Transmission and I–V characteristics of laterally-confined resonant tunneling structures

Microelectronic Engineering 51–52 (2000) 201–210 www.elsevier.nl / locate / mee

Transmission and I–V characteristics of laterally-confined resonant tunneling structures D. Csontos*, H.Q. Xu Solid State Physics, Lund University, Box 118, S-221 00 Lund, Sweden Abstract Several experiments on electron transport through quantum dots of various geometries have shown the existence of fine structure superposed on the conventional resonant tunneling current peaks. These features were attributed to resonant tunneling through the laterally confined states inside the quantum dots. In this work we investigate the effects on the electron transport induced by the inclusion of narrow connecting constrictions between the laterally confined double-barrier resonant tunneling structures and the reservoirs. We show that if the interfaces between the constrictions and reservoirs are made abrupt, longitudinal resonant states are formed in the constrictions, giving rise to additional fine structure in the current. Furthermore, the constrictions introduce a fast pinch-off of the current peaks.  2000 Elsevier Science B.V. All rights reserved. Keywords: Resonant tunneling structure; Scattering-matrix method; Transmission; I–V characteristics

1. Introduction As early as 1974 Chang et al. [1] succeeded in demonstrating electron resonant tunneling through molecular beam epitaxy grown Al x Ga 12x As / GaAs quantum wells (QWs). The growth and processing techniques have ever since improved dramatically, making possible the fabrication of ultra-small structures yielding quantum confinement to one dimension (1D), so called quantum wires, and zero dimensions (0Ds), so called quantum dots (QDs). The first evidence of quantum confinement in QDs was reported by Reed et al. [2]. The system they studied consisted of Al x Ga 12x As / Iny Ga 12y As columns, containing an epitaxially grown double-barrier resonant tunneling structure (DBRTS), with lateral confinement defined by electron-beam lithography and reactive-ion etching. Fine structure superimposed on the conventional double-barrier resonant tunneling peaks were observed in the measurements of the electron transport through the structure and was interpreted as resonant tunneling through a discrete spectrum of states inside the DBRTS. Similar effects and interpretation were subsequently reported by Tarucha et al. [3] in Al x Ga 12x As / GaAs DBRTSs, with lateral confinement *Corresponding author. E-mail address: [email protected] (D. Csontos) 0167-9317 / 00 / $ – see front matter PII: S0167-9317( 99 )00477-3

 2000 Elsevier Science B.V. All rights reserved.

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defined by Ga ion-beam implantation. These fine features of resonant tunneling were theoretically analyzed by Bryant [4,5], which supported the interpretation made by Reed et al. and Tarucha et al. A different approach was recently developed by Wernersson et al. [6] in which matrices of tungsten discs were embedded in n-type GaAs, about 60 nm above a Ga 0.5 In 0.5 P DBRTS. An opening was deliberately left empty of tungsten discs in the middle of the lattice. Due to the Schottky depletion around the discs, the current was assumed to be restricted to flowing through the opening only. This was proved by variation of the area of the opening, whereby they observed that the current scaled with the area. In a later paper, Wernersson et al. [7] investigated the transport characteristics for a similar structure with a gate. By variation of the gate voltage, and thereby the lateral confinement, they observed fine structure superimposed on the I–V characteristics. It was argued by the authors that this fine structure may be due to the lateral confinement in the double-barrier resonant tunneling region. It was however also noticed by the authors that this model does not explain many details of the measurement. This is because the structure of the measured device is far more complex than the structure of resonant tunneling through a laterally confined double-barrier system. For instance, the metal gate, which defines a 0D system in the double-barrier region, should lead to narrow constrictions in the emitter and collector. In addition, there may exist abrupt potential variations in the narrow constriction regions due to epitaxially well-defined interfaces formed during the fabrication process. To explain the details of the measured data, a deeper analysis and modeling by taking into account the complexity of the structure is therefore in demand. In this work we report on a theoretical calculation of electron transport through a laterally confined DBRTS connected to the emitter and collector reservoirs via two narrow constrictions of finite length. We have found that due to the presence of the narrow constrictions, the I–V characteristics of the system show fine structure, in addition to the fine resonant tunneling structure commonly observed in electron transport through a 0D system. Therefore the results of the calculation may shed light on the physical origin of the features observed by Wernersson et al. The rest of the paper is organized as follows: we will begin by briefly introducing the model and the method of calculations in Section 2. The results and discussion will be presented in Section 3. Section 4 contains a summary.

2. Theoretical model In the calculations we study the transport through a DBRTS connected via narrow constrictions of a finite length to two semi-infinite electron reservoirs and its corresponding structure without the narrow constrictions (the latter is shown for the three-dimensional (3D) case in the inset of Fig. 1a and for the two-dimensional (2D) case in the inset of Fig. 1b). In order to simulate the geometrical and potential variations of the structure, we divide the system into several strips, with different geometry and potential profiles, in such a way that the potential in each strip is constant along the transport direction. In the following we shall assume that the transport takes place along the x-direction. The ¨ Schrodinger equation in each strip i is then written as

H

F

G

J

"2 ≠2 ≠2 ≠2 2 ]] ]2 1 ]2 1 ]2 1V is ( y, z) C i (x, y, z) 5 ´C i (x, y, z), 2m* ≠x ≠y ≠z

(1)

where the transverse potential V is ( y, z) is assumed to be hard-wall. The effective mass, m*, is chosen

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to be 0.067m 0 (corresponding to GaAs), and is taken to be constant throughout the structure for simplicity. The electron wave function in each strip i, C i (x, y, z), can be expanded in terms of the transverse eigenfunctions of the strip according to

O FA

C i (x, y, z) 5

i ab

i

i

i

i

G

e ik ab (x 2x 0 ) 1 B iab e 2ik ab (x 2x 0 ) w iab ( y, z),

(2)

ab

]]]]]] i where k iab 5œ2m* / " 2 (´ 2 E iab ), x 0i is a reference point for strip i along the x direction and w ab ( y, z) i and E ab are the transverse eigenstates and eigenvalues inside strip i. By expanding the eigenstates w iab ( y, z) in terms of the transverse eigenstates, fmn ( y, z), in the reservoirs, i.e., by writing i w iab ( y, z) 5 o mn C mn ab fmn ( y, z), we obtain the following expression for the wave function in strip i

Of

C i (x, y, z) 5

mn

mn

( y, z)

OC

i mn ab

FA

i ab

i

i

i

i

G

i e ik ab (x2x 0 ) 1 B ab e 2ik ab (x2x 0 ) .

(3)

ab

i Here the expansion coefficients C imn ab together with the eigenvalues E ab can be obtained by solving the following eigenvalue equation [8]

O f(´ mn

mn

2 E iab )dm9n 9,mn 1 kfm9n 9 ( y, z)uV is ( y, z)ufmn ( y, z)lgC imn ab 5 0,

(4)

where ´mn denotes the eigenvalue of the eigenstate fmn ( y, z) in the reservoirs. To find the expansion i i coefficients A ab and B ab for each strip i, we make use of the usual matching conditions, i.e. the continuities of the wave function and its derivative, at each interface. The connection between two adjacent strips i and i 1 1 is then made by a transfer matrix M(i, i 1 1) [9]

SD

S D

Ai Ai 11 , i 5 M(i, i 1 1) B B i 11

(5)

i,i 11 i,i 11 where Ai,i 11 and B i,i 11 are matrices containing the expansion coefficients hA ab j and hB ab j, respectively. To obtain the expansion coefficients in the left and right reservoir we make use of the scattering-matrix method [8,10] in which the connection between the incoming and outgoing states is made by a scattering-matrix S(L, R)

SDS

DS D

AR S 11 (L, R) S 12 (L, R) L 5 B S 21 (L, R) S 22 (L, R) L,R

AL , BR

(6)

L,R

where A and B are matrices containing the expansion coefficients in the left (L) and right (R) L,R reservoir, hA L,R ab j and hB ab j. The derivation of the scattering-matrix S(L, R) for this problem is given in [8] for the 2D case. A generalization to the 3D case is straightforward and will be presented elsewhere. Having obtained the expansion coefficients through Eq. (6), the transmission for electrons incident from the left reservoir in the transverse mode m9n9 to the right reservoir in the transverse mode mn can be calculated from T m9n 9,mn 5 k Rmn /k Lm9n 9 uA Rmn u 2 , while the current at a given bias voltage and temperature is given by [11]

(7)

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2e I(V, mF , T ) 5 ] h

(R )

EO

T m 9n9,mn (V, E)f fL (E 2 ( mF 1 eV ), T ) 2 fR (E 2 mF , T )g dE,

(8)

m9n 9,mn L

where (R) indicates that the summation is to be done for those values of (m9n9, mn) for which k m 9n9 and k Rmn are real, mF is the chemical potential at zero source-drain voltage and fL,R are the Fermi distribution functions in the left and right reservoir, respectively. At zero temperature Eq. (8) reduces to 2e I(EF , V ) 5 ] h

E

E F 1eV

(R )

O

maxheV, E F j m9n 9,mn

T m9n 9,mn (E, V ) dE,

(9)

where EF is now the Fermi energy in the reservoirs. In the calculations we assume the potential drop to occur solely over the QD, since the main source of backscattering and, thus, the resistance occur at the barriers that confine the dot along the longitudinal direction. It should be noted that in principle, we need to model the emitter and collector reservoirs by taking their transverse size to infinity. However, in practice, we have to assume a finite transverse size for the reservoirs. In this work, a transverse size which is four times larger than the transverse size of the laterally confined structure is assumed for the reservoirs.

3. Results and discussion In the following we will present and discuss results of the calculation for the transmission and I–V characteristics for laterally confined DBRTSs with various geometries. In particular, we are interested in studying the effects of narrow constrictions, connecting the DBRTSs to the semi-infinite reservoirs, on the electron transport at finite bias. In order to explain the calculated I–V characteristics of the system, we will first present and discuss the results of the calculation for the transmission. We begin by considering the energy dependence of the transmission at zero bias for the electron transport through a DBRTS connected directly to two 3D semi-infinite reservoirs (hereafter denoted as a 3D-0D-3D system) as shown in the inset of Fig. 1a and its corresponding 2D structure as shown in the inset of Fig. 1b (hereafter denoted as a 2D-0D-2D system). Fig. 1a shows the transmission through the 3D-0D-3D system with a 0D structure of 50350 nm 2 in the lateral dimensions and d 5 25 and 40 nm in the longitudinal dimension. Here we see a number of Lorentzian-shaped peaks of various heights and widths. These peaks can be attributed to the electron tunneling through individual 0D states inside the QD and can be labeled by the quantum numbers (l, m, n) of the states, where l denotes the longitudinal quantum number and m, n the two transverse quantum numbers. It is seen that the peaks with m ± n have a height larger than unit. This is because the 0D structure has a square cross-section and the states in the structure with m ± n are double degenerate. It is also seen that the width of the peaks in each curve depends strongly on the longitudinal quantum number l. The peaks become broader for higher values of l. This is simply due to the fact that the wave function is less localized along the longitudinal direction for a 0D state with a higher quantum number l. Fig. 1b shows the transmission through the 2D-0D-2D system with a 0D structure of 50 nm in the lateral dimension and d 5 25 and 40 nm in the longitudinal dimension (the same as in the 3D-0D-3D system). Compared with the calculation as shown in Fig. 1a, similar features are found in the

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Fig. 1. Transmission through laterally confined DBRTSs at zero bias; (a) is for the 3D-0D-3D system with a 0D structure of 50350 nm 2 in the transverse dimensions and d 5 25 and 40 nm in the longitudinal dimension; (b) is for the 2D-0D-2D system with a 0D structure of size 50 nm in the lateral dimension and d 5 25 and 40 nm in the longitudinal dimension. The thickness of the barriers in both (a) and (b) is 1 nm. The peaks are labeled by the quantum numbers [(l, m, n) in (a) and (l, m) in (b)] of the resonant states in the 0D structures, through which the resonant transmission takes place.

transmission through the 2D-0D-2D system. These features can be explained in the same way as for the 3D-0D-3D system and the peaks in the transmission can also be labeled similarly by the quantum numbers of the corresponding resonant states of the QD. However, for the 2D-0D-2D structure, we only have to use two quantum numbers, (l, m), to label the transmission peaks, where l denotes the longitudinal quantum number and m the sole transverse quantum number of the resonant states of the QD. In addition, there is no degeneracy in the resonant states of the QD and all the transmission peaks seen in Fig. 1b have an intrinsic height of unit. Beside these differences, the overall similarity between the transmission of the 2D DBRTS and the transmission of the 3D DBRTS indicates that the two systems have the same fundamental properties of electron transport. Thus, to study these properties qualitatively, we need only to consider one of the two systems. Since the calculation for the 3D DBRTS is numerically very demanding, it is natural for us to continue the study by considering only the electron transport through the 2D DBRTS in the following. Fig. 2 shows the transmission calculated as a function of bias and energy for the 2D-0D-2D system as shown in the inset of Fig. 1b with d 5 25 nm. It is seen that with increasing bias, the peaks

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Fig. 2. Transmission as a function of bias and energy through the 2D-0D-2D system as in Fig. 1a with d 5 25 nm.

observed at zero bias are shifted towards lower energy and decrease in height until the peaks finally vanish. The pinch-off of the transmission occurs when a 0D state is aligned with the conduction band edge in the emitter reservoir. Note that kinks seen in the height of the transmission peaks as a function of the bias voltage are due to the fact that we have assumed a finite transverse size for the reservoirs. However, the appearance of these kinks has no effect on the fundamental properties of the electron transport through the systems studied in this work. We now consider what will happen as we add narrow constrictions with a length of 100 nm and a width of 50 nm in between the DBRTS and the 2D reservoirs (hereafter denoted as a 2D-1D-0D-1D-2D system). Fig. 3 shows the calculated transmission for the 2D-1D-0D-1D-2D system. It is seen that the transmission spectrum differs substantially from the one shown in Fig. 2 for the 2D-0D-2D system. In addition to the shift towards lower energy with increasing bias, we observe a strong oscillatory behavior and the oscillations increase in amplitude, with increasing bias. The origin of these oscillations is the formation of longitudinal resonant states in the narrow constrictions [13,14]. The oscillations appear when a resonant state in the QD scans through the energies of the longitudinal resonant states in the constriction attached to the DBRTS on the emitter side. Later, we will show that this oscillatory property of the transmission leads to a fine structure in the calculated current for the 2D-1D-0D-1D2D system. We now consider the I–V characteristics for the structures as considered in Figs. 2 and 3. In Fig. 4 we display the calculated current at the Fermi energies EF 53.0, 7.0 and 13.0 meV. The solid lines correspond to the current through the 2D-1D-0D-1D-2D system and the dashed lines to the current through the 2D-0D-2D structure. At EF 53.0 and 7.0 meV, only one lateral mode is open for conduction in the narrow constrictions with a width of 50 nm. Since the lateral confinement in the constriction and QD is identical, no cross-coupling will occur [5]. Thus, only direct-channel tunneling between the first lateral mode in the constriction and the quantum state in the dot with the lateral

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Fig. 3. Transmission as a function of bias and energy through a 2D-1D-0D-1D-2D system. The system is the same as in Fig. 2, except that the 0D structure is now connected to the 2D reservoirs via two narrow constrictions with a length of 100 nm and a width of 50 nm.

quantum number m51 can occur. The peak seen in Fig. 4a (solid curve) corresponds to the transition from the first lateral mode in the constriction to the lowest state in the dot with the quantum numbers l51 and m51 and no further peak is seen in the system for the applied bias voltages less than 40 mV. We denote this transition as 1→(1,1). The onset voltage of the current peak occurs as the Fermi level in the emitter side is aligned with the (1,1) state in the QD. For the 2D-0D-2D system we, however, see two peaks in the calculated current. This is because, in the absence of the narrow constrictions, it is possible for both the lowest state, (1,1), and the second lowest state, (1,2), in the QD to couple to the electron states in the reservoirs. For the second peak, the onset voltage occurs as the Fermi level in the emitter is aligned with the energy of the (1,2) state in the QD. Fig. 4b shows the calculation for the current at EF 57.0 meV. Still, at this Fermi energy, only one lateral mode is open for conduction in the narrow constrictions of the 2D-1D-0D-1D-2D system. Thus only one broadened peak corresponding to the 1→(1,1) transition is observed in the calculated current (solid line). This time, however, the broadened peak appears at lower voltage and superimposed on the downward slope of this current peak are a series of steps. These steps are not visible in the calculated current for the 2D-0D-2D structure (see dashed line in Fig. 4b). As we mentioned before (see Fig. 3), they originate from resonant tunneling through longitudinal resonant states in the constriction on the emitter side. Note that for the 2D-0D-2D system, the two peaks seen in the calculated current in the lower voltage region have the same physical origin as the corresponding ones in Fig. 4a. Fig. 4c shows the calculated current at EF 513.0 meV. A broadened peak is seen in the calculated current in the low bias voltage region for both the 2D-1D-0D-1D-2D structure and the 2D-0D-2D structure. In contrast to the calculations shown in Fig. 4a and b, a fine structure is seen on the upward

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Fig. 4. Calculated I–V characteristics for the 2D-0D-2D system as in Fig. 2 (dashed lines) and the 2D-1D-0D-1D-2D system as in Fig. 3 (solid lines) at (a) EF 53.0 meV, (b) EF 57.0 meV and (c) EF 513.0 meV. Some steps or peaks are labeled by the quantum numbers (l, m) of the resonant states in the 0D structure, through which the resonant tunneling takes place.

slope of the low-bias current peak in Fig. 4c. This fine structure appears in the calculated current for both the 2D-1D-0D-1D-2D system and the 2D-0D-2D system. However, on the downward slope of the peak, a fine structure is only seen in the calculated current for the 2D-1D-0D-1D-2D system. To explain the fine structure on the upward slope of the low-bias current peak, we emphasize the following facts: first of all, the first two lateral modes in the narrow constrictions of the 2D-1D-0D1D-2D system are open for conduction at EF 513.0 meV. Thus, the transmission through the first two lowest resonant states of the QD now becomes possible in the system under applied bias. Secondly, at

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EF 513.0 meV and zero bias, the first resonant state of the QD is occupied and far below the Fermi energy of the reservoirs (see Fig. 1b, the lower curve). When the bias applied to the two systems is increased, the two resonant states of the QD will gradually fall into the energy window delimited by the electrochemical potential EF 1 eV in the emitter and the electrochemical potential (i.e. the Fermi energy) EF in the collector. Thus, as the bias increases, the current will increase and a fine structure will appear on the upward slope of the current. Furthermore, since the second resonant state of the QD is closer to the Fermi level than the first resonant state of the QD at zero bias (see Fig. 1b, the lower curve), the second resonant state will move to the energy window for resonant tunneling before the first one with increasing bias. The first step (or peak) in the fine structure on the upward slope of the low-bias current peak is therefore associated with the resonant tunneling through the (1,2) state and the second step (or peak) with resonant tunneling through the (1,1) state (as indicated in Fig. 4c). The fine structure on the downward slope of the low-bias current peak, seen only in the calculation for the 2D-1D-0D-1D-2D system, originates again from resonant tunneling through longitudinal resonant states in the constriction on the emitter side. Finally, it should be noted that the calculated I–V characteristics presented in Fig. 4 show another interesting effect of the narrow constrictions on the electron transport: the low-bias current peak is pinched off more abruptly in the 2D-1D-0D-1D-2D system than the 2D-0D-2D system. This behavior is clearly seen in the calculations for all the three Fermi energies and can again be explained as a result of the absence of the cross-coupling [4,12] as follows: since the confinement in the constriction and QD of the 2D-1D-0D-1D-2D system is identical, no cross-coupling between the different modes in the constriction and QD can occur. Thus, the transmission through all lateral modes will be pinched off simultaneously with increasing voltage, yielding a sharp drop in the I–V curve.

4. Summary In conclusion we have studied the electron transport through laterally confined DBRTSs with and without being connected by narrow constrictions to two semi-infinite reservoirs. Fine structures superposed on the main resonant tunneling current peaks are observed in the calculated current for both cases. We have shown that the inclusion of the narrow constrictions has little effect on the upward slope of the resonant tunneling current peak in the low bias region, but introduces an additional fine structure and a fast current pinch-off on the downward slope of the low-bias current peak. This additional fine structure originates from the formation of the longitudinal resonant states in the narrow constriction on the emitter side, while the fast pinch-off is due to the absence of the cross-coupling between the lateral modes in the narrow constrictions and the 0D structure.

Acknowledgements The calculations were performed within the LUNARC computational facility at Lund University. This work was performed within the Nanometer Structure Consortium in Lund and has been supported by the Swedish research agencies TFR and SSF. The authors would also like to acknowledge a partial financial support from the Knut and Alice Wallenberg foundation.

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