi energy of the AlGaAs/GaAs heterostructure. All the possibly occupied states and their associated stagnation frequencies are listed in Table 3.

ωL/ωn
01234
l000000
−12/32/3, 0.2052/3, 0.373, 0.1192/3, 0.52, 0.192/3, 0.252, 0.124
−24/54/5, 0.3164/5, 0.543, 0.24/5, 0.73, 0.31, 0.1460.4, 0.213, 0.115
−36/76/7, 0.3916/7, 0.64, 0.260.39, 0.196
−48/98/9, 0.4450.7, 0.3, 8/90.454, 0.237
−50.4860.75, 0.347
−60.520.78, 0.38
−70.546
80.57

Table 3.

Stagnation frequencies ωL/ωevaluated in the quantum states nlat T=1.4K.

An incident electron subjected to an applied magnetic field Bmay enter any one of the occupied states listed in Table 3. The electronic resistance induced by Bdepends on the global transportation behavior across the quantum dot through all the allowable states. Magnetic stagnation slows down the electron’s angular rate and retards the passage of the electron. The angular motion is fully retarded and the electron is trapped in the quantum dot without contribution to the conductance, as θ̇ave0. To quantify the effect of magnetic stagnation, we define a magneto-stagnation function as following

SB=n,llnθ̇avenlB=n,llnωLω+lρ2aven,lE46

where the summation is taken over all the states listed in Table 3. The expression of ωL/ωas a function of Bhas already been given by Eq. (22). Upon comparing the prediction of Eq. (46) with the experimental results, we evaluate the constants in ωL/ωaccording to the experimental setup [4, 13], which gave ωc=eB/m=1.76BmeVand ω0=k/m=0.64meV. Using these data in Eq. (22) yields

ωLω=0.88B0.4096+0.7744B2E47

which in turn is substituted into Eq. (46) to express the magneto-stagnation function SBas an explicit function of B.

The electron’s total angular velocity via all admissible quantum states at T=1.4Kcan be estimated by the function SB. Because of SBas θ̇ave0, a large value of SBimplies that there is a high resistance to the electron’s angular movability. Accordingly, SBcan be reasonably treated as an alternative description of electronic resistance. Figure 5a and b illustrates the first evidence of this correspondence. At the two stagnation frequencies ωL/ω=1173/12and 2/3, corresponding to the two peaks of the resistance curve around B=0.22Tand B=0.65T, SBapproaches to infinity, even though only the state nl=11is considered in Figure 5a.

If magnetic stagnation takes place simultaneously in many states, its effect will be amplified. Stagnation frequencies such as ωL/ω=2/3, 4/5, and 6/7appear concurrently in different quantum states, as can be seen from Table 3. Because the stagnation function considers the superposition of lnθ̇avenlBcoming from all the allowable states, the value of SBis intensified at such stagnation frequency concurring in different states. According to the conversion formula Eq. (47), the magnetic field relating to the stagnation frequencies ωL/ω=2/3, 4/5, and 6/7is found to be B=0.65T, 0.97T, and 1.21T, respectively, which are just the locations of the three weak peaks of the resistance curve as shown in Figure 5b.

Figure 6 demonstrates the strong correspondence between the stagnation function SBand the resistance curve, where the resistance curve is superposed on the gray-level plot of the stagnation function SBwith the intensity of darkness representing the magnitude of SB. As can be seen, the gray-level distribution matches closely with the resistance curve and in that the dark bands of SBcorrectly locate the peaks of the resistance. The gray-level plot of SBhas several narrow dark bands and one broad dark band. The narrow dark bands come from the isolated stagnation frequencies at ωL/ω=2/3, 4/5, and 6/7, and their locations coincide with the three weak peaks of the resistance curve. The broad dark band of SBcovers the neighborhood of the strong peak of the resistance curve, which is formed by a series of closely distributed stagnation frequencies centered at ωL/ω=0.29, or equivalently, at B=0.22Tfrom Eq. (47). The agreement between the experimental data of magneto-resistance and the magneto-stagnation function SBconstructed from the quantum Hamilton dynamics Eqs. (19) and (20) is not surprising, if we recall that Eqs. (19) and (20) is fully determined by the wavefunction ψn,lrθ, which is responsible for the observed magneto-resistance in quantum dots.

Figure 6.

A gray-level plot of the stagnation function SB with the darkness intensity representing the value of SB is compared with the resistance curve. The resistance curve [4, 13] has a strong peak located around B=0.22T and three weak peaks at B=0.65T, 0.97T, and 1.21T. It appears that the locations of the three narrow dark bands coincide with the three weak peaks of the resistance curve, while the broad dark band covers the neighborhood of the strong peak of the resistance curve.

5. Conclusions

Parallel to the existing probabilistic description for a quantum dot by a probability density function ψψ, this chapter considered an alternative trajectory description according to a dynamic representation of ψconstructed from quantum Hamilton mechanics. The equivalence between a given wavefunction ψxand its dynamic representation ẋ=fxensures that the various quantum properties possessed by ψalso manifest in its dynamic representation. The established Hamilton dynamics for a quantum dot predicts that there are special magnetic fields, which can trap electrons within the quantum dot and cause a significant raise in the resistance. The comparison with experimental data validates this theoretical prediction. Apart from the magneto-transport considered in this chapter, many other features of a quantum dot, which were studied previously from a probabilistic perspective based on ψ, now can be reexamined from a trajectory viewpoint based on the dynamic representation of ψproposed here.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Ciann-Dong Yang and Shih-Ming Huang (April 4th 2018). A Quantum Trajectory Interpretation of Magnetic Resistance in Quantum Dots, Nonmagnetic and Magnetic Quantum Dots, Vasilios N. Stavrou, IntechOpen, DOI: 10.5772/intechopen.74409. Available from:

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