Simulating Charge Stability Diagrams for Double and Triple

Quantum Dot Systems

Ian E. Powell

(Dated: September 2, 2014)

Abstract

We recreate Sandia National Laboratories’ double quantum dot charge stability diagram simu-

lation using their rate equation approach and compare our simulation’s results to some touchstone

situations as well as their results. We find qualitative agreement between charge stability diagrams

and extend the code to construct the charge stability diagram for a triple dot structure. We note

how the triple dot charge stability diagrams change as the third gate voltage is increased.

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I. INTRODUCTION

A. Quantum Computing in Solid State Quantum Dots

The field of quantum computing has received a great deal of attention in the physics com-

munity due to its exciting prospective applications and recent progress made in constructing

systems with characteristics that allow for the formation of so called “qubits.” A qubit, short

for quantum bit, represents a two-level, quantum mechanical system. Canonical examples

of a qubit are a photon’s polarization–be it linearly horizontal or linearly vertical–or an

electron’s spin state–where down and up, or horizontal and vertical correspond to 0 and 1

in terms of binary. The difference, between a qubit and a classical bit is the qubit’s ability

to be in both 0 and 1 simultaneously–a superposition of up and down for the example of the

electron’s spin. It is this property of superposition, along with something called quantum

entanglement, that differentiates quantum computing from its classical counterpart.

David Divincezo of IBM famously constructed a list of requirements to construct quantum

computer. The “DiVincenzo Criteria” are: (1) the system must have a large number of well

defined qubits, (2) initialization to a known state for the system must be easily performed,

(3) a universal set of quantum logic gates must be developed, (4) qubit-specific measurements

must be able to be performed, (5) and long coherence times must exist for the qubits in

the system1–where coherence time describes how long a quantum state can exist on average

before being perturbed, and, in turn, altered by the environment.

Many different types of systems show promise in fulfilling said criteria–for example, super-

conducting systems have shown promise in quantum information processing due to the ease

of fabricating larger systems. Cold neutral atom systems’ qubits have some of the longest co-

herence times–offering promise for storing information for longer periods of time. Solid-state

systems in which spin states of electrons or nuclei represent the qubits have serious promise

in their scalability, due to extensive research done on the miniaturizing of semiconductors

for computing. This summer I worked on researching a solid state system–namely quantum

dots–for quantum computing purposes.

A quantum dot is generally any 3-D potential well, but in the context of solid-state

physics it is a structure made of semiconducting material that is small enough to exhibit

quantum mechanical phenomena. The dot’s bound states of electron and electron-hole, or

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excitons, are confined in all three spatial dimensions. The electronic properties of a single

dot, its discrete energy levels etc., are akin to that of an atom–hence the nickname for a

quantum dot is “artificial atom.” When another dot is introduced into the system and there

exists a strong, coherent, in-phase tunneling of electrons between the two dots, the system

behaves similar to a covalently bonded molecule.

We utilize double and triple quantum dots which have been fashioned onto a Si-SiO2

heterostructure as our means for quantum computing; for a schematic of a GaAs-AlGaAs

heterostructure see Fig. I.

FIG. 1. Schematic of a GaAs-AlGaAs heterostructure taken from [2].

At the interface between the two different materials a two dimensional electron gas (2DEG)

forms. To further confine the other two dimensions of motion we apply voltages to the

gates to create potential wells and trap some target number of electrons. We utilize a

quantum point contact (QPC) to determine the current through the dots and, in turn, the

occupancy of each dot. This is due to the fact that the QPC has a systematically varying

conductance as the occupation of the dots varies. The QPC, as opposed to some ammeter,

has the advantage of measuring the current of electrons through the dots indirectly–thereby

not perturbing our system of qubits, and thus not destroying any information stored in our

system.

There are two different types of qubits that are viable for double dot systems–spin qubits

and charge qubits. The bases for a double dot system’s spin qubits are the singlet and triplet

states–i.e. (S = 0, Sz = 0) = |0〉, (S = 1, Sz = 0) = |1〉. The bases for the charge qubits are

based on electron occupation of the dots–for example (0,1) occupancy could be |0〉 whereas

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(1,0) would be |1〉–for a helpful visualization of the wave function of our system written in

terms of our qubit bases see Fig. II. For the triple dot system we utilize the Heisenberg

FIG. 2. The “Bloch Sphere” visualization of a qubit’s wave function.

interaction and the “singlet” and “triplet-like” states as our bases–where, for example, |0〉

= |S〉 |↑〉, |1〉 = (2/3)1/2 |T+〉 |↓〉 - (1/3)1/2 |T0〉 |↑〉3.

For a given dot structure a charge stability diagram, or a “honeycomb diagram,” can be

formed–telling us at which gate voltages what electron occupancy is favored. To construct

this diagram we first model our double or triple dot as a system of capacitors and then follow

a rate-equation approach developed by a group working at Sandia National Laboratories

highlighted in the next section. Honeycomb diagrams have very distinct characteristics for

different sets of system parameters. For example, in a perfectly uncoupled double dot–in

the sense that the mutual capacitance between the dots → 0–we would expect a pattern of

rectangles to be formed in our stability plot. In a perfectly coupled system such that the

capacitance of each dot equals the mutual capacitance we would expect rectangles again,

however, now they would be symmetric about the Vg1 = Vg2 line where Vg1(g2) is the gate

voltage on dot 1(2)–for a visualization of these stability diagrams see Fig. III. We recreate

these figures–(a) → (c)–and others shown in Sandia National Laboratories’ paper4 to show

that we have successfully recreated Sandia’s simulation and continue to discuss how we

extended it for use in a triple dot system.

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FIG. 3. Different honeycomb diagrams for select cases of system parameters taken from [5].

B. Sandia’s Paper

The Paper written by Sandia Nation laboratories which we follow discusses a double

quantum dot structure that they extract biasing triangles from–structures that form in hon-

eycomb diagrams when a biasing voltage is present–and model using their rate equation

approach. With this model they can identify how different tunneling imbalances and tem-

peratures affect a charge stability diagram–namely how edges may be obfuscated or shifted.

Some of the figures from their paper are shown in the Results section below–highlighting

how different relative tunneling rates can alter the honeycomb diagram significantly.

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II. THEORY

A. Energy Of a Double Dot

The energy of a double dot modeled by a system of capacitors is

U =1

2~V ·C~V , (1)

where ~V is the voltage vector and C is the capacitance matrix. To find ~V we write the total

charge in each dot as the charge accumulated on each capacitor connected to the dot

Q1 = CL(V1 − VL) + Cg1(V1 − Vg1) + Cm(V1 − V2) + Cg21(V1 − Vg2) (2)

Q2 = CR(V2 − VR) + Cg2(V2 − Vg2) + Cm(V2 − V1) + Cg12(V2 − Vg1), (3)

where Q1(2) is the charge on dot 1(2), V1(2) is the voltage of dot 1(2), C1(2) is the capacitance

of dot 1(2), Cm is the mutual capacitance between dots, VL − VR is the bias voltage across

the system, and Cg21(g12) is the cross capacitance between gate 2(1) and dot 1(2)–for a circuit

diagram of the system see Fig. IV. We rewrite this in terms of the capacitance matrix and

FIG. 4. Capacitor model for a double dot with cross-gate capacitances.

voltage vector Q1 + CLVL + Cg1Vg1 + Cg21Vg2

Q2 + CRVR + Cg2Vg2 + Cg12Vg1

= C

V1

V2

, (4)

where

C =

C1 −Cm

−Cm C2

. (5)

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We now invert our capacitance matrix, and write our charge in terms of the occupancy of

each dot–N1(2), with Q1(2) = −|e|N1(2)–to find our voltage vector in terms of our capacitances V1

V2

=1

C1C2 − C2m

C2 Cm

Cm C1

× −|e|N1 + CLVL + Cg1Vg1 + Cg21Vg2

−|e|N2 + CRVR + Cg2Vg2 + Cg12Vg1

. (6)

We rewrite Equation (1) as

U =1

2C1V

21 +

1

2C2V

22 − CmV1V2, (7)

and plug in our values of V1 and V2 found using Equation (6) to find the electrochemical

potential of our double dot system.

B. Energy Of a Triple Dot

We model our triple dot as a system of capacitors as we did with our double dot–see Fig.

V. Accounting for our charge as we did with the double dot we yield

FIG. 5. Capacitor model for a double dot with cross-gate capacitances.

Q1 + CLVL + Cg1Vg1 + Cg21Vg2

Q2 + Cg2Vg2 + Cg12Vg1 + Cg32Vg3

Q3 + CRVR + Cg3Vg3 + Cg23Vg2

= C

V1

V2

V3

, (8)

where our capacitance matrix is now given by

C =

C1 −Cm1 0

−Cm1 C2 −Cm2

0 −Cm2 C3

. (9)

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We invert our capacitance matrix to find ~V as we did with the double dot, and utilize

Equation (1) to find our electrochemical potential expression

U =1

2C1V

21 +

1

2C2V

22 +

1

2C3V

23 − Cm1V1V2 − Cm2V2V3. (10)

We do assume in our model that there is negligible coupling between dots 1 and 3; this is

a reasonable assumption for some structures but not all. In the case that the coupling isn’t

negligible we would add terms to the capacitance matrix in Equation (9) and the charging

terms in Equation (8) as necessary.

C. Rate Equation Approach

To construct the charge stability diagram of a double or triple dot we utilize a rate

equation approach as highlighted in Sandia National Labs’ Paper.

For a dot system in state a we write the transition amplitude from state a to state b as

Γab = tabf(U(a)− U(b)), (11)

where tab is the tunneling rate between state a and state b, f is the Fermi-Dirac distribution

function, and U is the electrochemical potential of the system. For our simulation we consider

only single electron transitions between state a to b.

To understand how the probability of being in state a, Pa, changes with respect to time

we construct the rate equation

dPa

dt=

∑k 6=a

ΓkaPk −∑k

ΓakPa. (12)

The positive term accounts for all ways our system in some other state k may transition to

state a, and the negative term accounts for all ways our system in state a may transition to

some other state k in a time step dt. To find the steady-state solution we set Equation (9)

equal to 0 and construct the matrix equation

M~P = 0, where Mab =

Γba, a 6= b

−∑

k Γak, a = b,(13)

thus our probability vector, ~P , can be found by finding the nullspace of our M matrix.

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We then find the average occupancy of our dots via the weighted sum

N̄ =∑k

PkNk, (14)

where Nk is the number of electrons in each dot in state k. To visualize our charge stability

diagram we then take the total derivative of our occupancy with respect to our gate voltages

and make a grey-scale contour plot. Alternatively we can also visualize the honeycomb by

plotting the probabilities where they are ≈ .5.

D. The Algorithm

For the algorithm, we chose to iterate over situations where the system state can go from

a to b via a single electron. We construct a n × n matrix where n = number of states the

system can be in with a maximum number of electrons present. Utilizing a bit of statistics

we find that n = (N ′ + E)!/(E!N ′!), where N ′ is the number of dots in our system, and E

is the allowed maximum number of electrons in our system.

The algorithm goes as the following:

Iterating over the gate voltages, we choose an initial occupancy and find the transition

amplitudes from that state to all other states; we store these transition amplitudes in our

n×n matrix and repeat this for all other possible occupancies thereby yielding the probability

vector ~P–via the null space of this M–and thus the occupation associated with the system

associated with the gate voltage configuration. We store each probability vector and each

dot occupation for each voltage configuration until we have iterated over all chosen gate

voltages. The code for the double dot simulation can be found in The Appendix.

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III. RESULTS

A. Double Dot

We first recreate Fig. III (a) and (b) with arbitrary parameters and units using 3600

pixels in each plot shown–see Fig. VI and VII.

FIG. 6. Perfectly uncoupled system with a maximum of two electrons.

FIG. 7. Imperfectly coupled system with a maximum of three electrons.

Fig. 6 appears how we would expect it to–the diagonal line that separates the (1,1) and

(0,2) states is due to the fact that there are a maximum of two electrons in the system at

one time. Fig. 7 shows a reasonable structure as well with the formation of triple dots as

seen in Fig. III (b).

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We now recreate Sandia’s figures with their listed values, the tunneling rates between

source and left dot, left dot and right dot, and right dot and drain are listed as tL : tm : tR;

our plots’ axis are in units of volts–see Fig. VIII and Fig. IX.

FIG. 8. Simulation comparison using Sandia’s system parameters for tunneling rates 0.01 : 1 : 0.1

FIG. 9. Simulation comparison using Sandia’s system parameters with tunneling rates listed as

1 : 1 : 1.

Both our results and Sandia’s are in qualitative agreement as both exhibit similar struc-

ture, however an in depth quantitative comparison is yet to be done. One considerable

disagreement is that all of our structures are a factor of 10 larger than those in the Sandia

paper–this is likely due to a units issue that has eluded us or them.

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B. Triple Dot

Using Sandia’s original values for the first two dots and C3 = 55 aF, Cg3 = 1.7 aF,

Cg32 = 1.1 aF, Cg23 = 1.3 aF for the third dot, with no biasing voltage present, we create

two probability plots to highlight the structural change of the honeycomb diagram as Vg3

increases from 5 mV to 10 mV. The tunneling rates between source and left dot, left dot and

middle dot, middle dot and right dot, and right dot and drain are listed as tL : tcl : tcr : tR–see

Fig. X and Fig. XI.

FIG. 10. Honeycomb Diagram at Vg3 = 5 mV and T = .15K, with tunneling rates 1 : 1 : 1 : 1

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FIG. 11. Honeycomb Diagram at Vg3 = 10 mV and T = .15K, with tunneling rates 1 : 1 : 1 : 1

As Vg3 increases the pattern is shifted upwards and to the left as it becomes less likely

that our system would have a lower occupation as we would expect.

IV. CONCLUSION

We recreated Sandia National Laboratories’ double quantum dot charge stability diagram

simulation and compared our simulation’s results to their results qualitatively. When com-

pared, we found agreement between the charge stability diagrams’ geometries and extended

the code to simulate triple dot structures with weak dot 1-dot 3 couplings. Our results for

the triple dot were sensible and we plan to use the simulation to model charge stability

diagrams of devices we have made and use in the lab.

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ACKNOWLEDGMENTS

I would like to thank Professor Hong-Wen Jiang for mentoring me while I worked on this

project during my stay at UCLA, Blake Freeman and Joshua Schoenfield for all of their

useful discussions and guidance, and the NSF.

V. APPENDIX

iterations = 40 #total iterations = iterations squared

Vg1_max=.06

Vg1_min=0.

Vg2_max=.06

Vg2_min=0.

dV=(Vg1_max-Vg1_min)/iterations

print dV

V_0=0 #initialize

#tunneling factors

tS = 1. #tunneling from source to left dot

tM = 1. #tunneling from left dot to right dot

tD = .0 #tunneling from right dot to drain

T=1e-4

#Solve linear system.

#Double Dot, 2 electrons

N1_0=0. #starting number in dot 1

N2_0=0. # starting number in dot 2

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Vgate_1=[] #Gate voltage lists

Vgate_2=[]

elec=4. #max electrons in system

Gamma_array=np.zeros((500,500))

N_list=[] #initializing the list of states

q=0 #counter

r=0 #counter

Master_G_array=[]

for l in range(iterations+1):

for k in range(iterations+1):

q=0 #counter

#counter

N1_0=0. #starting number in dot 1

N2_0=0. # starting number in dot 2

Gamma_array=np.zeros((500,500))#initialize every iteration

inv_array=np.zeros((500,500))

for g in range(int(elec)+1):#changing configuration

for h in range(int(elec)+1):

r=0

q+=1

for i in range(int(elec)+1): #scanning over transitions for

a particular configuration

for j in range(int(elec)+1):

r+=1

# print r

if (g+h<=elec and i+j<=elec): #Avoid iterating over

situations where there are more than max electrons

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N_list.append([g,h])

#Criteria for an acceptable transition of one electron

if (((i!=g or j!=h) and (abs(g-i)+ abs(h-j)))<=elec

and (abs(g+h-i-j)<=1)):

#Tunneling direction logic in next if statements

if (i<g and h==j):

Gamma_array[q-1][r-1]=

Gamma_out(tS,g,h,i,j,l*dV,k*dV,T)

inv_array[q-1][r-1]=

Gamma_out_inv(tS,g,h,i,j,l*dV,k*dV,T)

if (g==i and h<j):

Gamma_array[q-1][r-1]=

Gamma(tD,g,h,i,j,l*dV,k*dV,T)

inv_array[q-1][r-1]=

Gamma_inv(tD,g,h,i,j,l*dV,k*dV,T)

if (h<j and g>i):

Gamma_array[q-1[r-1]=

Gamma(tM,g,h,i,j,l*dV,k*dV,T)

inv_array[q-1][r-1]=

Gamma_inv(tM,g,h,i,j,l*dV,k*dV,T)

if (g<i and h>j):

Gamma_array[q-1][r-1]=

Gamma(tM,g,h,i,j,l*dV,k*dV,T)

inv_array[q-1][r-1]=

Gamma_inv(tM,g,h,i,j,l*dV,k*dV,T)

if (h==j and i>g):

Gamma_array[q-1][r-1]=

Gamma_in(tS,g,h,i,j,l*dV,k*dV,T)

inv_array[q-1][r-1]=

Gamma_in_inv(tS,g,h,i,j,l*dV,k*dV,T)

if (j<h and g==i):

Gamma_array[q-1][r-1]=

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Gamma(tD,g,h,i,j,l*dV,k*dV,T)

inv_array[q-1][r-1]=

Gamma_inv(tD,g,h,i,j,l*dV,k*dV,T)

else:

None

else:

None

else:

None

if g==elec:

break

else:

None

Count_Deletes=0

#Deleting the extra trivial rows

for i in range(len(Gamma_array[0])):

if np.all(np.zeros(len(Gamma_array))-(Gamma_array[:,i-Count_Deletes])==0)

and np.all(np.zeros(len(Gamma_array))-(Gamma_array[i-Count_Deletes])==0):

Gamma_array=np.delete(Gamma_array,(i-Count_Deletes),axis=1)

#Delete Column

Gamma_array=np.delete(Gamma_array,(i-Count_Deletes),axis=0)

#Delete Row (square matrix)

inv_array=np.delete(inv_array,(i-Count_Deletes),axis=1) #Delete

Column

inv_array=np.delete(inv_array,(i-Count_Deletes),axis=0) #Delete

Row (square matrix)

Count_Deletes+=1

else:

None

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for m in range(len(Gamma_array)):

for n in range(len(Gamma_array)):

if (m==n):

Gamma_array[m][n]+=-sum(inv_array[m])

Master_G_array.append(Gamma_array)

Vgate_1.append(l*dV+Vg1_min)

Vgate_2.append(k*dV+Vg2_min)

print l*dV

1 David P. DiVincenzo, IBM (2000-04-13). “The Physical Implementation of Quantum Computa-

tion”.

2 R. Hanson et.al., “Spins in few-electron quantum dots” Reviews of Modern Physics. 79, October-

December 2007.

3 DiVincenzo et.al., “Universal quantum computation with the exchange interaction” Letters to

Nature, 2000.

4 Khoi T. Nguyen et.al., “Charge Sensed Pauli Blockade in a Metal-Oxide-Semiconductor Lateral

Double Quantum Dot” Nano Letters. 13 75, 5785-5790, 2013.

5 W.G. van der Wiel et.al., “Electron transport through double quantum dots” Reviews of Modern

Physics. 75, January 2003.

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