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Resonances in curved quantum wires
Volume 141, number 5,6
PHYSICS LETTERS A
6 November 1989
RESONANCES IN CURVED QUANTUM WIRES Pave! EXNER 1 Nuclear Physics Institute, Czechoslovak Academy ofSciences, 25068 1~ei near Prague, Czechoslovakia Received 12 December 1988; revised manuscript received 31 August 1989; accepted for publication 8 September 1989 Communicated by A.P. Fordy
A two-dimensional model of electron motion in a curved quantum wire of length 2D attached to a pair of macroscopic electrodes is studied. The wire is regarded as infinitely thin and supports a potential which is a combination ofa constant transversalmode energy and an attractive curvature-induced term. The system is shown to have at least one resonance which exhibits spectral concentration as D—~oo.
Recent!y developed fabrication techniques in microelectronics opened way to investigation of new quantum phenomena [1—8]and motivated also interesting mathematical problems [9]. Some ofthem concern electron motion in quantum wires, i.e., thin stripes of highly pure semiconductor material prepared on an insulating substrate. In view of the crystallic structure of the material, an electron moves in such a wire as a free particle of some effective mass m*, at least within the range of its mean free path which is typically 2—3 orders of magnitude larger than the wire thickness. In the simplest model therefore, the electron can be described as a free particle living on a strip of a constant width d with Dirichlet boundary conditions, It has been shown recently that bends on the strip can bind electrons [10]. The continuous spectrum of the electron Hamiltonian starts at the first transversal mode energy 2~=7t2/d2 (for simplicity, we put h2/2m* = 1); if the strip is curved and thin enough, then there is at least one eigenvalue below 2~(for an L-shaped strip, e.g., it equals 0.9321 [ll]).Fora strip with a smooth boundary, it can be seen rewriting the operator H= L~with Dirichlet boundary conditions into natural curvilinear coordinates s, u on the strip, —
Present address: Laboratory of Theoretical Physics, JINR, 141980 Dubna, USSR.
H=
—
-~-
(1 +uy)—2 2
V(s, u)=
_________ —
4(1 +uy)2
+
—
+
V(s, u),
2(1 uy” +uy)3
—
(la)
54 (1 u2y’2 +uy)4’ (lb)
where y(s) is the curvature of the boundary. Since 0 ~ u ~ d, for a sufficiently small d the operator (1) is close in a sense to the one-dimensional Schrödinger operator, d2 H~=
—
+A~ 3~~’(s)2,
(2)
—
at least for low-energy electrons cf. ref. [20], theorem 2. The operator (2) has always a bound state unless the strip is straight. The question naturally arises how such bound states can be observed. Our point in the present Letter is that it is actually a scattering problem, because in reality each quantum wire has a finite length (we denote it as 2D) and is attached to a pair of macroscopic electrodes as sketchedon fig. 1. Since a d, the bottom of the spectrum at the electrodes is much lower than 2~independently of the electrode matena!; one may suppose that it is zero. The eigenvalues referring to an infinitely long wire thus appear to be embedded into continuum for a finite D and one expects them to turn into resonances. To describe the associated scattering process, we
0375-9601/89/s 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
—
~‘
213
Volume 141, number 5,6
PHYSICS LETTERS A
~
6 November 1989
The first question is how to connect the segment to the half-plane. It can be done in a close analogy to the problem concerning a plane and a half-line [12]. In the present case only the even partial waves are present in the decomposition with respect to the connection point (but it makes no difference because only the s-wave can be coupled non-trivially to the segment) and also the normalization is different. The coupling is described by boundary conditions which we shall write down a little later. We can therefore limit our attention to the nontrivial part ofthe problem concerning the s-wave part of the wavefunction in the half-planes and write the electron wavefunction as w= (u1, f2, u3) with
Fig. I.
nian HD 2(ER÷, is of r, dr) the and form f 2( —D, D). The Hamiltou~eL 2eL
y~
/—u~j’—ru~—(4r2)~u
S)
HD
),
1\
f2 =~_f~+[2i_~y(s)2]f2 (u1)
(3)
the boundary conditions u3 _u~3~_ru~3_(4r2)_Iu3J L0(u~)=af2(RD)±bf~(R:D), with
___________________
x
L1(u1)=cf2(~D)±df~(±D)
(4)
Fig. 2.
adopt a few model assumptions. First of all, we shall regard the whole situation as two-dimensional and the electrodes as infinitely thick; they will be described as halfplanes with Neumann boundary condition on the borderline (cf. fig. 2). They are joined by a line segment of length 2D representing the quantum wire. The fact that in reality it has a finite thickness is expressed by keeping the transversalmode energy 2~as a fixed number; the potential supported by the segment is the sum of 21 and the curvature-induced attractive term ~y (s )2 We suppose that thefunction y is infinitely smooth, nonails. zero, 2for hasInafact, compact support and Iy(s)I~22V one may require y to be C4-smooth and to decay sufficiently fast as si [10]; it would make only the argument technically more complicated. Let us remark also that the Neumann conditions are essential for the present model since no other boundary conditions allow a nontrivial coupling of the halfplanes to the line segment cf. ref. [16] for details. It may not be so if the macroscopic wires are assumed to be three-dimensional [21]. —
~
—
214
at the junctions, where the minus (plus) sign corresponds to j= 1, (j=3), and Lk are the regularized boundary values U ( r) L0(u)=lim—, r—.O ln r L1 (u) = lim [u (r) L0 ( u) ln r] —
(5)
.
r-.O
The requirement of probability current conservation at the junctions selects a four-parameter family of the boundary conditions (4), namely a= i25/4e~)~ it sinfl’ b=
it
2514e~”~ sin /3’
c=
-~-
2514e ~
d=
-~-
-~-
It
it
(y—ln 2)— it.~/4 sin/I
,
2514e’~’~~ (y—ln 2)— it.~#/4 sin /1
(6a)
Volume 141, number 5,6
PHYSICS LETTERS A
6 November 1989
for real a, ö, ~andfl0, where y=O.57’7... is Euler’s constant and /=sin(a+~+~it)cosfi—sin(c~+~it),
fills iy(s) I
.r= sin (a+ ô)
tinued from the upper complex half-plane to For all large enough D, the function Q 1 ( D) simple poles in the lower half-plane parthas ofjust ~ atn
cos fl—sin
~=cos(a+ô+~it)
~,
<...
<21
(6b)
(7)
,
where n ~ 1. In order to find the resonances of HD and their relations to the eigenvalues (7), we use the factorization technique cf., e.g., refs. [13—15].We write the Hamiltonian as —
HD = TD
—
I
I (~I I)
,
(8)
where Tis the “kinetic” part and yD:=yr[_D, D]. The resonances are then poles of the function
Ql(z,D):=~~IyDI(HD_z)’(~IyDI)
(9)
continued analytically to the lower complex halfplane, which can be expressed by means of the perturbation and the “free” resolvent as Q 1(z,D)=I— [I+Q(z,D)]~ (l0a)
Q(z,D):=_iIyDI(TD_z)’(flyDI).
L~I=C\[P... u(z: Rez~21—~2+(Imz)2)] (11) for some ~j< (Aj—C~)”2.Then Q( D) can be con-
,
,
We have adopted in (4) the natural assumption that both junctions are the same. In fact, one should choose some particular values of a, b, c, d, motivating the choice by the low-energy limit of a more realistic solution to the problem of injection of electrons into the quantum wire. This is not an easy task and we bypass it by proving the existence of resonances for any boundary conditions ofthe described type; the scattering on the junctions plays then role of a background. The infinite wire is in our model described by the Hamiltonian (2). Under the stated assumptions about the curvature y, it has a finite number of simpie eigenvalues, 0< C1 <(2
and choose
~.
cosfl+cos(~+~it),
.~=cos(a+d)cosfl+cos~.
~22I,’2,
z~(D)=~(D)=~I(D)and
lim z~(D)=
(12)
for j= 1, n. Moreover, the resonances z1(D) exhibit spectral concentration: choose a one-parameter family (d3(D) : D>0) of positive numbers such that ...,
urn 53(D)= lim D-.oo
—0
(l3a)
D-~o~(D)—
and denote A1(D) = (~(D)—ô3(D), ~(D) +d~.(D), then the corresponding spectral measure converges strongly, s-lim D-.~EHD(4j(D)) =P~,
(l3b)
to the projection on the eigenspace referring to the eigenvalue e~. Sketch of the proof One uses a general factorization-technique theorem by Howland [14]. In order to express Q~ (z, D) from (10), one has to know the “free” resolvent (TD z) —‘. We introduce the aux—
iliary operator ~ which differs from TD by the boundary conditions which are changed to the separated ones, f2(~D)=0 and L0(u3)=0 for j=l, 3. The resolvent (TW~— z) ~ is a known integral operator; in particular, its “inner” part has the kernel K(x,y; z)= ch[,c(2D— Ix—yl )J—ch(,clx+yI) 2K sh (2,cD) (14) 1”2. The sought for —D
(lOb) (TD —z)’=
Hence the resonances are also points where the analytically continued function (lOb) is not invertible. Now we can formulate our main result. Theorem. Suppose that yeC~° (R) is non-zero, ful-
(T~ —z)~
4 + j,k=1 ~ 4UJk( z) I g3( z)> <~k(~‘) I
(15)
where the coefficients ~tJk(z) and the vectors g~(z) can be found from the requirement that (TD— z) ‘
215
Volume 141, number 5,6
PHYSICS LETTERS A
maps ic into the domain D ( TD). This yields a set of linear equations which can be solved explicitly [16]. The most important step in the application of Howland’s theorem is to check that Q ( z, d) tends to Q(z, on) corresponding to the operator (2) in the operator norm uniformly in Since (14) gives ~.
—
~yD(x)K(x, —~—
~y(x)
stitute of Mathematics, Ruhr University, Bochum, where this work was done, is gratefully acknowl-
e~x_~~I
y(y)
as D—~oo,where the r.h.s. is just the kernel of Q(z, ~x~) it remains to check the contribution from the finite-rank part in (15) is zero. It follows from the explicit form of ~tJk(z) and g~(z)mentioned above. Formulae (14) and (15) allow one to verify also the remaining hypotheses ofHowland’s theorem, in particular, compactness ofthe operator TD— z) x (~YD)for z from the resolvent set p(TD), existence of analytic continuation of Q( D) from the upper complex half-plane to ~ and the strong convergence (TD—z)’~(T~—z)’ as D—*oo for Im z= 0, where T~is the “free’ counterpart of the operator (2). Details of the proof can be found in ref. [16]. Let us finish with two remarks. First of all, we have not mentioned the rate of spectral concentration. A semiclassical estimate shows that it should be exponential, I’~(D)~exp[ —constX (2~_~,)I/2]. This is a positive feature from the experimentalist’s point of view because it gives hope for observation ofsharp resonances on not very long quantum wires, The last remark concerns the ways in which the resonance effect can be manifested. It certainly leads to a sheer variation in the transmission coefficient between the half-planes which is related to the conductance by the Landauer formula [18] — ~YD(
,
G = 2e2
T( )
h 1T(C)~
(16)
Tuning the applied voltage, one can change the electron energy, and by (15) the conductance of the whole structure. Let us mention an indirect but clear
216
indication for curvature-induced resonances in quantum wires in a recent experiment by Timp et al. [19]: they have found that the resistance of a manyprobe junction depends on the number of right-angle turns the electron must pass on its way between a pair of electrodes. The hospitality extended to the author at the In-
y; z) (~YD(Y))
2K
6 November 1989
edged. Thanks are due to Dr. P. ~eba for a useful discussion.
References [1] H. Sakaki, in: Proc. mt. Symp. on Foundations of quantum mechanicsin the94. light ofnew technology (Phys. Soc. Japan, Tokyo, 1984) p. [2] J.D. Bishop, J.C. Licini and G.J. Dolan, AppI. Phys. Lett. 46 (1985) 1000. [3] S. Datta eta!., Phys. Rev. Lett. 55(1985) 2344. [4]C.P.Umbachetal.,Phys.Rev.Lett.56 (1986) 386. [5]R.A.Webbetal.,PhysicaA 140 (1986) 175. [6] S. Dana and S. Bandyopadh~ray,Phys. Rev. Lett. 58 (1987) 717.
[7] H. Temkineta!., App!. Phys. Lett. 50(1987)413. [8] P. Exner and P. ~eba, Phys. Lett. A 129 (1988) 477. [9] P. Exner and P. ~eba, in: Schrodinger operators, standard and non-standard (World Scientific, Singapore, 1989) p. 85. [10]P. Exner and P. ~eba, J. Math. Phys. 30 (1989) No. 10. [11] P. Exner, P. ~eba and P. ~ovi~ek, On existence of a bound state in an L-shaped waveguide, Czech. J. Phys. B, to be published. [12]P. Exner and P. ~eba,J. Math. Phys. 28 (1987) 386, 2304. [l3]T. Kato, Math. Ann. 162 (1966) 258. [14] J.S. Howland, Trans. Am. Math. Soc. 162 (1971)141. [15] H. BaumgArtel and M. Demuth, J. Funct. Anal. 22 (1976) 187.
[161P.
Exner, A. mode! of resonance scattering in curved quantum wires, Ann. Phys. (Leipzig), to be published.
[17] space N.I. Akhiezer (Nauka,and Moscow, I.M. Glazman, 1969) p.Linear 106. operators in Hubert [18]R.Landauer,Phys.Lett.A85(!98!)9l. [19] G. Timp et al., Phys. Rev. Lett. 60(1988)2081. [20] P. Exner and P. ~eba, in: Stochastic methods in mathematics and physics (World Scientific, Singapore, 1989) p. 376. [211 P. Exner and P. ~eba, Czech. J. Phys. B38 (1988) 1095.
PHYSICS LETTERS A
6 November 1989
RESONANCES IN CURVED QUANTUM WIRES Pave! EXNER 1 Nuclear Physics Institute, Czechoslovak Academy ofSciences, 25068 1~ei near Prague, Czechoslovakia Received 12 December 1988; revised manuscript received 31 August 1989; accepted for publication 8 September 1989 Communicated by A.P. Fordy
A two-dimensional model of electron motion in a curved quantum wire of length 2D attached to a pair of macroscopic electrodes is studied. The wire is regarded as infinitely thin and supports a potential which is a combination ofa constant transversalmode energy and an attractive curvature-induced term. The system is shown to have at least one resonance which exhibits spectral concentration as D—~oo.
Recent!y developed fabrication techniques in microelectronics opened way to investigation of new quantum phenomena [1—8]and motivated also interesting mathematical problems [9]. Some ofthem concern electron motion in quantum wires, i.e., thin stripes of highly pure semiconductor material prepared on an insulating substrate. In view of the crystallic structure of the material, an electron moves in such a wire as a free particle of some effective mass m*, at least within the range of its mean free path which is typically 2—3 orders of magnitude larger than the wire thickness. In the simplest model therefore, the electron can be described as a free particle living on a strip of a constant width d with Dirichlet boundary conditions, It has been shown recently that bends on the strip can bind electrons [10]. The continuous spectrum of the electron Hamiltonian starts at the first transversal mode energy 2~=7t2/d2 (for simplicity, we put h2/2m* = 1); if the strip is curved and thin enough, then there is at least one eigenvalue below 2~(for an L-shaped strip, e.g., it equals 0.9321 [ll]).Fora strip with a smooth boundary, it can be seen rewriting the operator H= L~with Dirichlet boundary conditions into natural curvilinear coordinates s, u on the strip, —
Present address: Laboratory of Theoretical Physics, JINR, 141980 Dubna, USSR.
H=
—
-~-
(1 +uy)—2 2
V(s, u)=
_________ —
4(1 +uy)2
+
—
+
V(s, u),
2(1 uy” +uy)3
—
(la)
54 (1 u2y’2 +uy)4’ (lb)
where y(s) is the curvature of the boundary. Since 0 ~ u ~ d, for a sufficiently small d the operator (1) is close in a sense to the one-dimensional Schrödinger operator, d2 H~=
—
+A~ 3~~’(s)2,
(2)
—
at least for low-energy electrons cf. ref. [20], theorem 2. The operator (2) has always a bound state unless the strip is straight. The question naturally arises how such bound states can be observed. Our point in the present Letter is that it is actually a scattering problem, because in reality each quantum wire has a finite length (we denote it as 2D) and is attached to a pair of macroscopic electrodes as sketchedon fig. 1. Since a d, the bottom of the spectrum at the electrodes is much lower than 2~independently of the electrode matena!; one may suppose that it is zero. The eigenvalues referring to an infinitely long wire thus appear to be embedded into continuum for a finite D and one expects them to turn into resonances. To describe the associated scattering process, we
0375-9601/89/s 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
—
~‘
213
Volume 141, number 5,6
PHYSICS LETTERS A
~
6 November 1989
The first question is how to connect the segment to the half-plane. It can be done in a close analogy to the problem concerning a plane and a half-line [12]. In the present case only the even partial waves are present in the decomposition with respect to the connection point (but it makes no difference because only the s-wave can be coupled non-trivially to the segment) and also the normalization is different. The coupling is described by boundary conditions which we shall write down a little later. We can therefore limit our attention to the nontrivial part ofthe problem concerning the s-wave part of the wavefunction in the half-planes and write the electron wavefunction as w= (u1, f2, u3) with
Fig. I.
nian HD 2(ER÷, is of r, dr) the and form f 2( —D, D). The Hamiltou~eL 2eL
y~
/—u~j’—ru~—(4r2)~u
S)
HD
),
1\
f2 =~_f~+[2i_~y(s)2]f2 (u1)
(3)
the boundary conditions u3 _u~3~_ru~3_(4r2)_Iu3J L0(u~)=af2(RD)±bf~(R:D), with
___________________
x
L1(u1)=cf2(~D)±df~(±D)
(4)
Fig. 2.
adopt a few model assumptions. First of all, we shall regard the whole situation as two-dimensional and the electrodes as infinitely thick; they will be described as halfplanes with Neumann boundary condition on the borderline (cf. fig. 2). They are joined by a line segment of length 2D representing the quantum wire. The fact that in reality it has a finite thickness is expressed by keeping the transversalmode energy 2~as a fixed number; the potential supported by the segment is the sum of 21 and the curvature-induced attractive term ~y (s )2 We suppose that thefunction y is infinitely smooth, nonails. zero, 2for hasInafact, compact support and Iy(s)I~22V one may require y to be C4-smooth and to decay sufficiently fast as si [10]; it would make only the argument technically more complicated. Let us remark also that the Neumann conditions are essential for the present model since no other boundary conditions allow a nontrivial coupling of the halfplanes to the line segment cf. ref. [16] for details. It may not be so if the macroscopic wires are assumed to be three-dimensional [21]. —
~
—
214
at the junctions, where the minus (plus) sign corresponds to j= 1, (j=3), and Lk are the regularized boundary values U ( r) L0(u)=lim—, r—.O ln r L1 (u) = lim [u (r) L0 ( u) ln r] —
(5)
.
r-.O
The requirement of probability current conservation at the junctions selects a four-parameter family of the boundary conditions (4), namely a= i25/4e~)~ it sinfl’ b=
it
2514e~”~ sin /3’
c=
-~-
2514e ~
d=
-~-
-~-
It
it
(y—ln 2)— it.~/4 sin/I
,
2514e’~’~~ (y—ln 2)— it.~#/4 sin /1
(6a)
Volume 141, number 5,6
PHYSICS LETTERS A
6 November 1989
for real a, ö, ~andfl0, where y=O.57’7... is Euler’s constant and /=sin(a+~+~it)cosfi—sin(c~+~it),
fills iy(s) I
.r= sin (a+ ô)
tinued from the upper complex half-plane to For all large enough D, the function Q 1 ( D) simple poles in the lower half-plane parthas ofjust ~ atn
cos fl—sin
~=cos(a+ô+~it)
~,
<...
<21
(6b)
(7)
,
where n ~ 1. In order to find the resonances of HD and their relations to the eigenvalues (7), we use the factorization technique cf., e.g., refs. [13—15].We write the Hamiltonian as —
HD = TD
—
I
I (~I I)
,
(8)
where Tis the “kinetic” part and yD:=yr[_D, D]. The resonances are then poles of the function
Ql(z,D):=~~IyDI(HD_z)’(~IyDI)
(9)
continued analytically to the lower complex halfplane, which can be expressed by means of the perturbation and the “free” resolvent as Q 1(z,D)=I— [I+Q(z,D)]~ (l0a)
Q(z,D):=_iIyDI(TD_z)’(flyDI).
L~I=C\[P... u(z: Rez~21—~2+(Imz)2)] (11) for some ~j< (Aj—C~)”2.Then Q( D) can be con-
,
,
We have adopted in (4) the natural assumption that both junctions are the same. In fact, one should choose some particular values of a, b, c, d, motivating the choice by the low-energy limit of a more realistic solution to the problem of injection of electrons into the quantum wire. This is not an easy task and we bypass it by proving the existence of resonances for any boundary conditions ofthe described type; the scattering on the junctions plays then role of a background. The infinite wire is in our model described by the Hamiltonian (2). Under the stated assumptions about the curvature y, it has a finite number of simpie eigenvalues, 0< C1 <(2
and choose
~.
cosfl+cos(~+~it),
.~=cos(a+d)cosfl+cos~.
~22I,’2,
z~(D)=~(D)=~I(D)and
lim z~(D)=
(12)
for j= 1, n. Moreover, the resonances z1(D) exhibit spectral concentration: choose a one-parameter family (d3(D) : D>0) of positive numbers such that ...,
urn 53(D)= lim D-.oo
—0
(l3a)
D-~o~(D)—
and denote A1(D) = (~(D)—ô3(D), ~(D) +d~.(D), then the corresponding spectral measure converges strongly, s-lim D-.~EHD(4j(D)) =P~,
(l3b)
to the projection on the eigenspace referring to the eigenvalue e~. Sketch of the proof One uses a general factorization-technique theorem by Howland [14]. In order to express Q~ (z, D) from (10), one has to know the “free” resolvent (TD z) —‘. We introduce the aux—
iliary operator ~ which differs from TD by the boundary conditions which are changed to the separated ones, f2(~D)=0 and L0(u3)=0 for j=l, 3. The resolvent (TW~— z) ~ is a known integral operator; in particular, its “inner” part has the kernel K(x,y; z)= ch[,c(2D— Ix—yl )J—ch(,clx+yI) 2K sh (2,cD) (14) 1”2. The sought for —D
(lOb) (TD —z)’=
Hence the resonances are also points where the analytically continued function (lOb) is not invertible. Now we can formulate our main result. Theorem. Suppose that yeC~° (R) is non-zero, ful-
(T~ —z)~
4 + j,k=1 ~ 4UJk( z) I g3( z)> <~k(~‘) I
(15)
where the coefficients ~tJk(z) and the vectors g~(z) can be found from the requirement that (TD— z) ‘
215
Volume 141, number 5,6
PHYSICS LETTERS A
maps ic into the domain D ( TD). This yields a set of linear equations which can be solved explicitly [16]. The most important step in the application of Howland’s theorem is to check that Q ( z, d) tends to Q(z, on) corresponding to the operator (2) in the operator norm uniformly in Since (14) gives ~.
—
~yD(x)K(x, —~—
~y(x)
stitute of Mathematics, Ruhr University, Bochum, where this work was done, is gratefully acknowl-
e~x_~~I
y(y)
as D—~oo,where the r.h.s. is just the kernel of Q(z, ~x~) it remains to check the contribution from the finite-rank part in (15) is zero. It follows from the explicit form of ~tJk(z) and g~(z)mentioned above. Formulae (14) and (15) allow one to verify also the remaining hypotheses ofHowland’s theorem, in particular, compactness ofthe operator TD— z) x (~YD)for z from the resolvent set p(TD), existence of analytic continuation of Q( D) from the upper complex half-plane to ~ and the strong convergence (TD—z)’~(T~—z)’ as D—*oo for Im z= 0, where T~is the “free’ counterpart of the operator (2). Details of the proof can be found in ref. [16]. Let us finish with two remarks. First of all, we have not mentioned the rate of spectral concentration. A semiclassical estimate shows that it should be exponential, I’~(D)~exp[ —constX (2~_~,)I/2]. This is a positive feature from the experimentalist’s point of view because it gives hope for observation ofsharp resonances on not very long quantum wires, The last remark concerns the ways in which the resonance effect can be manifested. It certainly leads to a sheer variation in the transmission coefficient between the half-planes which is related to the conductance by the Landauer formula [18] — ~YD(
,
G = 2e2
T( )
h 1T(C)~
(16)
Tuning the applied voltage, one can change the electron energy, and by (15) the conductance of the whole structure. Let us mention an indirect but clear
216
indication for curvature-induced resonances in quantum wires in a recent experiment by Timp et al. [19]: they have found that the resistance of a manyprobe junction depends on the number of right-angle turns the electron must pass on its way between a pair of electrodes. The hospitality extended to the author at the In-
y; z) (~YD(Y))
2K
6 November 1989
edged. Thanks are due to Dr. P. ~eba for a useful discussion.
References [1] H. Sakaki, in: Proc. mt. Symp. on Foundations of quantum mechanicsin the94. light ofnew technology (Phys. Soc. Japan, Tokyo, 1984) p. [2] J.D. Bishop, J.C. Licini and G.J. Dolan, AppI. Phys. Lett. 46 (1985) 1000. [3] S. Datta eta!., Phys. Rev. Lett. 55(1985) 2344. [4]C.P.Umbachetal.,Phys.Rev.Lett.56 (1986) 386. [5]R.A.Webbetal.,PhysicaA 140 (1986) 175. [6] S. Dana and S. Bandyopadh~ray,Phys. Rev. Lett. 58 (1987) 717.
[7] H. Temkineta!., App!. Phys. Lett. 50(1987)413. [8] P. Exner and P. ~eba, Phys. Lett. A 129 (1988) 477. [9] P. Exner and P. ~eba, in: Schrodinger operators, standard and non-standard (World Scientific, Singapore, 1989) p. 85. [10]P. Exner and P. ~eba, J. Math. Phys. 30 (1989) No. 10. [11] P. Exner, P. ~eba and P. ~ovi~ek, On existence of a bound state in an L-shaped waveguide, Czech. J. Phys. B, to be published. [12]P. Exner and P. ~eba,J. Math. Phys. 28 (1987) 386, 2304. [l3]T. Kato, Math. Ann. 162 (1966) 258. [14] J.S. Howland, Trans. Am. Math. Soc. 162 (1971)141. [15] H. BaumgArtel and M. Demuth, J. Funct. Anal. 22 (1976) 187.
[161P.
Exner, A. mode! of resonance scattering in curved quantum wires, Ann. Phys. (Leipzig), to be published.
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