University of Iceland
Computational physics
Two exactly Coulomb interacting
electrons in a finite quantum wire
and an external magnetic field.
Author:
Ólafur Jónasson
Supervisor:
Viðar Guðmundsson
January 19, 2010
Contents
1 Introduction
1
2 The theory and program
1
2.1
The many-particle basis and Fermi operators . . . . . . . . . . . .
1
2.2
Making the MES basis . . . . . . . . . . . . . . . . . . . . . . . .
2
2.3
Calculating the matrix elements . . . . . . . . . . . . . . . . . . .
3
2.4
Calculating the charge density . . . . . . . . . . . . . . . . . . . .
4
2.5
Bottlenecks in calculations . . . . . . . . . . . . . . . . . . . . . .
4
2.6
The program . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
3 Convergence
6
3.1
Effects of limited MES basis . . . . . . . . . . . . . . . . . . . . .
6
3.2
Effects of limited grid . . . . . . . . . . . . . . . . . . . . . . . . .
7
3.3
Choosing a suitable η . . . . . . . . . . . . . . . . . . . . . . . . .
7
3.4
Conclusion on convergence . . . . . . . . . . . . . . . . . . . . . .
9
4 Physical results
4.1
4.2
12
Energy spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
4.1.1
General behavior . . . . . . . . . . . . . . . . . . . . . . .
12
4.1.2
Effects of magnetic field on lowest states . . . . . . . . . .
12
4.1.3
Energy splitting . . . . . . . . . . . . . . . . . . . . . . . .
17
Charge density . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
5 Conclusions
19
i
1
Introduction
The system I’m investigating consists of a GaAs-wire (200nm long) in a constant homogeneous magnetic field. I modified a program which calculated the
eigenvalues and eigenvectors of the Hamiltonian of the system without Coulomb
interaction. I modified the program in such a way that the calculations include
the Coulomb interaction for two or three electrons.
Without the interaction the system is not physically interesting because the
Coulomb interaction can drastically change quantities of interest. In this thesis I will explain the methods and results of my modification to the program.
My approach to the problem will be similar to that in the non-interacting case,
but obviously, some things need to be done differently. I will use second quantization reformulation of the Schrödinger equation [1]. One of the advantages of
second quantization for these calculations is that it incorporates the Fermi particle statistics so there is no need for the computationally/memory expensive task
of dealing directly with antisymmetric products of single-particle wave functions
or many-particle wave functions.
The calculations will be done “exactly”, that is, the only approximation is the
truncation of the single-electron states (SES) and many-electron states (MES)
basis, the finite grid used to store the SES wave functions and the error in the
numerical integration. How these errors effect the accuracy of the calculations
will be investigated by varying the size of the basis, grid and other external
parameters.
2
2.1
The theory and program
The many-particle basis and Fermi operators
For the calculations we need a MES basis. We will construct it from the noninteracting single-electron states. The MES are time-independent abstract state
vectors
|n1 n2 n3 · · · n∞ i = |Ψ1 i|Ψ2 i|Ψ3 i · · · |Ψ∞ i,
(2.1)
where n1 denotes how many-electrons are in SES 1, n2 how many in SES 2 etc. and
|Ψi i are single-electron states. Obviously the MES basis fulfills the orthogonality
and completeness conditions. We are only considering fermions (electrons) so
ni = 0 or ni = 1 (the Pauli inclusion principle). We will also use the Fermi
creation and destruction operators ĉ†k and ĉk with the properties
1
ĉ†k ck | · · · nk · · · i = nk | · · · nk · · · i
(
(−1)γk | · · · 0 · · · i
ĉk | · · · nk · · · i =
0
(
0
ĉ†k | · · · nk · · · i =
(−1)γk | · · · 1 · · · i
n = 0, 1
(2.2)
if nk = 1
if nk = 0
(2.3)
if nk = 1
if nk = 0
(2.4)
where
γk =
k−1
X
ni .
(2.5)
i=1
Using (2.1) - (2.5) the Hamiltonian of the system has been transformed into:
Ĥ =
X
a
Ea ĉ†r ĉs +
1X † †
ĉ ĉ hab |Vcoul | cdiĉd ĉc ,
2 abcd a b
(2.6)
where Vcoul is the Coulomb potential and Ea the energy of the non-interacting
MES defined by
X
Ea =
ni E i ,
(2.7)
i
where Ei is the energy of the i-th single-electron state. Of course (2.6) is equivalent to the original Schrödinger equation in first quantization but with this
formalism the anti-symmetry of the MES wave-functions are incorporated into
the Hamiltonian, making calculations much easier.
2.2
Making the MES basis
At this point we have already calculated the SES’s, so the task of making the the
basis is only a book-keeping problem. This is best explained with an example.
Suppose we have two electrons and 5 single-electron states. There are 52 = 10
ways of ordering these two electrons (the electrons are indistinguishable). An
example how to order them is shown in table 1
The pattern is obvious and is easy to generalize to a MES with NSES singleelectron states. A similar method can be used for more than two electrons but
the number of many-electron states NMES grows very fast with increasing number
of electrons according to (2.8)
NSES
NMES =
,
(2.8)
Ne
2
Table 1: Many-electron states for Ne = 2 and NSES = 5
#
State
1
2
3
4
5
6
7
8
9
10
|11000i
|10100i
|10010i
|10001i
|01100i
|01010i
|01001i
|00110i
|00101i
|00011i
where Ne is the number of electrons. This can be seen in table 2.
There is one problem with this method of making the MES’s. The energy of
state number i is not generally lower than that of state number i + 1. We can,
however, number the states any way we like after the energy of each state has
been determined. All that matters is that we have a consistent method of making
all possible states.
Table 2: NMES for some values of Ne and NSES
2.3
Ne \ NSES
8
16
24
32
40
48
2
3
4
28
56
70
120
560
1820
276
2024
10626
496
4960
35960
780
9880
91390
1128
17296
194580
Calculating the matrix elements
So far we have denoted the single-electron states as |ii, in order to make a distinction between many- and single-electron states we denote many-electron states
by |i). Now we need the matrix elements
X
(2.9)
Hµν =
hab |Vcoul | cdi(µ|ĉ†a ĉ†b ĉd ĉc |ν).
abcd
The right term is easy to calculate using the properties of the Fermi operators in
(2.2) through (2.4). For the left term we need to do a 4-dimensional integral [2]
Z
hab |Vcoul | cdi =
dr dr 0 Ψ∗a (r)Ψ∗b (r 0 )Vcoul (r − r 0 )Ψc (r 0 )Ψd (r),
3
(2.10)
where Ψi (r) is the SES wave-function of state |ii and
e2
VCoul (r − r 0 ) = p
,
κ (x − x0 )2 + (y − y 0 )2 + η
(2.11)
with
e2
aω
= ~Ωω ∗ ,
κ
a0
~2 κ
= 9.79 nm.
m∗ e2
(2.12)
drΨ∗a (r)Ibc (r)Ψd (r),
(2.13)
a∗0 =
It is convenient to write
Z
hab |Vcoul | cdi =
where
Z
Ibc (r) =
dr 0 Ψ∗b (r 0 )Vcoul (r − r 0 )Ψc (r 0 ) .
(2.14)
The η in (2.11) is a positive real number who purpose is to prevent the singularity
when r = r 0 . To find a suitable value for η we need to experiment a bit with it.
This will be done later.
2.4
Calculating the charge density
One of the most interesting observables in our system is the charge density Qs .
To calculate it I use
XX
X
φ∗a φb hµ ρ̂ĉ†a ĉb µi
φ∗a φb ĉ†a ĉb ⇒ hQ̂s (x)i = e
hQ̂s (x)i = e
nm
nm
=e
XX
ab
µ
φ∗a φb ρµν hν ĉ†a ĉb µi,
(2.15)
µν
where the φi are single-electron states.
2.5
Bottlenecks in calculations
One of the bottlenecks in the calculations is the right term on the right side of
2
(2.6) we have to do a quadruple sum for Nmes
matrix elements so the time it
takes to compute the sum (given that all the integrals hab |Vcoul | cdi have been
calculated) is proportional to
2
NSES
2
4
4
.
(2.16)
NMES NSES =
NSES
Ne
4
Another bottleneck is the 4-dimensional integral hab |V | cdi, which we have to do
for all combinations of a,b,c and d. The integral is split into two parts, (2.13)
and (2.14) of which (2.14) is is much more expensive to calculate with a reasonable choice of grid size and NSES . The time it takes to compute Ibc (r) for all
combinations of b and c is proportional to
2
NSES
Nx2 Ny2 ,
(2.17)
where Nx and Ny are the number of points in the x- and y-grid. This means that
if we double the amount of points in both x- and y-grid, the execution time is
multiplied by 16. We will later investigate how the grid size effects the accuracy
of the calculations.
The biggest bottleneck is calculating the charge density (see eq. (2.15)). The
computation amount is proportional to
2
NSES
2
Nx Ny
NSES
.
Ne
(2.18)
Luckily we are only interested in plotting the lowest states and ρµν ' 0 for large
µ and ν so work can be saved without losing much precision by only summing
over the first 200 − 300 µ and ν.
It is important to find a good balance between the choice of NSES and the grid
size. For example there is no point in having NSES huge if the error introduced
because of the finite grid is much larger than the error due to the limited NSES
and vice versa.
2.6
The program
The program for these calculations works exactly the same as for the non-interacting
case until after the single-electron states with the effects of the magnetic field and
geometry of the system have been calculated. Here, I will therefore just explain
the part of the program that is used only in the interacting case. I will begin
listing the functions and subroutines used and explain their function.
Fermi4CStructure: This function calculates (µ|ĉ†a ĉ†b ĉd ĉc |ν). It has input parameters µ, ν, a, b, c, d. Implementing this as a function is efficient because
each combination of the input parameters is only used once, so saving the
values to a rank 6 matrix would both be pointless and impossible due to
finite RAM. Inside this function I have defined two internal functions (the
Fermi destruction and creation operators).
make_coef: This subroutine uses the method explained in table 1 to make the
many-electron states for two electrons. The states are saved in a matrix
5
called coef whose rows are many-electron states. The rows are of length
NSES + 2. The two extra elements are the sign of the state (element number
−1) and existence (element number 0). If existance = 0 in a MES that
means a raising operator has been applied to an occupied state or destruction operator to an empty state.
make_coef3: Same as make_coef except it creates the states for 3 electrons.
MatElCoul: This subroutine has input parameters VCoul and η. It calculates
the 4-dimensional Coulomb integral (2.13) for all values of a,b,c and d and
saves it to the rank 4 matrix abVcd.
CoulInt: This subroutine calculates the quadruple sum in (2.9). It must be called
after MetElCoul because it uses its result.
make_gamma: This subroutines makes a rank 2 matrix whose elements gamma(i,k)
= γk for state |i), see (2.5). This matrix is used when calculating the charge
density.
Del_Qn2: This subroutine uses (2.15) to compute and write to a file the charge
density of a chosen number of many-electron states with and without e-e
interaction.
3
Convergence
Like mentioned in section 1, no theoretical approximation has been done. In this
section we will investigate how the finite MES basis (limited NSES ) and grid size
affects the accuracy of the calculations. We will also investigate the effects of
varying the size of η in (2.11). When one of those parameters is varied, all the
others will be held constant.
3.1
Effects of limited MES basis
If we have NMES many-electron states in our basis we will obtain NMES energy
(eigen) values from the calculations. The accuracy is highest for the ground state
and decreases as we go up in energy. This is a very convenient behavior for us
because we are only interested in the first 200 or so excited states. I ran the
program for two electrons and varied the value of NSES (see figure 1).
To get an idea of the precision of the first 200 states I plotted the relative
difference between the energy spectrum of the NSES = 48 case and all the other
32
40
|, |E48E−E
| etc. (see figure 2). From this data we can estivalues, that is | E48E−E
40
48
mate that the error is at most 0.6% if we use Nses = 48 for the first 200 states,
less than 0.01% for the first 20 states and less than 0.001% for the ground state.
6
Figure 1: Comparison of the energy spectrum for a few values of NSES . The energy
spectrum of the non-interacting basis is also plotted for comparison. Although the
energy spectrum is discrete, I drew lines between the values for easier comparison.
3.2
Effects of limited grid
The effects of the limited grid come into play when doing the Gaussian integration. Unlike the error due to the limited MES-basis, the grid error affects all
the states equally. We would therefore definitely not want the error to be higher
than 0.6% (that’s the maximum error due to the limited basis) but we also don’t
want to lose precision on the ground state so we are not gonna settle with error
of more than 0.001%.
I ran the program for varying grid size namely 80x40 (grid 1), 160x80 (grid 2),
320x160 (grid 3), 480x240 (grid 4) and 640x320 (grid 5) and plotted the relative
difference between the resulting energies from grid 5 and all the other ones (see
figure 3). From the data we see that grid 2 is sufficiently large but the difference
is computational time for grid 2 and grid 3 is very small so we might as well use
grid 3 (calculating the matrix elements is still a much more expensive procedure
for these grid sizes).
3.3
Choosing a suitable η
The effects of the choice of η are a little more complicated to assess, for, in my
experience it is also highly dependent on grid size and therefore requires special
7
(a) Nses = 16
(b) Nses = 24
(c) Nses = 32
(d) Nses = 40
Figure 2: Relative error plotted as a function of state number for multiple values of Nses
(a) Grid 1
(b) Grid 2
(c) Grid 3
(d) Grid 4
Figure 3: Relative error plotted as a function of state number for multiple grids.
8
attention. I ran my program for 5 values of η (0.1, 0.01, 0.001, 0.0001 and 0.00001)
and two grid sizes (grid 2 and grid 4). Like before I will calculate the relative
difference of the obtained energy values but I will begin by investigating how the
choice of η affects the integral (2.14) by plotting it for a chosen combination of b
and c (see figure 4).
For the smaller grid we start seeing some irregularities in Ibc when η is 10−4 or
less while it looks fine with the bigger grid. These irregularities I call η-artifacts.
These η-artifacts don’t start showing up on the bigger grid until η ≤ 10−5 and
even then the artifacts are barely noticeable. These artifacts obviously decrease
with bigger grid but unfortunately grid 4 is as big as we can have it without
drasticly increasing the program execution time. I therefore conclude that the
best choice of η is 10−4 along with grid 4.
Now we investigate the convergence of the energy spectrum with decreasing
η. We saw earlier that η = 10−4 with grid 4 gave the best resulting Ibc so I will
calculate and plot
Eη=10−4 − Eη=ηi (3.1)
Eη=10−4
for ηi = 10−1 , 10−2 and 10−3 (see figure 5). These results show that the error due
to the finite η is less than 0.001%, which is not much, but it it still the largest
cause of error.
3.4
Conclusion on convergence
We have investigated the convergence of the program’s results for all our numerical approximations and the results were positive (otherwise the program would
be worthless). We found that we have at least 5 correct significant figures on our
energy spectrum (0.001%relative error). Getting better precision would require us
to use a bigger grid which is not practical as we preferably want the calculations
to be over in our lifetime.
9
(a) η = 10−2 , grid 2
(b) η = 10−2 , grid 4
(c) η = 10−3 , grid 2
(d) η = 10−3 , grid 4
(e) η = 10−4 , grid 2
(f) η = 10−4 , grid 4
(g) η = 10−5 , grid 2
(h) η = 10−5 , grid 4
Figure 4: Ibc plotted with a = 1, b = 2 for a few η’s.
10
(a) ηi = 10−1
(b) ηi = 10−2
(c) ηi = 10−3
Figure 5: Relative error plotted as a function of state number for a few values of ηi .
11
4
Physical results
Now that we have gotten all the boring (but necessary) convergence calculations
out of the way we can finally start to look at the physically relevant data. We will
start by looking at how the e-e interaction affects the energy spectrum, especially
the energetically lowest 10 states. We will also investigate how the charge density
changes and compare it with the non-interacting case.
4.1
4.1.1
Energy spectrum
General behavior
We will start by investigating the energy spectrum of the lowest 10 states for a
broad range of magnetic field and comparing the results with the non-interacting
case. I did the calculations for a magnetic field in the range 0.0 − 4.0 T in increments of 0.1 up to 1.0 T and 0.5 up to 4.0 T. I used NSES = 40, η = 10−4 and
the grid size was 480 × 240. Each run took about 40 hours on Sol, running on 8
cores (practically) the entire time.
On figure 6 we can see the effects of the magnetic field on the energy spectrum.
Without the magnetic field the energy increases very evenly (exept for the first
two states), but with increasing magnetic field the states tend to group into groups
with similar energy. The energy spectrum begins to look more like a step function,
rather than continuous. This effect does not show up in the non-interacting case.
As expected, including the interaction raises the energy by about 1.5meV.
To see better the the grouping of the energy states I plotted the energy of the
20 lowest states as a function of magnetic field (see figure 7). On figure 7a we can
clearly see how the states tend to group together in higher magnetic field while
without the interaction (figure 7b) we see no such effects.
4.1.2
Effects of magnetic field on lowest states
Now let’s investigate the effects of the magnetic field on the lowest states by
plotting the energy of state n as a function of magnetic field. The results are
in figure 8. As we can see from the figure the energy increases with increasing
magnetic field, as expected (we are pumping energy into the system by applying
a magnetic field to it). We also see that the separation in the energy spectrum
between the interacting and non-interacting case stays practicly constant with
increasing magnetic field but the energy of the states increases. This means the
interaction plays a lesser role with increasing magnetic field. For example the
seperation of the ground state for B = 0.0 T is about 90% but only 14% for
B = 4.0 T.
12
(a) B = 0.0 T
(b) B = 0.2 T
(c) B = 0.4 T
(d) B = 0.8 T
(e) B = 1.0 T
(f) B = 2.0 T
Figure 6: Comparison of the energy of the 10 lowest states with the non-interacting
case. With interaction is in red and without in blue.
13
(a) With interaction.
(b) Without interaction.
Figure 7: Energy of the 20 lowest states as a function of magnetic field with and without
interaction.
14
(a) Ground state (state 1)
(b) First excited state (state 2)
Figure 8: Comparison of the energetically lowest 3 states as a function of magnetic field.
With interaction is in red and without in blue.
15
(c) Second excited state (state 3)
(d) Third excited state (state 4)
Figure 8: Comparison of the energeticly lowest 3 states as a function of magnetic field.
With interaction is in red and without in blue.
16
4.1.3
Energy splitting
Now let’s investigate how the system responds to excitation by calculating how
much energy is required to excite the system from a state to a higher one. The
higher this energy is, the harder it is to excite the system. I did this by calculating
the difference in energy between a few selected states as a function of magnetic
field, that is
∆1 = E2 − E1 ,
∆2 = E3 − E2 ,
∆3 = E4 − E3 .
(4.1)
See figure 9. From figure 9a we see that for B ' 0.25 T, ∆1 is biggest and then
decreases with increasing magnetic field, converging at around 0.2 meV. The effect
is similar for the non interacting case, but the energy is shifted. From figure 9b
we see a different behaviour. There is a maximum of ∆2 for no magnetic field and
then it decreases rapidly and goes very close to zero. This means the states E2
and E3 are close to being degenerate for large B. (This grouping of energy states
has been mentioned earlier in this section). This effect is not apparent without
the interaction. For ∆3 we see a similar behaviour as for ∆1 , there is a maximum
around 0.5 T and then ∆3 converges to around 0.15 meV while the behaviour
without the interaction is very different.
17
(a) E2 − E1
(b) E3 − E2
(c) E4 − E3
Figure 9: Energy splitting as a function of magnetic field. Red is for the interacting case
and blue for the non-interacting one
18
4.2
Charge density
We will start by looking at the charge density of the ground state for a few values
of magnetic field (see figure 10). From the figure we see that the charge density
is not strongly affected by the magnetic field as long it is lower than 0.5 T. In
strong magnetic field the charge density tends to smear out in the ±y-direction
(forming of edge states).
Let’s now how the first and second excited states are affected (see figures 11
and 12). From these figures we can see that the effects of the interaction is much
greater, especially in higher magnetic field. The first and second excited states are
not greatly modified in zero magnetic field but they is distinctively different in a
modest 0.2 T magnetic field. The “hump” in the middle of figure 11d disappears
when the interaction is included. This is because of the electron electron repulsion
which makes areas of high charge density energetically unfavorable. The magnetic
field seems to have very different effects on the interacting case compared to the
non-interacting one. For examples on figure 12 we see that without the magnetic
field both the interacting and non-interacting charge densities have areas of high
density. With increasing magnetic field the interacting density seems to smear
out and spread more uniformly while the non-interacting case retains its areas of
high density.
5
Conclusions
We have used numerical techniques to calculate the energy spectrum and charge
distribution of two electrons in a 200nm GaAs wire in a constant homogeneous
magnetic field including the Coulomb interaction between the electrons and compared the results with the non-interacting case.
According to section 3 the results converged very nicely when the basis/grid
was enlarged and the biggest cause of error was the non-zero value of η (see
section 3.3). In section 4 we saw how the lowest states are not greatly affected by
the magnetic field nor the Coulomb interaction but when we go higher in energy
the system is highly sensitive to the interaction and magnetic field.
There is a lot more to be studied in this system. One example is adding the
third electron. The program is capable of doing the calculations for three electrons
(and it is easy to generalize it to any number of electrons) but the basis has to
be shrunk considerably (see table 2) so calculations beyond 4-7 electrons are not
very practical. Another example is adding a time dependent perturbation which
would force us to truncate the many-electron basis from a few thousand states
to a few hundred to make numerical calculations possible.
19
(a) With interaction, B = 0.0 T
(b) Without interaction, B = 0.0 T
(c) With interaction, B = 0.5 T
(d) Without interaction, B = 0.5 T
(e) With interaction, B = 2.0 T
(f) Without interaction, B = 2.0 T
(g) With interaction, B = 4.0 T
(h) Without interaction, B = 4.0 T
Figure 10: Plot of charge density of the ground state for a few values of magnetic field,
with and without interaction.
20
(a) With interaction, B = 0.0 T
(b) Without interaction, B = 0.0 T
(c) With interaction, B = 0.2 T
(d) Without interaction, B = 0.2 T
(e) With interaction, B = 0.5 T
(f) Without interaction, B = 0.5 T
(g) With interaction, B = 4.0 T
(h) Without interaction, B = 4.0 T
Figure 11: Plot of the charge density of the first excited state for a few values of magnetic
field, with and without interaction.
21
(a) With interaction, B = 0.0 T
(b) Without interaction, B = 0.0 T
(c) With interaction, B = 0.2 T
(d) Without interaction, B = 0.2 T
(e) With interaction, B = 0.5 T
(f) Without interaction, B = 0.5 T
(g) With interaction, B = 4.0 T
(h) Without interaction, B = 4.0 T
Figure 12: Plot of the charge density of the second excited state for a few values of
magnetic field, with and without interaction.
22
References
[1] Fetter and Walecka. Quantum Theory of Many-particle Systems, pages 4–21.
McGRAW-HILL BOOK COMPANY, 1971.
[2] Viðar Guðmundsson. Gme, interaction in the system, 2009.
23