Download Undergraduate Project in Physics Yuval Zelnik Advisor: Prof. Yigal Meir

Undergraduate Project in Physics
Yuval Zelnik
Advisor: Prof. Yigal Meir
Department of Physics
Ben Gurion University
Spontaneous formation of magnetic impurities in disordered metals
Abstract:
Recent work has demonstrated that a magnetic impurity, a quasi localized state, forms
generically near a quantum point contact. We plan to explore whether a disordered metal,
where quantum point contacts form naturally near saddle-points of the potential,
contained such magnetic impurities. This may explain several recent experimental
observations.
Theory
Electrons in metals are a many body problem, a problem that is impossible to solve
analytically in any but the simplest potentials. We therefore resort to simulations in order
to find the approximate solutions of the system. We will use several more assumptions in
order to simplify the calculations, as will be explained below.
The Jellium Model
We shall build the following Hamiltonian as our starting point:
pˆ 2 1
e2
(1) Hˆ = ∑ i + ∑ r r + Hˆ e,b + Hˆ b ,b
2 i , j ≠i rˆi − rˆj
i 2M
Where the electron-background interaction is:
r
r
r r nˆ ( r ) nˆb ( r ' )
2
ˆ
(1.1) H e ,b = −e ∫ dr ∫ dr '
r r
rˆ − rˆ
And the background-background interaction is:
r
r
1 2 r r nˆb ( r ) nˆb ( r ' )
ˆ
(1.2) H b ,b = e ∫ dr ∫ dr '
r r
2
rˆ − rˆ
The electron charge density operator is:
N
r r
r
(1.3) enˆ ( r ) = e∑ δ rˆ − rˆi
i =1
The background charge density is:
(
)
r
(1.4) enˆb ( r ) = en
Here we made the simplification that the background charge is uniformly distributed,
jelly like (hence the name of the model).
It can be shown [1] that in this model the combination of the electron-background
interaction and the background -background interaction cancel out the q=0 (momentum)
term in the electron-electron interaction.
This means that these terms simply contribute to the Energy of every electron, which is
not surprising considering the uniform charge of the background. From now on we will
not consider these two interaction terms, and use a simplified form of the Hamiltonian:
pˆ 2 1
e2
(2) Hˆ = ∑ i + ∑ r r
2 i , j ≠i rˆi − rˆj
i 2M
And if we want to add an external potential we will get:
pˆ 2 1
e2
(3) Hˆ = ∑ i + ∑ r r + Vext ( ri )
2 i , j ≠i rˆi − rˆj
i 2M
Slater Determinant
Because the electrons we describe are fermions, it follows that the total wave function
must be anti-symmetric. In order to build an anti-symmetric total wave function we use
the Slater Determinant.
For an anti-symmetric wave function of two electrons, we will get:
1
( ϕ1 1 ϕ2
2
Which is a determinant of a 2x2 matrix.
For N electrons we similarly get:
(4)
Ψ =
(5) Ψ =
1
N
2
− ϕ1
ϕ1
ϕ1
1
2
...
ϕ1
2
ϕ2
ϕ2
ϕ2
1
2
...
N
ϕ2
N
1
)=
1 ϕ1
2 ϕ1
...
...
...
...
ϕN
ϕN
1
2
ϕ2
ϕ2
1
2
1
2
...
ϕN
N
If we define the simple (not anti-symmetric) total state as:
(6) n1 , n2 ...nN ≡ n1 1 n2 2 ... nN N
And a permutation operator as:
(7) Pˆp n1 , n2 ...nN ≡ n p1 , n p2 ...n pN
Then we can define an anti-symmetric operator which acts on the simple total state and
turns it into an anti-symmetric total state (Slater Determinant) as:
1
P
(8) Aˆ
( −1) Pˆp
∑
N p∈S N
This operator is hermitian (AH=HA) and it happens that: Aˆ 2 = N !Aˆ
We will mark the simple total state as: ϕ and the anti-symmetric determinant as: Ψ
Hartree-Fock method
We go back to the Hamiltonian from (3):
pˆ 2
1
e2
(9) Hˆ = ∑ i + Vext ( ri ) + ∑ r r = Hˆ 0 + Hˆ 1
2 i , j ≠i rˆi − rˆj
i 2M
Where:
pˆ 2
(9.1) Hˆ 0 ≡ ∑ i + Vext ( ri )
i 2M
1
Hˆ 1 ≡ ∑ U (| ri − rj |)
2 i , j ≠i
e2
U (| ri − rj |) ≡ r r
rˆi − rˆj
If the single electron wave functions ni are eigen states of Ĥ 0 , then so is the slater
determinant from these single electron states, Ψ .
Using Perturbation theory to the first order, in the base where Ĥ 0 is diagonal:
1
1
Hˆ 1 ≅ Ψ ∑ U (| ri − rj |) Ψ = ∑ ϕ A†U (| ri − rj |) A ϕ =
2 i , j ≠i
2 i , j ≠i
=
1
1
ϕ U (| ri − rj |) A2 ϕ = ∑ ϕ U (| ri − rj |) N !A ϕ =
∑
2 i , j ≠i
2 i , j ≠i
=
1
P
( −1) n1 , n2 ...nN U (| ri − rj |) n p1 , n p2 ...n pN
∑
∑
2 i , j ≠i p∈S N
=
1
P
( −1) ∏ nk | n pk
∑
∑
2 i , j ≠i p∈S N
k ≠i , j
(10)
Since
∏
k ≠i , j
ni , n j U (| ri − rj |) n pi , n p j
nk | n pk = 1 and the other element equals zero except for two cases
( i, j = pi , p j and i, j = pi , p j ), we get:
1
Hˆ 1 ≅ ∑ ⎡ ni , n j U (| ri − rj |) ni , n j − ni , n j U (| ri − rj |) n j , ni ⎤
⎦
2 i , j ≠i ⎣
Written more explicitly:
⎡
⎤
1
2
2⎢
2
2
*
ˆ
H1 ≅ ∑ ∑ ∫ dr dr ' | ϕiσ (r ) | | ϕ jσ ' (r ') | U (| r − r ' |) − δσ ,σ 'ϕiσ (r )ϕ jσ ' (r ')*ϕ jσ ' (r )ϕiσ (r ')U (| r − r ' |) ⎥
4244444444
3⎥
⎢ 1444442444443 14444444
2 i , j ≠i σ ,σ '
Direct
Exchange
⎣
⎦
(11)
Where σ , σ ' are the spin indices, and the integrals are two dimensional for our specific
purposes.
As written, the first term is called the Direct (or Hartree) term, and the second the
Exchange (or Fock) term.
The Direct term can be seen as the classical electro-magnetic repulsion between the
electrons, while the Exchange term has no classical equivalent, and is purely Quantum
Mechanical. It can be seen as the attractive force between electrons of different spins,
caused by the so called exchange hole, the lack of electrons of the same spin in the
vicinity.
If we take the full Hamiltonian’s expectation value:
⎛ p2
⎞
< ψ | H |ψ >= ∑ ∫ d 2 rϕi*σ (r ) ⎜
+ Vext (r ) ⎟ ϕiσ (r ) +
i ,σ
⎝ 2M
⎠
(12)
1
d 2 rd 2 r 'U (| r − r ' |) ⎡⎣| ϕiσ (r ) |2 | ϕ jσ ' (r ') |2 −δσ ,σ 'ϕi*σ (r )ϕ jσ ' (r ')*ϕ jσ ' (r )ϕiσ (r ') ⎤⎦
∑
∑
∫
2 i , j ≠i σ ,σ '
Using the variation principle in (13) we get:
(13)
δ
δψ
⎡
2
2⎤
⎢< H > −∑ ε i ∫ d r | ϕiσ (r ) | ⎥
i
⎣
⎦
⎛ p2
⎞
+ Vext (r ) ⎟ ϕiσ (r ) + ∑ ∫ d 2 r ' | ϕ jσ ' (r ') |2 U (| r − r ' |)ϕiσ (r )
⎜
j ,σ '
⎠
(14) ⎝ 2 M
−∑ ∫ d 2 r 'ϕ jσ (r ') * ϕiσ (r ')U (| r − r ' |)ϕ jσ (r ) = ε iσ ϕiσ (r )
j
Using Fourier series expansion:
ϕiσ (r ) = ∑ aiσ k eikr
(15)
ϕ jσ (r ) = ∑ b jσ k eikr
k
k
U (| r − r ' |) = ∑ uk e
ik ( r − r ')
k
ε iσ ∑ aiσ k eikr = ∑
k
k
(16) + ∑ ∫ d r '
2
j ,σ '
−∑ ∫ d r '
(17)
∑
b*jσ ' k b jσ ' k 'uk '' aiσ k '''ei( k ''+ k ''')r ei( − k + k '− k '')r '
∑
b*jσ k aiσ k 'uk ''b jσ k '''e (
ikr
k
∑ bσ
j ,σ ' k , k ', k ''
k2
aiσ k eikr + ∑ aiσ k vk 'ei( k + k ')r
2M
k ,k '
i k '' + k ''') r i ( − k + k ' − k '') r '
e
k , k ', k '', k '''
ε iσ ∑ aiσ k e
+∑
k
k , k ', k '', k '''
2
j
Vext (r ) = ∑ vk eikr
*
j 'k
k2
=∑
aiσ k eikr + ∑ aiσ k vk 'ei( k + k ')r
k 2M
k ,k '
b jσ ' k 'uk − k ' aiσ k ''ei( − k + k '+ k '')r − ∑
∑ b σ aσ
i ( − k + k ' + k '') r
*
j k i k ' k − k ' jσ k ''
u
b
e
j k , k ', k ''
So that for a given i,σ we get a set of coupled equations, for each value of q:
(18) ε iσ aiσ q =
q2
aiσ q + ∑ aiσ k vk ' + ∑ ∑
2M
k ,k '
j k , k ', k ''
k + k '= q
− k + k ' + k ''= q
⎛
⎞
*
*
⎜ ∑ b jσ ' k b jσ ' k 'uk ' − k aiσ k '' − b jσ k aiσ k 'uk ' − k b jσ k '' ⎟
⎝ σ'
⎠
⎡
⎤
q
(19) ε iσ aiσ q = ⎢δ q ',q
+ δ q ', q − k vk + δ q ', q + k − k ' ∑ b*jσ ' k b jσ ' k 'uk '− k − δ q ', q + k − k ' ∑ b*jσ k b jσ k 'uq '− k ⎥ aiσ q '
2M
j ,σ '
j
⎣
⎦
2
If we take the matrix inside the square brackets, and diagonize it, we will get eigen values
representing the energies of the different electrons, and the eigen states of the electrons
themselves. If we start with a guess for the solution of the wave functions of the
electrons, and use equation (19) iteratively, we can hope to find a self-consistent solution
for the problem. This is a solution that when given as an input to equation (19) is very
close to the solution that is received as the output of the same equation (with the
mentioned input).
This self-consistent solution will be our approximation to the true solution of the system.
Simulations
The simulations were carried out using Matlab.
The algorithm that was written was given an external potential and a guess for the
starting wave functions of the electrons. For all the simulations the starting guess for the
wave functions was plane waves. That is, given periodic boundaries, the starting guess
for the wave function was:
(20) ϕiσ (r ) = Aeiki r = Ae
i k xi 2 + k yi 2 r
In each iteration, after finding the new wave functions, a merge was made between the
old and new wave functions, so that at each iteration the wave functions we changed only
by little. Most of the simulations we carried out where 10% of the new wave functions
were mixed with 90% of the old ones.
In order to stop “spin drifting” (since there was no “spin mixing” the switch of electrons
between spin up and down was very fast), only one electron was allowed to change its
spin direction on every iteration.
For efficiency reason the combinations of plane waves that were used to build the
electron's wave functions were only plane waves with low momentum (Which are the
only ones relevant).
For the interaction force, we used the Yukawa interaction instead of the usual EM
interaction:
α e− β |r −r '|
(21) U (| r − r ' |) =
| r −r '|
When β is close to 0, we get back to the usual EM interaction.
Checking the Direct and Exchange terms
In order to check the code, the solution was tested for a situation with no external
potential. Here the electrons remain as plane waves since nothing is breaking the
symmetry of the system. For this problem, it is therefore possible to calculate the energies
analytically.
Assuming plane waves and no external potential:
ϕiσ (r ) = Aeik r
Vext (r ) = 0
i
(22)
U (| r − r ' |) =
α e− β |r −r '|
| r −r '|
Using (14) we get:
2
− β |r − r '|
α e − β |r −r '| ik r
− ik r '
(23) ki Aeik r + ∑ ∫ d 2 r ' | Aeik r ' |2 α e
Aeik r − ∑ ∫ d 2 r ' Ae
Aeik r '
Ae = ε iσ Aeik r
j
i
2M
(24)
j ,σ '
| r −r'|
j
i
j
i
j
| r −r'|
ki 2
α e − β |r |
α e− β |r| i( ki − k j )r
e
+ A2 N ∫ d 2 r
− A2 ∑ ∫ d 2 r
= ε iσ
2M
|r|
|r|
j
N ≡ ∑1
j ,σ '
i
To solve the second integral we use a Fourier Transform of the interaction potential,
which has a known solution [1], in two dimensions we have:
⎧ α e − β |r | ⎫
(25) FT ⎨
⎬=
⎩ |r| ⎭
2πα
β 2 + k2
We therefore get:
(26)
ki 2
α
+ 2π A2 N − A2
2M
β
2πα
β 2 + ( ki − k j )
2
= ε iσ
N ≡ ∑1
j ,σ '
Comparing these results for 16 electrons of the lowest kinetic energy (8 electrons in each
spin direction) showed an error of less then 0.1%, which originates from the discrete
nature the simulation.
Using a QPC potential
We used a QPC potential such as described in [5] to check for magnetic impurities using
the Hartree-Fock method:
⎛
⎜
⎜
V
V0
0
(27) VQPC ( x, y ) =
+ m* ⎜ ω y +
*
⎛ xω m* ⎞
⎛
⎜
2 xω x m
2 cosh 2 ⎜ x
cosh
h
⋅
⎟
⎜
⎜
⎜
⎜
2V0 ⎟⎠
2V0
⎝
⎝
⎝
Using the following paramaters:
m = 0.1
We get the potential:
h = 100
ω y = 0.1
ωx = 5
2
⎞
⎟
⎟ y 2 m*ω y2 y 2
−
⎟
2
⎞⎟ 2
⎟⎟
⎟
⎠⎠
V0 = 1000
The other parameters used were:
mass = 10−8
Lx = 40
Ly = 80
α = 500
β = 0.01
N electrons = 8
This gave a semi-consistent solution. After about 200 iterations, the total energy stopped
declining, and reached a semi-stable state, with fluctuations. After 600 more iterations
this has not changed much. Below is the graph for the total energy as a function of the
iteration number:
After 700, 750 and 800 iterations, the electrons wave function densities were:
(Each row shows the 8 wave functions of the electrons, from left to right are the 4 spin up
electrons and the 4 spin down electrons, from low energy to high energy):
Spin up after 700 iterations
Spin down after 700 iterations
Spin up after 750 iterations
Spin down after 750 iterations
Spin up after 800 iterations
Spin down after 800 iterations
The sum of the spin up, spin down, and the difference between them (left to right) was:
After 700 iterations
After 750 iterations
After 800 iterations
As can be clearly seen, there is a noticeable difference in the total density between the
spin up electrons and the spin down electrons, which causes magnetic impurities.
However, the system is not stable, and these differences fluctuate between a difference in
the upper part of the QPC, and a difference in the lower part.
Using a modified potential
With a modified potential from the given above, such that the 3 different terms in
equation (27) are multiplied by 500, 0.01 and 0.1 respectively, and using the following
parameters:
m = 0.1
h =1
ωy = 5
ω x = 0.1
V0 = 1
We get the potential:
The other parameters used were:
mass = 10−9
Lx = 40
Ly = 80
α = 2000
β = 0.01
N electrons = 8
This gave a self-consistent solution. After less then 100 iterations, the total energy
stopped declining, and reached a stable state with very small fluctuations. Continuing a
further 50 iterations showed no strong change. Below is the graph for the total energy as
a function of the iteration number:
Spin up after 100 iterations
Spin down after 100 iterations
Spin up after 120 iterations
Spin down after 120 iterations
Spin up after 150 iterations
Spin down after 150 iterations
The sum of the spin up, spin down, and the difference between them (left to right) was:
After 100 iterations
After 120 iterations
After 150 iterations
The solution is obviously quite stable, and it can be seen that the spin up electron’s
density is higher in the middle of the saddle, and lower at the valleys.
Here we have a magnetic impurity in stable self-consistent solution.
Disordered media
Due to time constraints, no work was done on a disordered potential, as was planned.
References
1. G. F. Giuliani, G. Vignale, “Quantum Theory of the Electron Liquid”, Cambridge
University Press, 2005.
2. P. L. Taylor, O. Heinonen, “A Quantum Approach to Condensed Matter Physics”,
Cambridge University Press, 2002.
3. L. Pauling, E. B Wilson, “Introduction to Quantum Mechanics”, 1935.
4. T. Rejec, Y. Meir, “Magnetic impurity in quantum point contacts”, Nature, Vol. 442
#900, 2006.
5. K. Hirose, Y. Meir, N.S. Wingreen, “Local Moment Formation in Quantum Point
Contacts”, Physical review letter, Vol. 90 #2, 2003.