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The theory of coulomb blockade in semiconductor devices, focusing on recent experiments with double quantum dots and the regime of activated conduction. The periodic dependence of conductance on the voltage applied to a gate electrode, caused by electrostatic energy arising from a change in charge of the grain. The document also presents the results of theoretical studies on the effects of temperature and asymmetry on the coulomb blockade peaks.
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Academic and^ Research^ Staff
Professor Patrick A. Lee, Dr. Konstantin Matveev
Technical and Support Staff
Margaret E.^ O'Meara
Sponsor
Joint Services Electronics Program Grant DAAH04-95-1-
Project Staff
Professor Patrick A. Lee, Dr. Konstantin Matveev
The conductance through a small grain weakly coupled to^ metallic^ leads^ shows^ periodic^ depend- ence on the voltage^ applied^ to^ a^ gate^ electrode. This phenomenon is observed in^ both^ metallic'^ and semiconductor 2 devices, and is commonly^ referred to as the Coulomb blockade. This behavior^ of^ con- ductance is caused by electrostatic energy^ arising due to a change in the charge^ of^ the^ grain (quantum dot) by an electron tunneling through^ it.
The Coulomb blockade is typically observed in structures with a well defined quantum dot, which^ is separated from the leads by tunneling barriers. Recently, the experimental observation of^ Coulomb blockade has^ become^ possible^ in^ more^ sophisti- cated devices. An experimental group^ at^ Harvard has studied the Coulomb blockade in^ double quantum dots.^3 In a number^ of^ other^ experiments,^4 the phenomenon of Coulomb^ blockade^ was observed at unusually high^ temperatures^ T-^^100 K, where the transport may^ be^ due^ to the^ activation
of electrons over the barriers. In the past year, our research group worked on the theoretical problems raised by^ these^ experiments.
2.1.1 Transport through^ Double^ Quantum Dots
In these experiments, 3 the Coulomb blockade oscil- lations of conductance through a system of two quantum dots (figure^ 1)^ were^ studied.^ The^ contacts between the^ quantum^ dots^ and^ the^ leads^ were^ in the weak tunneling regime,^ with^ conductances G,r << e 2 /h. On the other^ hand,^ the^ contact between the dots was tuned^ by^ adjusting^ the^ gate voltage Vo in such a way that the corresponding transmission coefficient To scanned the whole region from 0 to 1. As a result, a series of conduc- tance dependences on the gate voltage, at^ different values of To, were measured.^5 The most basic^ fea- tures of the experimental data, including^ the^ split- ting of the Coulomb blockade peaks^ when^ To increases from 0 to 1, were explained in^ our^ pre- vious work. 6 We have now extended the^ theoretical description to include the effects of temperature and the small asymmetry of the double dot system on the heights and shapes of the Coulomb^ blockade peaks observed^ in^ the^ experiments.^ 3
The experiments have demonstrated^ that^ small^ tun- neling between the dots gives rise^ to^ a^ small^ split- ting of the Coulomb blockade peaks.^ At^ non-zero
1 H. Grabert and M.H. Devoret, Single Charge Tunneling, (New York: Plenum Press,^ 1992).
2 M.A. Kastner, Rev. Mod. Phys. 64: 849 (1992).
3 F.R. Waugh et al., Phys. Rev. Lett. 75: 705 (1995); F.R. Waugh et^ al.,^ Phys.^ Rev.^ B^ 53:^^1413 (1996).
4 K. Murase et al., Microelectr. Eng. 28: 399 (1995); E. Leobandung et^ al.,^ Appl.^ Phys.^ Lett.^ 67:^^2338 (1995);^ E.^ Leobandung^ et^ al., Appl. Phys. Lett. 67: 938 (1995).
5 K.A. Matveev, L.I. Glazman, and H.U. Baranger, Phys. Rev. B 54: 5637 (1996).
6 K.A. Matveev, L.I. Glazman, and H.U. Baranger, Phys. Rev. B 53: 1034 (1996).
Figure 1. Schematic view of the double quantum dot system. The dots are formed by applying negative voltage to the gates (shaded); the solid line shows the boundary of the 2D electron gas (2DEG). V! and V, create tunnel barriers between the dots and the leads while Vo controls the transmission coefficient through the constriction connecting the dots.
temperatures, the^ splitting^ may^ or^ may^ not^ be resolved depending on^ the^ relation^ between^ the temperature T and the charging energy Ec. We have studied the whole temperature dependence of the split peaks analytically; the results are pre- sented graphically in figure 2.
In the experiments with double dots, one usually tries to make the two quantum dots symmetric. However, a small asymmetry is unavoidable, and is reflected in the actual experimental data. For example, when the gate voltage is chosen in such a way that an electron can tunnel to the first dot without changing its energy, tunneling^ to^ the^ other dot can still cost some^ electrostatic^ energy.^ This energy itself depends on^ the^ gate^ voltage^ X,^ and leads to a periodic in X suppression of the peak heights. Our (^) results are summarized in figure 3, which strongly resembles the actual experimental data.^7
0.
10 -8 -6 -4 -2 0 2 4 Gate voltage X
6 8 10
Figure 2. The evolution of the split peaks with temper- ature. The gate voltage X is plotted in units of the dis- tance between splitted peaks. The peak splitting is
7 F.R. Waugh et al., Phys. Rev. Lett. 75: 705 (1995); F.R. Waugh (^) et al., Phys. Rev. 6 53: 1413 (1996).
54 RLE Progress Report Number 139
/.
metry of the barrier, -1 <^ A^ <^ 1.^ For^ symmetric^ bar- riers A = 0, and^ the^ peaks^ are^ similar^ to^ those^1 o^ of the blockade of tunneling conductance. It^ is^ inter- esting, however, that the activation energy EcX^ for the conductance (1) at off-peak values of the gate voltage is smaller than in the tunneling case 10 by^ a factor of two.
For an asymmetric barrier **A ***^ 0,^ and^ the^ peaks^ (1) have an unusual asymmetric shapes. This effect is specific to^ the^ activated^ regime.^ Thus,^ by^ mea- suring the^ Coulomb^ blockade^ peaks,^ one^ may^ be able to identify the mechanism of^ transport through the quantum dot.
Matveev, K.A., L.I. Glazman, and H.U. Baranger. "Coulomb Blockade of Tunneling Through a Double Quantum Dot." Phys. Rev. B 54: 5637 (1996).
Matveev, K.A., and L.I. Glazman,^ "Coulomb Blockade of Activated Conduction." Phys. Rev. B 54: 10339 (1996).
Matveev, K.A., L.I.^ Glazman,^ and^ H.U.^ Baranger. "Tunneling Spectroscopy of^ Quantum^ Charge Fluctuations in the Coulomb Blockade." Phys. Rev. B 53: 1034 (1996).
10 L.I. Glazman and R.I. Shekhter, J. Phys. CM 1: 5811 (1989).
56 RLE Progress Report Number^139