Tunneling through a quantum channel with impurities: An exactly solvable model

Physica E 4 (1999) 211–219

Tunneling through a quantum channel with impurities: An exactly solvable model Chang Sub Kima;∗ , Arkady M. Sataninb a Department

b Department

of Physics, Chonnam National University, Kwangju 500-757, South Korea of Theoretical Physics, Nizhny Novgorod State University, Nizhny Novgorod 603091, Russia Received 7 September 1998; accepted 12 January 1999

Abstract The quasi-one-dimensional nanochannel with two attractive impurities are considered theoretically to investigate the electron localization and transmission. We solve the single-electron problem exactly within the short-range interaction and the phase-coherent limit. Consequently, we determine the wave functions and the energy eigenvalues of the special electron states. The special states are the discrete levels in the continuum, thus forming an arti cial molecule in the electron waveguide. Further, we derive the analytical expression for the transmission amplitude and present a detailed analysis of the electron transmission through the quantum waveguide. ? 1999 Elsevier Science B.V. All rights reserved. PACS: 73.40.Gk; 73.23.Ad; 73.20.Dx Keywords: Transport and tunneling; Quantum waveguide; Impurity; Fano resonances

1. Introduction According to the Mott conception of the theory of electron localization, the propagating electron states (energy of electron occurs as continuum) and localized states (energy occurs as discrete levels) are separated each other by a de nite energy, mobility edge [1]. It is generally accepted that in transport phenomena participate only the propagating states. On the other hand, the basic principles of quantum theory admit the discrete energy levels in the con∗ Corresponding author. Tel.: +82-62-530-3361; fax: +82-62530-3369. E-mail address: [email protected] (C.S.Kim)

tinuum [2]. The examples of the latter case were discussed in atomic systems [3,4]. Thus, it seems very interesting and worthwhile to consider as to whether the discrete levels appear above the mobility edge in low-dimensional systems and to investigate transmission of electron through such discrete levels. The recent quasi-one-dimensional (Q1D) nanostructures open up a new possibility of taming quantum states and thus provide an arena for study of novel quantum coherent eects (see for instance Ref. [5]). The conductance of the quantum waveguides, in uenced by the electron diraction in the constrictions and by the electron interference with the impurities, may be expressed through the Landauer–Buttiker transmission formula [6,7]. In a perfect waveguide

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without impurities, the lowest subband level plays a role as the mobility edge for electron transmission. Only those electrons with energies greater than the mobility edge can propagate. In this regard the upper subbands energies in waveguide form an in nite series of quasi-mobility edges: the transmission enhances with the allowed number of channels. When impurities are present in the waveguide the localized states occur below the mobility edge, whereas above the mobility edge but below a higher quasi-mobility edge only resonant states are possible as a consequence of the coherent interaction of localized modes with propagating ones. The problem of resonant interaction is closely connected to the problem of conductance erosion due to impurities, that has been intensively studied both experimentally [8–10] and theoretically [11–21]. In particular, in Refs. [12–21] it was shown that a single attractive impurity gives rise to the Fano transmission line shape, possessing the asymmetric resonance and antiresonance structure together [22] (see Ref. [23] for a review of Fano resonances). Using the nite channel approximation it was shown recently that the widths of Fano resonance structure may go to zero [24]. In this work we study theoretically the coherent interaction of Fano resonances in the framework of a short-range interaction model in the electron waveguide [12,13]. We take an in nitely long, narrow quantum constriction with two attractive point impurities as a model system for electron transmission and solve the one-electron Schrodinger equation exactly to obtain the electron wave functions. We show that the electron wave functions may be normalized for energies above the mobility edge, accordingly the discrete levels appears in the continuum. In addition, we provide an exact expression for the electron scattering amplitudes and study the problem of transmission through the channel. We show that the electron transmission manifests very interesting, qualitatively new features as the transmission energy is tuned to the discrete levels. We also argue that the distinctive type of electron states might be prepared at the same energy, corresponding to the dierent boundary conditions. The paper is organized as follows. In Section 2 we solve the Schrodinger equation for an electron in the chosen nanochannel model. We show here that it is possible to create a discrete level in the continuum. In Section 3 the scattering matrix for electron trans-

mission through the channel is obtained. We investigate the coherent interaction between two Fano resonances and discuss the analytic structure of the scattering amplitude for the critical parameters of the system where the discrete levels appear. Finally, in Section 4 we summarize our results and present an experimental implication of the obtained novel coherent eects. 2. Localized states in quantum channels: “Fano molecule” We consider the electron motion in an in nitely long, Q1D nanochannel with width W . The channel is assumed to be placed in xy-plane along the x-axis in the space between y = 0 and W . The quantum states in a perfect channel may be described by (0) n; k (x; y)

= eikx ’n (y);

(1)

where k is the wave vector of electron motion along the channel, ’n (y) is the wave function in a transverse con nement potential Vc (y). The corresponding electron energy is given as En(0) (k) =

˜2 k 2 + En(0) ; 2m

n = 1; 2; : : : ;

(2)

where m is the eective mass of electron and En(0) is the subband energy of the transverse motion, for instance for the hard-wall potential, En(0) = ˜2 =2m(n=W )2 . In general, the wave function in the channel with impurities can be represented as (x; y) =

∞ P n=1

n (x)’n (y);

(3)

where the longitudinal wave functions found from the Schrodinger equation −

˜2 @2 2m @x2

n (x)

+

∞ P n0 =1

Vnn0 (x)

n (x)

are to be

n0 (x)

(4) =(E − En(0) ) n (x); R where Vnn0 (x) = dy’n (y)V (x; y)’n0 (y) is the matrix elements for the impurity potential V (x; y). We assume that two single-level impurities are located at (x; y) = (±L=2; ys ) in the waveguide, where the origin of the in nite waveguide has been chosen at the middle point (x ≡ 0) along x and the bottom of the

C.S. Kim, A.M. Satanin = Physica E 4 (1999) 211–219

waveguide (y ≡ 0) along y. We model the impurity potential according to Refs. [12,13] as V (x; y) = −

˜2  ((x − L=2) + (x + L=2))(y − ys ); m (5)

where  is the coupling constant of the electronimpurity scattering, characterizing the short-range electron-impurity interactions. Then, the continuity condition for wave functions n and their derivatives 0 n at the impurity locations brings about ¿ n

¡ n ;

=

0¿ ¡ n n0

= −2

∞ P

n0 =1 ¿ ¡ n and n

vnn0

¿ n0 ;

(6)

denote left and right wave functions where in two adjacent region around impurity locations x = ±L=2, and the matrix elements vnn0 are de ned to be vnn0 = ’n (ys )’n0 (ys ):

(7)

Now, we prove that for an electron energy in interval E1(0) ¡ E ¡ E2(0) the Schrodinger equation allows the normalized solution with the discrete eigenvalues. The symmetric solution to Eq. (4), possessing the even parity n (x) = n (−x), in the region where the scattering potential is zero can be constructed for n = 1 as 1 (x)

= a1 cos(k1 x);

1 (x)

= 0;

|x| ¡ L=2;

|x| ¿ L=2

(8)

and for n¿2 as evanescent modes, n (x)

= an cosh(|kn |x);

|x| ¡ L=2;

(9) = cn e−|kn | |x| ; |x| ¿ L=2; q where kn = 2m(E − En(0) )=˜2 . Applying matching conditions (6)–(9), we get n (x)

cos 1 = 0; |kn |e|n | an = 2

(10) ∞ P n0 =2

vnn0 cosh(|n0 |)an0 :

(11)

where n are the phases of the electron waves, de ned to be n = kn L=2:

(12)

Similarly, the antisymmetric solutions, n (x) = − n (−x), can be obtained and the discrete levels are determined from sin 1 = 0

(13)

|kn |e|n | an = 2

∞ P n0 =2

vnn0 sinh(|n0 |)an0 :

213

(14)

The simultaneous solutions to Eqs. (10) and (11) specify the symmetric localized states in the continuum. And, the discrete levels can be determined from Eqs. (13) and (14) for the antisymmetric case. Here, we focus our analysis only on the symmetric states. Since the solutions are speci ed by the coupled equations, we are confronted with the problem with two spectral variables. We shall use the energy E and the distance L as the relevant variables, (E; L). First, solution of Eq. (10) is given as ˜2 2 (2j + 1)2 ; j = 0; 1; 2; : : : : (15) 2mL2 And, after some algebra, Eq. (11) is reduced to ∞ cosh| | P n −|n | 2 ’n (ys ): (16) e 1 = 2 |kn | n=2

Ej = E1(0) +

An asymptotic analysis show that the sum in Eq. (16) is logarithmically divergent with the number of terms in the series which renders us to invoke a regularization procedure. We take a physical route for this using the following consideration. The divergence stems from the short-range interaction potential in the transverse direction, ∼ (y − ys ) in Eq. (5), accordingly from the use of Eq. (7) for the matrix elements. However, in real systems the impurity interaction is nite-ranged, and the corresponding matrix elements vnn0 are the decreasing functions of n and n0 . Accordingly, there exists a natural cut-o in the summation in Eq. (16). Thus, by introducing the nite-size impurity interaction in the transverse direction, extended over ∼ Wa but still in the short-range interaction limit Wa W (note that the delta-function representation of the impurity interaction along the propagation direction does not cause a trouble), only a number of N ∼ W=Wa terms are essential in the summation in Eq. (16). Then, the discrete eigenvalues corresponding to the localized states Eqs. (8) and (9) are the simultaneous real roots of Eqs. (15) and (16) with a nite number of terms taken. The graphical solutions for this problem is given in Fig. 1, where we have incorporated one-hundred channels and the other system parameters chosen are  = 0:361

and

ys = 0:440W:

The intersections specify the critical levels {Ec } and corresponding distances {Lc } between impurities.

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lated bound state level. For the antisymmetric states solution exist only for L ¿ La since for the smaller distance than La the antisymmetric level is pushed out of the wells. We have obtained that the minimal length La = 5:47W . It seems relevant to mention that when the in nite square-well is considered as a con nement potential it is possible to write the spectral equations as the systems of equations depending only on two dimensionless parameters: e = E=E1(0) and l = L=2W . It means that the width of channel W could be also chosen as the second spectral parameter instead of L. 3. Transmission through the discrete levels in the continuum In this section, we derive an exact expression for the transmission coecient and present a detailed analysis associated with the discrete levels of the arti cial molecule obtained in the previous section. To this end, we write the solutions of the Schrodinger equation for electron waves in each spatial regions as Fig. 1. Graphical solutions for two spectral variables (E; L): the upper gure is drawn in the considered full energy interval 0) (0) (0) E1 6E6E2 and the lower gure is for details near E2 ; where the intersections between solid curves (dotted curves) specify the symmetric (antisymmetric) localized states; the energy and the (0) distance are normalized with respect to E1 and W , respectively.

These levels are evidently discrete and lie in the continuum, i.e. above the mobility edge. Since a discrete level indicates a long-living electron state, we have just created an arti cial molecule composed of two impurities. Existence of such localized states is due to the coherent interaction of electron waves in the space between two impurities, playing roles as electronic mirrors. We shall investigate this point more clearly in the next section in terms of Fano resonances. By this reason, we may call the arti cial molecule “Fano molecule”. It is worthwhile to notice in Fig. 1 that an in nite series of the levels are arranged between two limiting values, Emin = 3:42E1(0) and E∞ = 4:00E1(0) . The former corresponds to solution of Eq. (16) for L = 0. The other limiting solution is the case when L → ∞ in Eq. (16), which corresponds to the iso-

n (x) n (x) n (x)

= An eikn x + Bn e−ikn x ; = an e

ikn x

+ bn e

ikn x

;

= Cn e

−ikn x

x ¡ − L=2; −L=2 ¡ x ¡ L=2; (17)

;

x ¿ L=2:

When we denote Tn0 n as the transmission coecient from channel n to n0 , the total transmission coecient can be expressed using the Landauer–Buttiker formula, P kn0 P Tn0 n = |tn0 n |2 ; (18) T= 0 n;n n;n0 kn where the transmission matrix tn0 n (and also the re ection matrix rn0 n ) is de ned through the relations among the wave amplitudes as P P Bn = rnn0 An0 : (19) Cn = tnn0 An0 ; n0

n0

By substituting Eq. (17) into Eq. (6) and after eliminating intermediate amplitudes an and bn , and also with use of a few steps of manipulations, we obtain the desired transmission matrices as ’n (ys )’n0 (ys ) tnn0 = nn0 − 2 ikn Ds Da ×(sin n sin n0 Ds + cos n cos n0 Da )

(20)

C.S. Kim, A.M. Satanin = Physica E 4 (1999) 211–219

215

and rnn0 = −2

’n (ys )’n0 (ys ) ikn Ds Da

×(−sin n sin n0 Ds + cos n cos n0 Da );

(21)

where the newly de ned functions are Ds = 1 + 2

N cos  P n in 2 e ’n (ys ); ik n n=1

(22)

Da = 1 − 2

N sin  P n in 2 e ’n (ys ): k n n=1

(23)

Here, it is important to note that formulae (20) and(21) are exact and the sums in Eqs. (22) and (23) run over all channel including the evanescent modes. One can verify that as distance L → 0 Eqs. (20) and (21) become tnn0 = nn0 − 2

’n (ys )’n0 (ys ) ikn D

and

(24)

rnn0 = −2

’n (ys )’n0 (ys ) ikn D

Fig. 2. Transmission coecient in the energy interval (0) (0) E 1 ¡ E ¡ E2 when the distance between impurities is L = 7:64W ; where the inset is the transmission near the second (0) (0) mobility edge E2 = 4 (the energy is in unit of E1 ). The dimensionless coupling constant is chosen as  = 0:361 and the y-coordinate of the impurity location is ys = 0:440W measured from the bottom of the waveguide. Also, the number of channels taken is N = 100.

with

and

N ’2 (y ) P n s : D = 1 + 2 ikn n=1


a = −2

This result was previously obtained in [14] by using the Dyson equation approach. In the following, we shall restrict our interest in the rst energy window (E1(0) ; E2(0) ) for incident electron energy. We will examine the analytic structure of the transmission amplitude Eq. (20) in the complex energy plane. In the energy window chosen, only n = 1 channel is opened and all other channels n¿2 are closed, accordingly the total transmission amplitude is simply given by t11 that can be spelled out from Eq. (20) as t11 (E)
s
a −2(’21 (ys )=ik1 )(sin2 1
s +cos2 1
a ) ; (25)
s
a where cos 1 i1 2 e ’1 (ys ) + 1
s = 2 ik1 =

− 2

N cosh| | P n −|n | 2 e ’n (ys ) |k | n n=2

sin 1 i1 2 e ’1 (ys ) + 1 k1

− 2

N sinh| | P n −|n | 2 e ’n (ys ); |k | n n=2

respectively. In Fig. 2 we depict the transmission coecient T (E) = T11 = |t11 |2 as a function of energy E for a chosen L = 7:64W . Same values have been used for N; , and ys as in Fig. 1 throughout the calculations. A speci c value of the eective mass m is not required for the numerical data since the mass-dependence is scaled out in terms of E1(0) = ˜2 2 =(2mW 2 ). One can see that much familiar Breit–Wigner resonances appear at the various quasi-bound states E = E˜ − i , represented by i ; (26) E − E˜ + i with the full transmissions at the resonance peaks, ˜ and the nite lifetimes of the corresponding quaE, sibound states, ∼ ˜= . Interestingly, however, for energies near the quasimobility edge, E2(0) = 4E1(0) , a dierent structure appears. In order to investigate this additional structure we have magni ed this region in t11 (E) =

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the inset where we observe the interesting asymmetric resonance and antiresonance structure. By analyzing Eq. (25) this new structure can be formulated as an analytical expression, E − E0 ; (27) E − E˜ + i which is identi ed as Fano line shape [22]. The resonance structure is characterized by complex poles Ep = E˜ − i to Eq. (25) or complex solutions to
s
a = 0. More precisely, there exist two types of poles which can be calculated from
s = 0 and
a = 0. The former speci es the symmetric resonant states and the latter speci es antisymmetric resonant states of the electron between two impurities. Here, we write down
s = 0 explicitly for later purposes, t11 (E) ∼

cos 1 i1 2 e ’1 (ys ) + 1 2 ik1 − 2

N cosh| | P n −|n | 2 e ’n (ys ) = 0: |kn | n=2

(28)

The antiresonance structure, or Fano dip, at E0 is determined by the condition of vanishing numerator in Eq. (25), ’21 (ys ) (sin2 1
s + cos2 1
a ) = 0; (29) ik1 whose solution is complex in general, but it may be possible to have real zero-energies at which the transmission quenches. Now, it is evident to be able to write the Fano structure as Eq. (27), in particular in the weak coupling limit  1 since the poles Ep and zero-energies E0 are located closely near resonances. In the inset in Fig. 2 we see the Fano structure located near the symmetric state Es = 3:994E1(0) and Fano dip near the antisymmetric state Ea = 3:999E1(0) of the double impurity system. Very interestingly, in our studied system we are able to observe the Breit–Wigner and Fano resonances together and complicated interactions among them. The Breit–Wigner resonance is connected with the interference of the electron waves between two impurities, whereas the Fano resonance is associated with the interaction of the quasibound states with continuum. Further study shows that one can isolate one resonance structure from another by changing the system parameters. As an illustration, in Fig. 3 we display the transmission for a particular distance L = 0:063W .
s
a − 2

Fig. 3. Fano resonances in the transmission for distance L = 0:063W , where the Breit–Wigner resonances are suppressed. The other parameters are the same as in Fig. 2.

In this case all Breit–Wigner resonances disappear, so as the asymmetric peak from the symmetric resonance state. And, the Fano structure near the antisymmetric state is pushed out of the considered energy window completely. Accordingly, only Fano dip around the symmetric level is seen at E0 = 3:984E1(0) . We have con rmed analytically that in this case the Breit–Wigner poles are located far away from the real energy axis so that the resonances are dominated by the Fano type. Next, we give connection between the obtained transmission formula and the discrete levels discussed in Section II. In Fig. 4 we present the electron transmission for several molecular size: L = 7:38; 7:45; 7:51; and 7.59 in units of W , near the critical level Ec = 3:995E1(0) that is the seventh symmetric discrete level in Fig. 1. The result shows that as L reaches the critical value Lc from smaller distances (Fig. 4(a) → Fig. 4(c)) the Fano resonance and antiresonance near the symmetric state collapses, leaving a nite transmission of ∼ 0:315 at Ec . In Fig. 4(d) we see that the Fano resonance structure reappears after the distance is gone through the critical value. In Section 2 we analyzed that the symmetric, arti cial molecular levels are formed at critical values (Ec ; Lc ) when Eqs. (15) and (16) have simultaneous solutions. Our analysis shows that the simultaneous solutions to Eqs. (15) and (16) satisfy Eqs. (28) and (29) automatically.

C.S. Kim, A.M. Satanin = Physica E 4 (1999) 211–219

217

Fig. 4. Transmission coecient T as a function of energy E: (a) L = 7:38W , (b) L = 7:45W , (c) L = 7:5116W , and (d) L = 7:59W . The other parameters are the same as in Fig. 2.

It means that the arti cial molecular levels correspond to the situation where the poles Ep in the lower complex plane and the zero-energies E0 on the real axis approach to the common values Ec on the real energy axis in the Fano transmission formula. Thus, the same energy levels Ec belong to the dierent class of the electron states: the localized states Eqs. (8) and (9) and the propagating states Eq. (17).

In Fig. 5, we present the transmission in the full energy window for the critical distance Lc = 7:5116W , where the inset is redundant to Fig. 4(c). Contrary to Fig. 3, the Breit–Wigner line shapes dominate the resonance structures but the dip in the transmission around the antisymmetric level of the Fano molecule. The resonance–antiresonance structure near the symmetric level is missing. The results show that it is possible to manipulate two distinctive type of

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C.S. Kim, A.M. Satanin = Physica E 4 (1999) 211–219

faster than the distance |E˜ − E0 |. Consequently, q becomes very large near the critical state. However, it should be noticed that at a critical state, e.g. the symmetric critical level in Fig. 4(c), the line shape cannot be represented by Eq. (30). The correct limit is given by Eq. (27) as T (Ec ) ∼ const. 4. Conclusion

Fig. 5. Transmission coecient T as a function of energy E at the critical distance Lc = 7:5116W ; where the Breit–Wigner resonances are dominant except the Fano dip around the antisymmetric resonance state. The other parameters are the same as in Fig. 2.

resonances in Q1D channel by varying the system parameters. Finally, we provide a connection between our results and the Fano function studied in Ref. [21] in a dierent context. For the system with the inversion symmetry, which is the case of the present work, it was shown that the transmission line shape can be cast into T () =

1 ( + q)2 ; 1 + q2  2 + 1

(30)

˜ where  ≡ (E − E)= and q ≡ (E˜ − E0 )= . 1 If the asymmetry parameter q is small, Eq. (30) behaves as ∼ 2 =(2 + 1), producing only a dip in the transmission. In the opposite limit, the transmission becomes ∼ 1=(2 + 1), manifesting a peak. Accordingly, there exists a crossover in the transmission from a small q to a large q behavior. An estimation shows that q ∼ 0:19 around the symmetric resonance level in Fig. 3, which is relatively small. That is why only a dip is shown up. We believe that similar arguments may be applied to other gures presented. Interestingly, our careful analysis con rms that, as the system parameters are tuned to a critical state Ec , the line width approaches zero 1 Note that zero-energies E indicate the original bound states 0 in Ref. [21]. However, this dierence goes away in the weak coupling limit considered in the present work.

Within the short-range interaction limit we have studied a model Q1D nanochannel with two attractive impurities. Consequently, we have tamed an arti cial molecule whose discrete levels lie in the continuum. The formation of such special electron states was attributed physically to the coherent interaction of Fano resonances near the symmetric and antisymmetric double impurity levels. Also, we have presented the exact expression for the transmission coecient of the system and have investigated the electron transmission through the system. We have shown that one can have a control over two distinctive resonances, the symmetric Breit– Wigner and the asymmetric Fano resonances, by changing the system parameters such as the width of channel or the distance between the impurities. And, the interaction between Fano resonances leads to the novel coherent eects such as merge of the resonance-antiresonance and the disappearance of the resonances. In particular, the electron transmission through the arti cial molecular discrete levels was investigated in detail and was seen to give the nite values. Our studied model permits an unusual degeneration of quantum states: one state belongs to the localized states whereas the other belongs to the propagating states. The eect of disappearance of resonances and tunneling through the discrete levels in the continuum may be observed in the high-mobility nanostructures in the ballistic regime. For example, in a typical Q1D channel in GaAs=A1x Ga1−x As heterostructures with width W the mobility edge is well approximated as E1(0) = 2 ˜2 =2mW 2 . Using the results obtained in this work, we estimate that the minimal critical distance and the lowest discrete energy are Lc = 0:578W and Ec = 3:99 E1(0) , respectively. The short-range impurity considered may be arti cially fabricated in the quantum waveguide using the recent nanotechnology [25].

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Acknowledgements This work was supported by a CNU Research Fund and also in part by the Ministry of Education of Korea through Grant No. BSRI-97-2431. AMS acknowledges the support from Russian Basic Research Foundation, Grant No. 97-02-169231. References [1] N.F. Mott, E.A. Devis, Electronic processes in noncrystalline materials, Claremont Press, Oxford, 1979. [2] J. von Neumann, E. Wigner, Z. Phys. 30 (1929) 465. [3] F.H. Stillinger, D.R. Herrick, Phys. Rev. A 11 (1975) 446. [4] H. Friedrich, D. Wintgen, Phys. Rev. A 31 (1985) 3964. [5] D.K. Ferry, H.L. Grubin, C. Jacoboni, A.-P. Jauho (Eds.), Quantum Transport in Ultrasmall Devices, Plenum NATO Series B 342, New York, 1995. [6] R. Landauer, Philos. Mag. 21 (1970) 863. [7] M. Buttiker, Phys. Rev. B 35 (1987) 4123. [8] P.L. McEuen, B.M. Alphenaar, R.M. Weeler, R.N. Sack, Surf. Sci. 229 (1990) 312.

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