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6.2. Device simulation 105

5

0

10

15

20

25

30

Figure 6.8: Single-particle levels ( |i |2 , i = 0 . . . 34; spin-degenerate). Thecut has been taken in the xy-plane 8 nm below the GaAs/AlGaAs interfacein the 0DEG. The lead channels are situated to the left and the right of eachindividual plot.

third and the fourth level which can also been seen in the 2D plots of theindividual wavefunctions labeled 2 and 3 in the xy-plane (Fig. 6.8). Thedistorted shape of the quantum dot potential is also re ected in the shape of

the individual wavefunctions shown in Fig. 6.8. One observes a tendency toform quasi-one-dimensional wavefunction- scars . These scars are related tothe classical trajectories of particles entering the dot, then bouncing within,and nally exiting. Two pronounced families of scars are visible. The rst(S1) is related to the wavefunctions labeled 4, 6, 9, 12, 15, 19, 23, 28, 33and the second (S2) to 5, 8, 11, 16, 20, 26, 31. The scars of one family havealways the same shape and position in the quantum dot, however, the numberof nodes of the wavefunctions increases with increasing energy. The classicaltrajectories for S1 and S2 are schematically shown in Fig. 6.9. The occurrenceof scars is an indication that the quantum dot is already acquiring properties


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106 Chapter 6. Simulation examples

of a disordered ballistic 1 structure in which conductance uctuations can beunderstood as interference of phase-coherent electrons traversing the dot viaa number of distinct classical paths. This is the dominant regime in large,highly populated ( N 200) quantum dots.

S2

S1

Figure 6.9: Approximate classical paths of the two wavefunctions scars S1 and S2.

Figure 6.10 shows the tun-neling rates calculated using thewavefunctions in Fig. 6.8, ver-sus the eigenenergies at a par-ticular value of the gate voltage(V g = 400 meV). The tunnel-ing rates belonging to one par-ticular scar family are linked bydotted lines. It can be seen thatespecially the rates related toS1 have almost the same valuewhich can be explained by the shape of the wavefunctions which is similar,leading to similar values for the overlap integral between the lead and the

0.002 0.000 0.002

Energy (eV)

10 2

100

102

10 4

10 6

Tunneling rate

k (s

1)

1

5

10

1824

29 35

S1

S2

Figure 6.10: Tunneling rates, i. e. elastic couplings of the wavefunctions inFig. 6.8 to the leads versus the single-particle eigenenergies. Some of thestates with the strongest coupling are labeled for comparison. The dotted lines link the rates which belong to one of the scar families (S1 and S2).

1In ballistic structures the elastic mean free path l of the particles exceeds the structure size.This, however, is always the case in small quantum dots at low temperatures where the mean freepart can be a few microns.


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0.8 0.6 0.4 0.2Gate voltage V g (V)

0

1x10 5

2x10 5

3x10 5

4x10 5

Conductance G (e2/h)


48

52

56

60

64

Electron number N

T=0.7 K T=1.0 K

Figure 6.11: Electron charging N (dotted line) and conductance G (solid line) in units of the conductance quantum e2 /h versus the gate voltage V g at two different temperatures.

quantum dot wavefunction. The highest values, however, are related to the

wavefunctions which are more uniformly distributed over the quantum dotarea, showing some alignment along the x-axis (see for example states 18, 24,29 and 35).

Figure 6.11 shows the electron charging of the quantum dot and the con-ductance versus the gate voltage. The envelope of the conductance peaks ex-hibits some modulation which is more pronounced at the lower temperature(T = 0 .7 K). Over the given range the quantum dot is charged, successivelyincreasing the number of electrons from 49 to 66. One of the main reasonsfor envelope modulation is that at nite temperature more channels are ther-mally accessible then at T = 0 K, i. e. when N is in the middle of a shell,more channels are available for charge transport than if N is just entering orleaving the shell. In Fig. 6.11 a pronounced step in the peak height occursat V g 400 mV. On the other side, the conductance peaks show an over-all tendency to linearly increase their peak height with an increasing numberof electrons in the dot. This can be explained by the stronger coupling of thequantum dot states to the leads because of the extended size of the 0DEG. Alsovisible is the thermal dependence of the conductance minimum between thepeaks. In agreement with the predictions of Section 3.4.2 the minimum of theconductance peaks approaches the maximum if the temperature is increased.

At this point, one notes that the values of the conductance is very low evenat the conductance peaks which have heights of a few 10 5 e2/h . Indeed, the


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0.0

1.0

2.0

3.0

4.0

Conductance G (e

2/h)


86

90

94

98

102

106

110

Electron number N

T=0.8 K T=1.0 K

Figure 6.12: Electron charging N (dotted line) and conductance G (solid line) in units of the conductance quantum e2 /h versus the gate voltage V g at two different temperatures.

noise level in conductance measurements experiments of this structure [72]is to be found at 10 3 e2 /h which means that above results are below therange which can be accessed in standard measurements. Therefore, the dopingdensity in the doping sheet was changed and the number of electrons in thequantum dot was adjusted to values around N = 85 at a negative gate bias of V g = 800 mV. Now, the conductance peaks exhibit values of the order of the conductance quantum which is GQ = 4 e2/h . The reason for this is thealready mentioned lateral extension of the 0DEG if the dot is getting lled withmore electrons. The electron-wavefunction coupling to the leads becomesstronger resulting in higher tunneling rates, Fig. 6.13. The overall values of the tunneling rates are some four magnitudes larger then in the previous case.This corresponds to the experimental observations [72] which show the onsetof the oscillations at V g = 1000 mV with around N = 80 electrons in thequantum dot. The 2DEG below the gates starts to get depleted at negative gate

voltages of around 200 mV. Above this value at around 200 . . . 100 mVat the control gate, the dot starts to extend below the gate. Experimentally,one observes a signi cant change in the spacing of the Coulomb-blockadepeaks. The spacing becomes smaller because of the much larger capacitanceof the extended quantum dot. In the simulations, the region where the electrongas is modeled as a 0DEG has to be extended below the control gate if thishappens. The calculations here presented explicitly exclude this case and theconductance calculation only includes the range where the gate regions arefully depleted. However, the beginning breakdown is always detected becauseof a sudden change in the conductance characteristics.


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0.002 0.000 0.002 0.004Energy (eV)

102

10 4

10 6

108

1010

Tunneling rate

k (s

1

) S1

S2

1

5

8

24 2934

4052 5945

47

Figure 6.13: Tunneling rates, i. e. elastic couplings of the dot wavefunctionsto the leads versus the single-particle eigenenergies. The dotted lines link therates which belong to one of the scar families (S1 and S2).

The modulations visible in both curves in Fig. 6.12 are due to coherent-resonant transport through the quantum dot. The quantum dot is acting asa resonator and the electrons are re ected by the quantum point contactssimilar to a Fabry-Perot interferometer. The period of the oscillations is

N

N

N 1

N + 1

V g

V g

E Fe

2

C

Figure 6.14: Schematic of the chemicalpotential as function of V g . The Fermilevel in the leads, E F , is indicated.

roughly 300 mV. The oscillationscan not be attributed to shell ll-ing effects for two reasons. First,shell- lling is mainly observableat low temperatures, usually dom-inating the conductance character-istics at a few tens of mK. Sec-

ond, both the upper and the lowerenvelope of the peaks are modu-lated which is a signi cant featurealso present in the measurements[72] and clearly identi ed as dueto coherent-resonant transport. Aquantum-mechanical interpretationcan be given considering the tunnel-ing rates for different wavefunctionsymmetries. It has already been ar-


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gued that those states which are more situated in the center of the quantumdot, showing some alignment with the x-direction contribute the most to thetransport by having the highest tunneling rates p . Strong coupling can beobserved for example for the states labeled 24, 29, and 34 (Fig. 6.13). If onetakes the two-fold spin-degeneracy of the states into account, a period for theenvelope modulation of around 10 peaks should be visible in Fig. 6.12. In bothcurves, which are taken at different temperatures, modulations of roughly thisperiod can be observed.

Figure 6.14 shows a schematic of the quantum-dot electron number N and the quantum-dot chemical potential as a function of the gate voltageV g according to the orthodox theory at T = 0 K. The chemical potentialapproaches the Fermi level, E F , in the leads at the step from electron numberN 1 to N . The occurring step in the chemical potential can serve as ameasure for the self-capacitance of the quantum dot, i. e. = e2 /C .

Figure 6.15 shows the gate-voltage dependence of the single-particleeigenvalue spectrum close to the quantum-dot chemical potential . Thesawtooth-like structure re ects the changes in the electrostatic and the chem-ical potential at the electron charging steps. The lling of a level pushes theenergies of the already lled levels energetically lower. This requieres addi-


1.8

2.0

2.2

Energy (meV)

48

52

56

Electron number N

Figure 6.15: Single-particle eigenvalue spectrum (dotted lines) and quantumdot chemical potential (solid line) versus gate voltage at T = 0 .8 K. The step-like curve is the electron number in the dot. The Fermi energy in the leads isE F = 0 eV. The occupation of a previously empty level occurs when the levelcrosses the chemical potential.


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tional energy and the charging energy is increased. Another effect is relatedto the degeneracies of the levels. Some of the levels are degenerate at somevalues of V g but the degeneracy lifts with increasing gate voltage (Fig. 6.16).Other levels are non-degenerate, becoming degenerate at some valueof V g andeventually becoming non-degenerate again. This effect is related to changesin the shape of the quantum-dot potential due to subsequent lling of stateswith different symmetry (see Fig. 6.8). This leads to oscillations in in theproportions of the spatial dimensions of the quantum-dot con nement, i. e.space-symmetries of the potential are established and broken again. Also vis-ible are shells of various sizes which also change with the gate voltage.

0.8 0.7 0.6 0.5 0.4 0.3 0.2Gate voltage V g (V)

4

3

2

1

0

1

2

Energy (meV)

Figure 6.16: Complete single-particle eigenvalue spectrum versus gate volt-age over the whole voltage range at T = 0 .8 K.


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0

5

10

15

Gate capacitance C

g (aF)

Figure 6.17: Gate capacitance C g from numerically differentiating theCoulomb staircase N (V g ) at T = 0 .8 K in Fig. 6.12.

The gate capacitance C g , i. e. the capacitive coupling of the control gateto the quantum dot may be de ned as

C g =dQdV g , (6.3)

where Q is the quantum dot charge. Therefore, the gate capacitance is calcu-lated by numerically differentiating the Coulomb-staircase, Fig. 6.17. Anothermethod to evaluate the gate capacitance is via the spacing of the Coulomb-blockade peaks. The difference V g between the N th and the (N + 1) th peak is used and the gate capacitance is calculated as

C g (N ) =e

V g=

eV g (N + 1) V g(N )

. (6.4)

The capacitance is now a function of N . Both methods give similar results.The gate capacitance increases from 5 aF to a value of around 7 aF over therange of the conductance calculations. These values are to small if comparedwith the measured value which is C g = 32 aF [72]. Consequently, the spacingof the conductance peaks is to wide compared with the measured results. Adiscussion of possible reasons for this discrepancy is given in the next section.If one now considers a factor 32 aF/ 5 aF 6 by which the calculated spectraare stretched in comparison to the measured ones, these results correspondwith each other.

The concept of capacitances is only meaningful for conductors, whoseelectrostatic potential can be speci ed by one number, the voltage. The con-cept of self-capacitances for a semiconductor quantum dot whose dimensions


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87 90 93 96 99 102 105 108Electron number N

1.1

1.2

1.3

1.4

1.5

Self capacitance C

dd

(N) (fF)

Figure 6.19: Quantum dot self-capacitance C dd as a function of the electronnumber N at the dot at T = 0 .8 K.

entiating the Coulomb-staircase with respect to the chemical potential, i. e.

C dd

(V g) =

dQ

d. (6.7)

The capacitance is now a function of the gate voltage V g (Fig. 6.20). Strikingfeatures in the capacitance curve are the two singularities which result fromthe two extrema in the quantum-dot chemical potential left and right fromthe conductance peak. Also, the self-capacitance is negative between the twosingularities in the range of the gate voltages where Coulomb blockade islifted and the quantum dot charge is shifted from N to N + 1 . The negativecapacitances re ect the highly unstable regime during the charging of the dot.The dot is free to oscillate between two groundstate energies which correspondto the two electron numbers. Its most stable con guration is acquired betweenthe blockade peaks, where the electron number is integer at low temperatures.In this regime the chemical potential is almost linear and consequently theself-capacitance is almost constant. There, the value of the self-capacitanceis around 100 . . . 150 aF which corresponds to the classical electrostatic limit(for the GaAs quantum dot = 13 , d = 200 nm, C = 2 0 d 150 aF).

In the classical limit for large, highly populated quantum dots with metal-lic properties the self-capacitance is much higher then the values observedhere. The consequence is that the steps in the chemical potential vanish sincethey are proportional to C 1 and the chemical potential becomes a constant.Consequently, the capacitance in the metallic regime is constant as well sincethe steps in the charging characteristics of the quantum dot vanish and the in-


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6.3. Comparison with measurements 115

0.80 0.78 0.76 0.74Gate voltage V g (V)

1.0

0.5

0.0

0.5

Capacitance C (fF)

1.9

2.0

2.1

Chem. pot. (meV)

87.5

88.0

88.5

89.0

89.5

E

lectron number N

Figure 6.20: Quantum dot self-capacitance C dd as a function of the gatevoltage V g at T = 0 .8 K.

crease of the island charge is linear. This resolves the problems of the negativecapacitances and makes them a unique feature of single-electron charging andCoulomb blockade invisible in the classical (metallic) regime.

6.3 Comparison with measurements

The measured linear-response conductance as a function of the control-gatevoltage V C for the simulated structure is shown in Fig. 6.21 [72]. All othergates were kept at constant voltages de ning the operating point of the device.The conductance shows dramatic oscillations of two orders of magnitude. Thepeak height is of the order of the conductance quantum ( e2/h ). This measure-ment was performed at a very low temperature of 25 mK. The number of electrons in the quantum dot was estimated to be N = 75 . . . 250. The sim-ulated results shown in Fig. 6.12 in the previous section are obtained with aquantum dot with an electron number of N = 87 . . . 110 and as such the sit-


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Figure 6.21: Coulomb-blockade oscillations measured as a function of thecontrol-gate voltage V C . All other voltages are kept constant. (Taken from

[72]. Courtesy T. Heinzel.)

uation should closely resemble the measured one. The main difference to theexperimental results is the much larger period of the conductance oscillations.This corresponds to the much lower simulated gate capacitance of 5 . . . 7 aFcompared to the measured one which is around 32 aF. A possible reason forthis discrepancy is the assumption of Fermi-level pinning at the exposed partsof the GaAs surface. More generally speaking, the exposed surface has a largeinuence on the general electrostatics of the device, also in uencing the cou-pling of all other gates amongst each other and the coupling of the gates to thesubstrate. This is especially evident for narrow split-gate structures as the oneused here [75].

Figure 6.22 shows the conductance oscillations over a much wider rangeof gate voltages and for several temperatures. One observes pronounced enve-lope modulations of the conductance peaks. The period of these modulationsis around 50 . . . 80 mV. The simulated data (Fig. 6.12) show a similar mod-ulation of the peak height, however, the period is again larger as with themeasured data. Both the measured and the simulated modulations are due tocoherent-resonant transport between the two QPCs, i. e. because of a Fabry-Perot like behavior of the quantum dot, acting as the cavity. Even though thesimulations could not be performed for temperatures lower then 800 mK, the


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6.3. Comparison with measurements 117

Figure 6.22: Coulomb-blockade oscillations measured as a function of thecontrol-gate voltage V C at various temperatures. The conductance curvesexhibit strong modulations due to coherent-resonant transport. (Taken from[72]. Courtesy T. Heinzel.)

same trend in the height of the conductance peaks for different temperatures

was observed. In agreement with the theoretical predictions the measurementsshow a smaller amplitude for the conductance oscillations and a general trendto higher conductance values between the peaks. This trend is con rmed bythe simulations (Fig. 6.11 and Fig. 6.12). Also in agreement with the mea-surements, the envelope modulations become less pronounced.


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Chapter 7

E come quei che con lena affannatauscito fuor dal pelago alla rivasi volge all acqua perigliosa e guata ...

Just as a swimmer, still with panting breath,now safe upon the shore, out of the deepmight turn for one last look at the dangerous waters, ...

D A N T E , Inferno, I

Achievements, failures andprospectsThis work describes the theoretical foundations and the numerical algorithmsused in a simulation software for single-electron devices. A prototype simula-tor has been implemented and its capabilities are tested at a particular example,a GaAs/AlGaAs heterostructure SET. To some extend, the approach chosen isvery limited because of its restriction to the quasi-equilibrium. Nevertheless,in the context of device simulation which relies on the drift-diffusion equa-tions, inappropriate to simulate single-electron tunneling, this work meansconsiderable progress.

Some aspects should be especially emphasized. First, the inclusion of size quantization and many-body effects in the solution of a non-linear Pois-son equation and the implementation of stable numerical algorithms has beenachieved. This allows for a self-consistent evaluation of single-electron tun-neling. Second, a sound method for the calculation of the tunneling rates forsoft electrostatic barriers which is based on the transfer-Hamiltonian method isincluded. Finally, these results are used in a free-energy minimization proce-

119


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120 Chapter 7. Achievements, failures and prospects

dure which allows for the computation of the discrete charging of the quantumdot and the linear-response characteristics of a SET at nite temperatures andfor realistic geometries.

The software, for which the preliminary name SIMNAD (Simulation andModeling of Nanodevices) is used, is embedded in the TCAD suite of ISE AG(Zurich). This allows for the utilization of the ISE grid editor TED ISE whichis especially suitable for tensor-product grids. Furthermore, the visualizationof results can be done by graphical software such as INSPECT ISE since alloutput les are compatible with the ISE datastructures.

Since the development of the SIMNAD simulation software was initializedin 1997, at lot of structures, some even with chances to nd device appli-cations, have been presented. It more and more seems to be clear that any-thing that could be of practical interest for logical circuits should be basedon silicon. Simulation of silicon single-electron transistors is well within thescope of the SIMNAD software. The calculation of the quantized levels for asix-valley bandstructure is possible in all three con nement models (0DEG,

1DEG, and 2DEG) implemented in SIMNAD and for a < 100 > -substrateorientation. However, silicon SETs, most of them in SOI-technology, haverather complicated three-dimensional geometries that require huge grids andsometimes cause severe numerical problems. Also, the simulation of siliconquantum dots is often not possible using the single-subband approximation(Eq. (4.38) and Eq. (4.39)) that allows for a considerable speed-up of simula-tions of GaAs/AlGaAs heterostructure SETs.

Apart from silicon, GaAs/AlGaAs based heterostructures are still rstchoice when properties of con ned electrons in zero-dimensional electrongases are investigated. The simulation of these structures can be carried outin a numerically stable and reasonably fast manner. However, there is stillneed for further investigation of the role of the exposed surface and the dop-ing sheet including the doping levels in order to become really predictable. Itshould also be mentioned that the numerical algorithms implemented still failto converge at very low temperatures below a few hundred mK. This regime,however, is experimentally of particular interest. At higher temperatures, ef-fects related to the single-particle spectrum are often disguised by the thermalbroadening of the levels.

Further work should start by dealing with the shortcomings of this work.Certainly, the direction of further work depends on what the simulation soft-ware will be used for. If a TCAD-oriented tool is planned, some work has to be


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121

devoted to a more advanced discretization scheme at unstructured grids whichgoes beyond the simple tensor-product/ nite-difference approach used in thepresent software. The use of automatic meshing, however, requires the formu-lation of adequate meshing criteria, especially suitable for the Schr odinger-Poisson system. At this point the author has no knowledge that a stable, self-consistent scheme for the Schr odinger-Poisson equation system employing abox discretization for instance on a general, unstructured grid has ever beenimplemented.

On the physics side, the inclusion of magnetic elds would be desirablesince the application of magnetic elds reveals a lot about the electronic struc-ture of the electronic gases in quantum dots. A whole literature of experimen-tal data related to the effects of magnetic elds in quantum dots is available.Some work on the discretization of the vector potential at tensor-product gridsusing nite differences has been done in the course of this project [80]. Atest implementation using the object-oriented program structure of SIMNADis already available.

The SIMNAD code has the capability for providing the advanced applica-tion oriented research in academia as well as in industry with more insight intoone of the most promising branches of modern device physics. The single-electron concept and single-electron devices are certainly beyond the ultimatescaling limits of the conventionalCMOS, however, they may proof to be its ul-timate replacement. Computer simulations and complex TCAD experimentswill be of key importance with this technology. Goal of this work was cer-tainly not to clarify the question whether or not the single-electron conceptwill have any applications. It is only reasonable to expect that the basic re-search undertaken today will eventually nd use in further microelectronic- ornanoelectronic devices. Some of the effects discussed in this work may playa more signi cant role for conventional devices, intentionally or not. Oth-ers may only have the potential for niche applications. However, that single-

electron devices will never have an impact on the industry is certainly a verdictto early to be proclaimed. The current status of nanodevices seems somewhatlike the situation in the 1940s when there was no reason to believe that vacuumtubes could ever be displaced.


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Appendix A

Thermodynamic propertiesof localized systems

Grand canonical systems at a temperature T are characterized by a Fermienergy E F , a particle number N , a volume V , and a total energy E . In equi-librium the volume V = V 0 , the particle number N = N 0 , and the energyE = E 0 are strictly constant. The entropy of the system S 0 = kB T ln 0 ismaximal, therefore being constant as well. 0 is the dimension of the classicalphase space, i. e. gives the number of accessible states of the system.

The system (Fig. A.1) is subdivided into a reservoir R with a set of thermo-dynamic variables N , V , E , , S and a small subsystem A characterizedby N,V,E, , S . The subsystem A is embedded in the much larger reservoir

Rand

V 0,

N 0, and

E 0are properties of the system as a whole (

A + R).The

subsystem exchanges energy and particles with the reservoir. The particlenumber in the reservoir changes according to the law of equilibrium changes

dE = T dS P dV + E F dN . (A.1)

The important assumption is made that the subsystem A is so small that T and E F of the reservoir remain xed even though the subsystem uctuatesamong its possible states which are labeled by . It is furthermore assumedthat E E 0 and N N 0 . Even if some of the states | may violatethe above condition, it is only a small number which can safely be ignored.

123


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124 Appendix A. Thermodynamic properties ...

A

R

N,V,E

T, E F , N , V , E

Figure A.1: Reservoir- (R) subsystem (A)model and thermodynamic variables.

Using Eq. (A.1), one obtains forthe constant derivatives with re-spect to entropy S and the par-ticle number N

E S = T E N

= E F .(A.2)

Within this model the Gibbsdistribution for the occupationnumbers of the small subsys-tem in particle exchange withthe reservoir can be derived. Thenumber of possible states in A isdenoted and that in R is denoted . Therefore, the total number of statesin A + R is = t . In equilibrium the entropy is maximal, t = 0 andS t = S + S = S 0 . The probability for the system as a whole of being in aparticular state is P eq = 1 / 0 . The number of choices for the reservoir statesgiven the subsystem is in one particular state | = |N , E is . Witht = and = 1 (since the system is in state | by assumption), theprobability of this state can be written as

P =0

, (A.3)

i. e. out of 0 possible states of the total system, there are states that ndA in state | .

The reservoir entropy S = kB T ln depends on the energy and theparticle number in the subsystem

S = S (E 0 E , N 0 N ), (A.4)

and with S 0 = kB T ln 0 it follows that

S 0 S = kB ln0

= kB ln P . (A.5)

It has to be emphasized that S 0 S is not the entropy of the subsystem A.The entropy of the subsystem is zero, since A is in a given state | . The


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125

equilibrium entropy of A + R is the average of S over all possible states.The thermodynamic average of a quantity f is

f =

P f , with

P = 1 , (A.6)

which for the entropy S = S 0 S leads to

S = kB

P ln P , P = exp k 1B (S 0 S ) . (A.7)

With the law of equilibrium changes, Eq. (A.1), one writes

S = S (E 0 E , N 0 N )

= S (E 0 , N 0) S E V ,N

E S N V ,E

N

= const E T

+ E FN T

, (A.8)

which substituted into Eq (A.7) gives the Gibbs distribution

P = Z 1

exp E N E F

kB T

Z =

exp E N E FkB T

, (A.9)

using the normalization condition for the P and Z is the partition functionof the grand canonical ensemble, i. e. the grand canonical partition function .The thermodynamic averages of the energy E = E and the particle numberN = N of the subsystem A are calculated using the Gibbs distribution inEq. (A.6) giving

E =

E P and N =

N P . (A.10)

Inserting the Gibbs distribution into the equation for the entropy of the systemS , Eq. (A.7), one obtains

kB T ln Z = E T S NE F = F NE F = , (A.11)

and consequently to an explicit form of the grand canonical potential forthe subsystem A, i. e.

= kB T ln

exp E N E FkB T

(A.12)

or = kB T ln Z. (A.13)


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126 Appendix A. Thermodynamic properties ...

All thermodynamicvariablesat xed V,T,E F of the system can now be calcu-lated by thermodynamic averaging of E,S,N if the energies and the electronnumbers {E , N } for all states | of the subsystem A are known.

If the subsystem contains a xed number of electrons and no particle ex-change between the reservoir and the subsystem occurs, the system has to betreated as a canonical ensemble . Instead of keeping the chemical potentialxed, a constant electron number according to N = N for all | is assumedand the thermodynamic variables of the system are determined by thermody-namic averaging at a particular N,T,V . The thermodynamic potential is theHelmholtz free energy F (N )

F (N ) = kB T ln

exp E

kB T (A.14)

or F (N ) = kB T ln Z (N ), (A.15)

with the canonical partition function Z(N) and the Gibbs distribution

P = Z 1

(N )exp E

kB T

Z (N ) =

exp E

kB T . (A.16)

To illustrate the principle difference between the Gibbs distribution and theFermi-Dirac distribution for single-particle levels, the following example isused. Consider a two level system {E 1 , E 2} with N = 1 particles. Accordingto Eq. (A.16) the occupation probability for the level is given as

P = P (E |N = 1) =exp E kB T

exp E 1

kB T + exp E 2

kB T

. (A.17)

If cast into a slightly different form, this results in

P = 1 + expE E F

kB T 1

with E F =12

(E 1 + E 2) and T =12

T, (A.18)

which corresponds to a Fermi-Dirac distribution, however, at a ctitious tem-perature T , which is half the true temperature of the electronic system. The


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127

Fermi energy E F can be de ned as the average of the two energies E 1 andE 2 . For higher electron numbers and the number of levels approaching in n-ity, the Gibbs distribution becomes equivalent to the Fermi-Dirac distribution.Numerical analysis con rming this fact for semiconductor quantum dots canbe found in [31].


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Appendix B

Transition rates

Subject of this appendix are the transition rates between the many-particlegroundstates in the quantum dot. The calculation is based on time-dependentperturbation theory in lowest order in the tunneling terms.

B.1 Time-dependent perturbation theory

The evolution of the Hamiltonian operator H with time is described by thetime-dependent Schr odinger equation

i t

|(t) = H| (t) , (B.1)

where |(t) is the time-dependent state vector of the system. The time-evolution of the state vector from an initial state |(t i ) to a nal state |(t f )can be written as

|(t f ) = U (t f , t i )|(t i ) , (B.2)

with a time propagator U of the form

U (t f , t i ) = exp i

t ft i dt H (t) , (B.3)129


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130 Appendix B. Transition rates

which reduces to

U (t f , t i ) = exp i

H(t f t i ) (B.4)

if the Hamiltonian operator is not explicitly time-dependent. The propagator

is usually evaluated via expansion of the exponential function in powers of the argument. The direct evaluation of the time propagator is in general avery dif cult, if not impossible task. A very common procedure is to splitthe Hamiltonian into an time-independent part H0 for which the stationarySchr odinger equation H0 |(0)i = E

(0)i |

(0)i can be solved exactly and an

time-dependent part Hp (t)

H = H0 + Hp (t). (B.5)

The time-dependent part is treated as weak perturbation to the system. Thetime propagator is expanded in terms of Hp in the basis of eigenstates of H0 .The total propagator is given as a series

U (t f , t i ) = U 0(t f , t i ) +

n =1

1i

n

U n (t f , t i ), (B.6)

where U 0 is the propagator of the unperturbed system

U 0(t f , t i ) = exp i

H0(t f t i ) (B.7)

and U n the n th order correction due to the presence of the perturbation

U n (t f , t i ) = t ft i dtn t nt i dtn 1 . . . t 2t i dt1 U 0 (t f , tn )Hp (tn ) U 0(tn , tn 1)Hp (tn 1) . . . Hp(t1)U 0(t1 , t i ). (B.8)

A small perturbation of the system due to Hp can be assumed if the matrixelements of the operator are small. If, furthermore, the system is consideredto evolve in short-time periods then

(0)i 2 |H p |(0)i 1 (t f t i )/ 1. (B.9)

Consequently, the series converges very fast and only the lowest terms in nneed to be considered. Since tunneling is always a small perturbation to theisolated quantum dot, i. e. the tunneling terms are always small compared to


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B.1. Time-dependent perturbation theory 131

the other energy scales in the system, only the rst order term in the expansionof U in terms of the perturbation Hp needs to be considered. The perturbationdoes not explicitly depend on time, it rather depends on the time differencet = t f t i . The probability amplitude, A(t), for one tunneling event isdened as

|(0)i t |(0)f : Af,i (t) = (0)f |U (t)|(0)i . (B.10)The unperturbed operator (0th order perturbation) generates only the trivialtime dependence but does not allow for transitions

|(0)i (t) = exp i

E (0)i t |(0)i (0) . (B.11)

However, one is interested in transitions between different many-particleeigenstates where i = f . In this case, the lowest perturbation gives the proba-bility amplitude

A(1)f,i (t) = (0)f |

1i

t

0dt1 U 0 (t t1)Hp U 0(t1 )|(0)i , (B.12)

which becomes after integrating over t1

A(1)f,i (t) = M f,i1

E (0)f E (0)i

exp i

E (0)f t exp i

E (0)i t

with M f,i = (0)f |H p |

(0)i , (B.13)

where M f,i is the perturbation calculated as the matrix element of the eigen-functions of the unperturbed Hamiltonian H0 . The probability W (1)f,i (t) thatthis perturbation has caused the transition is |A(1)f,i (t)|2 , the absolute squareof the probability amplitude. This leads to the nal form of the transitionprobability

W (1)f,i (t)t

= |M f,i |2(1/ 2)sin 2 (E (0)f E

(0)i )

t2

(E (0)f E (0)i )

t2

2 . (B.14)

Assuming long enough times re ected in the uncertainty relation betweentime and the energy difference (E (0)f E

(0)i )t , the following limes leads

to the transition rate which is de ned as

f,i = limt

W (1)f,i (t)t 1 (B.15)


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132 Appendix B. Transition rates

and therefore

f,i =2

|M f,i |2(E (0)f E (0)i ). (B.16)

B.2 Effective transition rates

The expression derived above is now used to calculate the transition rates forthe many-particle dot states. The Hamiltonian of the unperturbed system con-sists of a contribution due to the quantum dot and one contribution for thesource-side reservoir and the drain-side reservoir respectively

H0 = Hdot + H s + Hd . (B.17)

The eigenvalues E doti , E si , E di and the eigenfunctions |doti , |si , |di foreach operator are solutions to the time-independent Schr odinger equation forthe particular subsystems. The total energy is E (0)i = E doti + E si + E di andthe total wavefunction factorizes as |(0)i = |doti |si |di . The tunnelingHamiltonian which acts as a small perturbation to the system is

Hp = H ts + H td with H ts/ d = Ht ,s/ d + H

t ,+s/ d , (B.18)

where one term (-) describes the tunneling out of the dot

H t ,s/ d =k,m

T s/ dk,m cs/ d ,k cm (B.19)

and the term (+) describes the tunneling into the dot

H t ,+s/ d

=k,m

T s/ dk,m

cm

cs/ d ,k

(B.20)

summing over all dot states and lead states. Therefore, it is assumed that tran-sitions between the dot states occur solely if an electron is added or removedfrom the dot. Transitions which keep the number of electrons xed and couldbe due to electron-phonon interactions are neglected here.

Transitions which reduce the number of electrons in the quantum dot aretransitions that move one electron from the dot to the source or drain reser-voir. Inserting the tunneling Hamiltonian from Eqs. (B.20) and Eq. (B.19)


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B.2. Effective transition rates 133

into Eq. (B.13) leads to

M s/ d ,f,i =k,m

T s/ dk,m sf | df | dotf |c

s/ d ,k cm |

doti |di |si

M s/ d ,f,i =k,m

T s/ dk,m Df,i,m A

s/ d ,f,i,k B

d / s,f,i , (B.21)

with the matrix elements

D f,i,m = dotf |cm |doti

As/ d ,f,i,k = s/ df |c

s/ d ,k |

s/ di

B d / s,f,i = d / sf |

d / si .

The expression for M s/ d ,f,i contains a product of an operator acting on the leftor right reservoir Hilbert space only ( cs/ d ,k ) andan operatoracting exclusivelyon the dot Hilbert space ( cm ). The matrix elements are inserted into Eq. (B.16)for the transition rate. The effective transition rate is calculated by summingover all possible nal states in the reservoirs which can be lled leaving aninitial dot state empty

W D ,f,i =s,d

f

f,i . (B.22)

The following relations are used

s/ di |s/ df = i,f (completeness of |

s/ di )

s/ di |cs/ d ,k cs/ d ,k |

s/ df =[1 f s/ d (E

s/ dk E

s/ dF )]k,k

doti |cm cm |

dotf = f d (E dotm )m,m ,

where f dot is the occupation distribution function for the quantum dot andf s/ d is the Fermi-Dirac distribution function for the reservoirs which containsthe reservoir chemical potential and the temperature. Since the initial and thenal state in the dot are characterized by the electron number N , the followingnotation is adopted for the transition rate W , W D ,f,i = W

s/d (N, N 1).Finally, one arrives at the expression

W s/d (N, N 1) =m

s/ dm (E s/ dk )[1 f s/ d (E

s/ dk E

s/ dF )]f dot (E

dotm )

with s/ dm (E s/ dk ) =

2

|T s/ dk,m |2(E dotf E doti + E

s/ dk ) (B.23)


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147/161

Appendix C

Density-functional theory

The many-particle problem of interacting electrons in an external potentialhas been a key problem since the introduction of quantum mechanics in thelate 1920s. A rst solution in the form of a mean- eld theory was providedby the Hartree theory in which the electrons are described by an effectivesingle-particle Schr odinger equation and the effect of the other electrons, thescreening , is included via an effective potential, the Hartree potential. Whilethe Hartree theory still failed to include quantum mechanical interactions, thelater introducedHartree-Fock theory expanded the concept and included quan-tum mechanical exchange.

The Hartree-Fock theory proved to be useful for systems with small num-bers of particles. However, the expansion of the many particle wavefunctionin Slater-determinants, which even for small systems leads to huge numeri-

cal problems, made the application to solid-state systems prohibitive [46]. Anearly alternative to Hartree-Fock theory was the Thomas-Fermi theory [47, 48]which gave, on a heuristic basis, a description of the physical properties of anelectron gas by use of a density-functional. However, it took until 1964 thatan formally exact theory of the many-particle problem of a system of inter-acting electrons was formulated. In their pioneering paper Hohenberg andKohn 1 [49] proved that the ground state properties of an arbitrary system of interacting particles is uniquely represented by its charge density.

1W. Kohn received the Nobel prize in Chemistry for his work on density-functional theory in1998.

135


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136 Appendix C. Density-functional theory

In the following section a short account of the theory developed by Hohen-berg, Kohn and Sham, the density-functional theory (DFT) is given. A moredetailed presentation of density-functional concepts can be found in [50].

C.1 Hohenberg-Kohn theorem

Consider a system of interacting particles described by the many-particleSchr odinger equation with the Hamiltonian operator

H = T + V + W and H| = E | (C.1)

where T is the kinetic operator, V the operator of the external potential, W theoperator of the electron-electron interaction which, at least for semiconductorsystems, can assumed to be Coulombic, is the many particle wave-function,and E the total energy of the system. The Hohenberg-Kohn theorem states inits rst part that the external potential V = V = |V| is always a uniquefunctional of the true ground state charge density 2 n

V = V [n] + const . (C.2)

In a second statement a (total) energy functional E V [n] is postulated whichis a unique functional of the charge density as well and which, assumingcharge conservation, can be minimized giving the total energy of the system,E V = E V [n] and E V [n] < E V [n ] if n is not the ground state density.Therefore, the ground state density determines the external potential and themany-particle Hamilton operator and consequently all ground state propertiesof the electron gas in a unique way. The restriction to the ground state prop-erties, however, is one of the mayor weaknesses of density-functional theory.

The properties of excited states are beyond the scope of the basic theory.The second part of the Hohenberg-Kohntheorem enables one to explicitly

calculate the total energy of the ground state. Introducing the following formof the total energy functional

E V [n] = dr n(r )V (r ) + F [n] (C.3)2Originally density-functional theory was only introduced for electron gases, however, it is

applicable to hole gases as well.


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C.2. Kohn-Sham equation 137

where F [n] is a universal functional independent of the external potential. Itcontains the kinetic contribution to the total energy and the electron-electroninteraction, and therefore is the same for all kinds of electronic systems

F [n] = T s[n] +12

dr dr

n(r )n(r )|r r |

+ E xc , (C.4)

where T s is the functional of the kinetic energy for non-interacting electronswith the charge density n , the second term is the Hartree energy E H , and thelast term is the so called exchange- and correlation energy which is de ned as

E xc = T T s[n] + W E H . (C.5)

The exchange- and correlation energy contains the difference between the ex-pectation value of the kinetic operator in the many-particle Hamiltonian, T ,and the energy T s of the non-interacting system and the difference between theexpectation value of the operator of the electron-electron interaction W andthe classical Hartree energy E H . This de nition of E xc ensures the decompo-sition of F [n] according to Eq. (C.4). The two functionals T s and E xc are not

known from the beginning. E xc is given within appropriate approximations asthe local-density approximation (LDA).

C.2 Kohn-Sham equation

The Hohenberg-Kohn theorem provides the basis for the calculation of thetotal energy using a variational principle with respect to the charge density.One now tries to nd an appropriate representation of this charge density byassuming a system of non-interacting particles with the density n s . Accordingto the Hohenberg-Kohn theorem the total energy functional is given as

E V s [n s] = T s[n s] + dr ns(r )V s (r ) (C.6)

where n s can be represented by a sum over squared single-particle wavefunc-tions i (r ) weighted by an occupation factor N i

n s =

i=0N i |i (r )|2 . (C.7)

It can be shown (Kohn-Sham theorem) that for every system of interactingparticles there exists a single particle potential V s so that the ground state


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138 Appendix C. Density-functional theory

charge density n of the interacting system equals that of the non-interactingsystem n s . For the effective single-particle potential V s an effective single-particle Schr odinger equation, the Kohn-Sham equation 3 can be formulated

2

2m

2 + V s(r ) i (r ) = i i (r ). (C.8)

The system of interacting particles in an external potential V is now projectedonto a system of non-interacting particles in an effective single-particle poten-tial V s . Equation(C.8) can be solved exactly if the effective potential is known.The kinetic energy functional of non-interacting particles can be written as

T s =

2

2m i dr i (r ) 2i (r ). (C.9)Variation of the functional of the total energy Eq. (C.3) with respect to thecharge density and consideration of the Kohn-Sham equation leads to an ex-plicit formulation for the effective potential

V s(r ) = V (r ) + dr n(r )|r r | + V xc (r ). (C.10)The second term is the Hartree potential V H , which is related to the elec-trostatic potential as V H = q. The exchange-correlation potential isformally given by

V xc (r ) =E xc [n]n(r )

. (C.11)

Equations (C.7), (C.8) and (C.10) describe in an unique manner the system of non-interacting particles which is related to the system of interacting particlesvia the identical groundstate chargedensity. These equations have to be solved

self-consistently. Usually, one begins with an assumed n(r ), constructs V s(r )from Eq. (C.10) and Eq. (C.11), and nds a new n(r ). The Kohn-Sham totalenergy (ground-state energy) is given by

E [n] =

i=0

N i i 12 dr n(r )V H (r ) + E xc dr n(r )V xc (r ). (C.12)

The interpretation of the single-particle eigenvalues i and the single-particlewavefunctions i which are the solutions of the Kohn-Sham equation remains

3We restrict to the vacuum problem with m being the mass of the free electron.


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C.3. Local-density approximation 139

a problem. They describe a system of non-interacting (quasi-) particles, but nodirect physical meaning for the system of interacting particles can be attachedto them 4. Nevertheless, they are interpreted as single-particle (orbital-) ener-gies which suf ciently accurate describe the bandstructure in the crystallinesolid-state.

C.3 Local-density approximation

The relatively straightforward formulation of the Kohn-Sham formalism of DFT is only possible because all many-particle properties of the electron gaswhich are dif cult to describe, especially the exchange- and correlation inter-action between identical particles are relocated in the functional E xc . Thisfunctional is not explicitly known and, as already mentioned, one depends onappropriate approximations.

The most widely used approximation is the local-density approximation(LDA)

E LDAxc = dr xc (n) n ( r )n(r ), (C.13)where xc is a function of the density (not a functional) which is de ned asthe exchange- and correlation energy per electron for a homogeneous elec-tron gas. It locally approximates the density of the inhomogeneous electrongas. This approximation is certainly good in systems where the charge den-sity is spatially nearly constant. However, LDA proved to be an astonishinglygoodapproximationeven in systems where the chargedensity is strongly inho-mogeneous such as molecules (and of course quantum dots). The exchange-and correlation contribution to the effective potential is given according toEq. (C.11)

V xc (r ) = nd

dn+ 1 xc (n)

n ( r )(C.14)

The function xc has been determined for an interacting homogeneouselectrongas by Ceperly and Alder [52] using a Quantum-Monte-Carlo method. In thiswork a parameterized form for the three-dimensional problem is used, whichwas given by Perdew and Zunger [53].

4In fact they are the Lagrange-parameters of the variational scheme used to derive the Kohn-Sham effective potential Eq. (C.10).


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153/161

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Curriculum VitaeAndreas Scholze was born in Arnstadt (Germany), on June 14, 1969. After

nishing secondary school in 1989, he studied Physics at the Universities of Jena (Germany) and Edinburgh (Scotland). He specialized in solid-state the-ory submitting a thesis on density-functional based analysis of diamond sur-face reconstructions. After graduating from the University of Jena, he joinedthe Integrated Systems Laboratory (IIS) of the Swiss Federal Institute of Tech-nology in Z urich (ETHZ) working towards the Doctoral degree in ElectricalEngineering. During the summer of 1997 he stayed as a visiting research as-sociate at the Beckman Institute for Advanced Science and Technology at theUniversity of Illinois in Urbana-Champaign (UIUC) with the group of Prof.K. Hess. He is currently working within the framework of a national project(MINAST) in the eld of nanodevice simulation. The main focus of this work is on the establishment of numerical methods and algorithms for the simula-tion of single-electron devices at the solid-state level.

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