Magnetic dynamics of
superconducting thin films and
devices
Jørn Inge Vestgården
Thesis submitted for the degree of Philosophiae Doctor
Department of Physics
University of Oslo
September 30, 2007
Contents
1 Introduction
2 Domain wall and vortex matter
2.1 Background . . . . . . . . . . . . .
2.2 Ferrite garnet films . . . . . . . . .
2.3 London superconductors . . . . . .
2.4 Vortex at an interface . . . . . . .
2.5 Thin magnetic rods . . . . . . . . .
2.6 Modeling a charged Bloch wall . .
2.7 Bloch wall-vortex interaction force
2.8 Vortex matter response . . . . . .
2.9 Discussion of Bloch wall width . .
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3 Thin film flux dynamics
3.1 Background . . . . . . . . . . . . . . . . . . .
3.2 Thin film flux dynamics . . . . . . . . . . . .
3.3 Survey of sample geometries . . . . . . . . . .
3.4 Electric field and polarization . . . . . . . . .
3.5 Inversion of Biot-Savart law in Fourier space .
3.6 An alternative simulation formalism . . . . .
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4 Landau-Zener transitions in superconducting qubits
4.1 Background . . . . . . . . . . . . . . . . . . . . . . . .
4.2 The Landau-Zener Hamiltonian . . . . . . . . . . . . .
4.3 Zener’s solution . . . . . . . . . . . . . . . . . . . . . .
4.4 Charge qubit . . . . . . . . . . . . . . . . . . . . . . .
4.5 Noise in solids . . . . . . . . . . . . . . . . . . . . . . .
4.6 Bloch notation . . . . . . . . . . . . . . . . . . . . . .
A Domain wall, calculations
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47
i
B Qubit, calculations
B.1 Integral equations for zp and zq . . . . . . . . . . . . . . .
B.2 Adiabatic transform . . . . . . . . . . . . . . . . . . . . . .
51
52
52
C Formulas
C.1 Forward and inverse Laplace transforms . . . . . . . . . . .
C.2 Parabolic Cylinder Functions . . . . . . . . . . . . . . . . .
57
57
57
Bibliography
59
List of papers
64
ii
Chapter 1
Introduction
Overview
Superconductivity was discovered as early as in 1911. Since then the phenomenon has been subject to considerable interest, in fundamental science
as well as for advanced electronic applications. The interest is due to the
vast number of highly distinct magnetic and electronic properties found in
no other states of matter.
This thesis is a theoretical investigation of selected problems about
magnetic dynamics and superconducting devices. Reading the thesis requires a basic level on superconductivity, as found in textbooks [1, 2, 3, 4].
This thesis operates at several length scales and it shows how dramatically
magnetic properties change when length scale changes. At macroscopic
length scale, ∼ mm, dynamics of type-II superconductors is described by
the classical Maxwell equations. However, the resistivity is non-constant
and even highly non-linear, giving both magnetic and electrical properties
far from that of ordinary metals. At smaller length scales, < µm, one
sees that magnetic field is not continuous, but consist of quantized magnetic flux lines, all carrying exactly the same quantity of magnetic flux,
φ0 = h/2e = 2.07 × 10−15 Wb, where h is Planck’s constant and e is the
elementary charge. I.e., the lines drawn in school text books to illustrate
magnetic fields are actually real physical objects in type-II superconductors.
The special magnetic and electronic properties of superconductors are to
high degree exploited for nanoscale devices. Single vortices can be manipulated and moved around with small electromagnets or permanent magnets.
Devices based on single vortices can, e.g., interpret the presence of a vortex
as a bit of information. Due to the discrete vortex nature such a bit is ex1
Chapter 1. Introduction
Figure 1.1: Magnetic flux distribution in a type-II superconducting square
thin film, simulated with the algorithm described in Sec. 3 and plotted in
the style of magneto-optical images, where intensity represents Bz .
tremely persistent and provides high fault tolerance devices. Furthermore,
the energy cost of moving a vortex is low compared to the energy costs
of semiconductor based devices, enabling high performance computing. In
this way superconductors can be used to build classical computers. But
even more exciting, the quantum nature of superconductivity means that
one can make tunable quantum two levels systems. Such two level systems
are called qubits and they are the building blocks for quantum computers.
The advantage of having quantum computers in a solid state environment
is that one can exploit the knowledge from classical computers, and it is
easy to connect leads and control the system with e.g., gate voltages. The
disadvantage is that temperature must be extremely low in order for the
qubit to maintain its coherent quantum state. Even then, noise is a major
problem and noise reduction is a major research area within the quantum
computing community.
The common factor that binds the topics of this thesis together is the
possible applications in superconducting devices.
The work on flux distribution in superconducting thin films, chapter 3,
shows how important the basic sample shape is with regards to magnetic
properties. Even more interesting, just small asymmetries introduced in
the sample can totally modify the dynamical properties. As discussed in
papers 2 and 3, a small indentation of the edge, or a strategically placed nonconducting hole can guide flux away from or to specific regions of a superconducting device. Having good models of thin films is important since most
devices are based on films, while calculations often assume thick samples,
2
which simplifies the mathematical treatment. Both papers 2 and 3 show
that the properties of thin superconductors are highly different from those
of thick superconductors, and especially so with regards to magnetic properties.
The work on single vortices, chapter 2 and paper 1, is motived by the
need for manipulation of single vortices in devices. This manipulation is
e.g. executed by small electromagnets or strategically introduced defects, as
antidots or permanent magnets. The domain wall in chapter 2 has several
desirable properties when it comes to vortex manipulations. First, it has
strong magnetic field gradients, which is the very basics when it comes to
interaction with vortex matter. Second, it is a permanent magnet so that
no disturbing leads and currents are required. Third, contrary to most
permanent magnets it is easily movable. Forth, being a part of a magnetooptical setup it is actually possible to immediately monitor the actions as
they take place. All these properties make domain walls a potentially useful
building block in single-vortex based devices.
The work on qubits, chapter 4, is motivated by the need for models that
handle true dynamics of quantum devices, not just their static properties.
In order to do computations qubits must maintain their coherence while
evolving dynamically. Thus we focus on qubits whose energy states are
dynamically interchanged by the control system. The operation of switching energy levels of the qubit is called a Landau-Zener transition, and the
outcome is either a flip of the qubit state, or a superposition of the two
states, all depending on the interchanging rate. The state after transition
is hence strongly dependent on how the operation is performed, and the
outcome cannot be approximated with simple adiabatic models, generalizing from the static case. Even more so, as found in paper 4, for models
including environment coupling, the long-time behavior of driven qubits
depend dramatically on details of the operations performed.
Hence, superconducting devices operate at many length scales and exploit a wide range of properties only found in superconductors. This thesis
argues that there is a need for understanding true dynamics, not just approximations based on equilibrium and stationary properties.
3
Chapter 1. Introduction
Road-map
This work is divided in three main parts.
a) Interaction between a domain wall and single vortices is the topic of
paper 1 and chapter 2. Paper 1 describe the concept of a charged domain wall and tells how this particular model can explain a magnetooptical experiment. Chapter 2 is about how superconductors and
superconducting vortices react when they get in close contact with a
magnet, where the magnet in this case is a long domain wall residing inside a magnetic film a short distance above the superconductor
surface. The presence of the domain wall has two effects. First, it
induces Meissner currents. Second, it interacts with vortex matter.
Both these effects are explained in detail in chapter 2.
b) Thin film flux dynamics is the topic of papers 2 and 3, and chapter 3. Paper 2 describes how a small indentation of the edge affects
flux penetration in a strip. Paper 3 describes a method to handle
samples with non-conducting holes. Chapter 3 is on flux penetration in macroscopic superconducting thin films, when the films are
exposed to a gradually increasing external magnetic field. Since the
magnetic field penetrates from the edges, the overall sample shape is
of special importance and the chapter gives an overview of how the
magnetic field distributes in a variety of samples, like squares, rectangles, disks, and rings. Also more complicated shapes including one
or many non-conducting holes are included. Such holes and patterns
are important since they might be used for flux guidance in devices.
Chapter 3 also discusses charge distribution of the superconductors
and a short outline of an improved thin-film simulation formalism.
c) Superconducting qubits is the topic of paper 4 and chapter 4. Paper 4 covers Landau-Zener-like transitions in qubits influenced by
telegraph noise. The emphasis is on how the functional form of the
external driving affects the transitions. Chapter 4 covers background
on Landau-Zener transitions and noise in superconducting devices.
Appendix B covers formalism on qubits and telegraph noise.
4
Chapter 2
Domain wall and vortex
matter
2.1
Background
This chapter is connected to paper 1 and the topic is interaction between
a magnetic domain wall and superconducting vortices. The physical setting of this section comes from magneto-optical experiments performed by
Pål Erik Goa et al. [5, 6, 7, 8]. Due to the amazing spatial resolution
of the magneto-optical images single vortices were visible and the experiments demonstrated real time monitoring and simultaneous manipulation
of vortex matter.
Paper 1 is concentrated around to the two images of Fig. 2.3. The
images shows the tip of a domain wall, where the domain wall is clearly
attractive for the vortex matter. The interesting problem with the images
is that the domain wall should by calculations be repulsive, not attractive.
This motivates a closer study of the structure of the domain wall itself,
and the proposed solution to the problem is to include also the in-plane
magnetization, giving what is called a charged Bloch wall [9].
In this chapter I will hence go through models for all separate entities of the problem in detail: the magnetic film, the domain wall, the superconductor and the vortex matter. Then I will show how the different
parts interacts, e.g., how the superconductor screens the charged Bloch
wall (Fig. 2.6), the interaction force on a vortex (Fig. 2.7), and how vortex
matter is perturbed by the presence of the wall (Fig. 2.8).
The mathematical formalism of this chapter is also quite general so that
it can be utilized for related problems regarding single vortices or elongated
5
Chapter 2. Domain wall and vortex matter
Figure 2.1: Sketch of a charged Bloch wall in a ferrite garnet film. The
term ’charged’ refers the non-parallel in-plane magnetization, ϕ 6= 0, which
creates the excess magnetic ’+’ charges at both sides of the wall.
magnets as micro-manipulators.
2.2
Ferrite garnet films
Magneto-optical experiments on superconductors are performed by placing
a magnetic indicator film at the top of the sample of interest. The indicator
film is usually an in-plane magnetized ferrite garnet crystal. The effect
which is utilized for the imaging is the Faraday rotation. Faraday rotation
means that the rotation of the polarization direction of light shone through
the crystal depends on the z-component of the magnetization, which in turn
is related to the external magnetic field. In this way one can create images
of the magnetic field penetrating a surface, e.g., from a superconductor. For
a general review on magneto-optical imaging see Ref. [10] and for magnetooptical setup for single vortex resolution see Ref. [7].
Large magnetized films tend to split in a number of domains in order
to reduce their free energies. For magneto-optical imaging this has two
consequences. First, since the small background z-component of the magnetization varies from domain to domain, different parts of the indicator
film have different color scales. This is why the domains can be clearly
seen in images like Fig. 2.2, where the domain wall itself is to small to be
visible. Second, the domain wall acts as an effective magnet. The domain
wall width is typically of order micrometer or less and it interacts with the
superconductor and vortex matter at this length scale, as seen in Fig. 2.3.
The domain wall in this image is perpendicular to the film and such domain
walls are called Block walls. For more information please see Refs. [9, 11]
for background on magnetic domains or Refs. [12, 13, 14] for more on the
6
2.2 Ferrite garnet films
Figure 2.2: Two anti-parallel domains separated by a zigzag domain wall.
Bloch walls in ferrite garnet films and the relation to superconductors and
vortex matter.
Fig. 2.1 sketches a simplified model of a Bloch wall. There are two antiparallel in-plane domains separated by a domain wall of finite thickness
2W . The main simplification is to treat the magnetization as constant
when it in reality turns smoothly [11]. The unusual thing with the model
is that in-plane magnetization makes an angle ϕ with the wall. This angle
relates to the fact that the wall is a part of a larger zigzag pattern, as
seen in Fig. 2.2. The common model for Bloch walls found in text books
have ϕ = 0, since the magnetization normally relaxes and aligns with the
wall in order to reduce its stray field. In fact, a ϕ 6= 0 gives rise to an
energetically expensive magnetic monopole-like stray field, and hence such
walls are called charged Bloch walls [9]. Charged Bloch walls can only exist
for in-plane anisotropic films, where the anisotropy prevents an alignment of
the in-plane magnetization. Fig. 2.1 illustrates the perpendicular wall with
a magnetic dipole, i.e., equal amounts of magnetic ’+’ and ’−’ charges at the
top and bottom of the wall. The ϕ 6= 0 is illustrated with excess magnetic
’+’ charges at both sides of the wall. Together with the fact that the field
from a single vortex also can be approximated with a magnetic charge [15],
we can easily find when the respective forces on the vortex are attractive
or repulsive. The setup in Fig. 2.1 is chosen so that the perpendicular
magnetizations repels while the in-plane magnetization attracts the vortex.
Note that the charges in Fig. 2.1 are so that the wall and vortex would
appear with opposite polarity of the domain wall in an magneto-optical
images, just as in Fig. 2.3.
The simple argument about magnetic charges can tell the sign of each
of the two force contributions on a trial vortex, but cannot tell which one is
dominant. In order to find that, we need to look at each of the two forces
in detail. We label F ⊥ = F ⊥ (x) as the force on a trial vortex originating
from the perpendicular Bloch wall. Correspondingly, F k = F k (x) is the
force from the in-plane charges.
We must also be careful not to forget the superconductor itself. Because
7
Chapter 2. Domain wall and vortex matter
before
after
Figure 2.3: Magneto-optical images with single vortices visible as bright
dots. Central in the left figure is the black domain wall, which corresponds to the tip of a zigzag wall, see Fig. 2.2. In the right image the
wall has been removed, leaving a frozen in vortex distribution. Dimensions
are 70µm×70µm. Images are taken by P. E. Goa in the same series of
experiments as Refs. [5, 6, 7].
of induced Meissner currents there will be a net interaction between the
superconductor and the vortex matter. However, these forces will turn out
to be exactly equal to the direct force between the magnet and vortex.
2.3
London superconductors
Superconductors have two intrinsic length-scales: the correlation length
ξ and the penetration depth λ. The former sets length scale of spatial
variations of the order parameter, Ψ, while the latter sets the length scale
of magnetic properties. London theory means to ignore spatial variation
of
√
Ψ, i.e., ξ = 0. Type-II superconductors are characterized by λ > ξ/ 2 and
consequently London theory is the extreme type-II limit.
Mathematically, London theory is formulated by the London equation
∇ × ∇ × H + λ−2 H = 0,
(2.1)
where H is the magnetic field and λ is London penetration depth.
The London equation can also be reformulated by the vector potential
−∇2 A + λ−2 A = 0,
(2.2)
where we operate in London gauge, ∇ · A = 0. As soon as the vector
potential is determined, the other quantities as magnetic flux density, B =
8
2.4 Vortex at an interface
z/λ
5
0
−5
−8 −6 −4 −2
0 2
r/λ
4
6
8
Figure 2.4: The field lines of a superconducting vortex near a planar interface plotted as contour lines of rAϕ , Eq. (2.5), with quadratic spacing
[15].
∇ × A, and Meissner current density, j = −A/µ0 λ2 , are readily given. The
permeability of vacuum is µ0 = 4π × 10−7 N A−2 .
2.4
Vortex at an interface
Let us now consider a vortex residing inside a half space superconductor.
Deep in the bulk the vortex is like an Abrikosov vortex whose radius is
the penetration depth λ. Far outside the superconductor the field, on the
other hand, looks like the field from a magnetic monopole [15]. Both these
features are visible in Fig. 2.4.
The problem of how a vortex in a bulk superconductor breaks through
a surface was solved by J. Pearl in 1966 [16].1 This solution is of course
useful for the problem of interaction between the domain wall and vortex
matter, so I will include a short derivation of it here, and write it on a
form convenient for our purpose. Let us start with the London equation,
Eq. (2.2), with one flux quantum as a source,
−∇2 A + λ−2 A = λ−2 Φ(r),
1 This
(2.3)
work must not be confused with the thin film Pearl vortices from 1964 [17].
9
Chapter 2. Domain wall and vortex matter
where A is magnetic vector potential. The source function of the vortex is
Φ(r) =
φ0 1
eϕ .
2π r
(2.4)
where ∇ × Φ = φ0 δ2 (r) and the magnetic flux quantum is φ0 = h/2e. The
derivation of A from Eq. (2.3) is in appendix A and the result is
Φk
Ak (z) =
(λτ )2
τ
τ +k
e−kz
k
eτ z
1 − τ +k
, z≥0
, z<0
(2.5)
√
where k = (kx , ky ), τ = λ−2 + k 2 , and Φk = −φ0 k12 (ẑ × ik) is the Fourier
transform of Eq. (2.4). Note that Eq. (2.5) split in a z-dependent and a
z-independent term for z < 0 . The z-dependent term is a surface term and
the z-independent term is the familiar Abrikosov term.
The circulating currents are easily obtained by the second London equation j = (Φ − A)/µ0 λ2 , where j = j(r, z). The Fourier transform is
Φk
k
1
2
τz
jk =
(λk) +
.
(2.6)
e
µ0 λ2 (τ λ)2
τ +k
The interaction with other vortices is given by the superconducting
Lorentz force f = φ0 j × ẑ. RIntegrated over the vortex length this gives
0
the interaction force, Fvv = L dz f (z). From the above calculations of j,
the Fourier components of the vortex-vortex force is
1 1 1
ik
φ20
vv
,
(2.7)
|k|L + 2
Fk =
µ0 λ2 |k|τ 2
λ τ |k| + τ
√
where again τ = λ−2 + k 2 . The first term is the Abrikosov term which
depends on vortex length L. The second term is the surface term, which is
independent of thickness, provided L ≫ λ. The surface term is independent
of λ and for vortices a large distance r apart the surface force scales as 1/r2 ,
contrary to the exponential decay of the Abrikosov term.
2.5
Thin magnetic rods
This section contains building blocks to construct solutions for elongated
magnets above a bulk superconductor. Details and derivations are in appendix A.
Let us consider a magnetized rod which is infinite in y-direction. The
vector potential has then only one component, A = Aŷ, where A = A(x, z).
10
2.5 Thin magnetic rods
The superconductor is in the half space z < 0 and the magnetic rod is
somewhere in the vacuum z > 0. The London equation and Ampère’s law
read as
λ−2 A − ∇2 A = 0
, z ≤ 0,
(2.8)
−∇2 A = µ0 (∇ × M)y , z ≥ 0,
where M is a magnetic source term. We will below look at the special cases
of thin rods magnetized in x and z-direction.
For some particular choices of M and λ the vector potential and magnetic field can be expressed by elementary functions. E.g., when λ → 0
and for bar magnet, the solutions is trivially obtained from the free space
solutions by putting a mirror magnet inside the superconductor [12]. However, we need solutions for nonzero λ and express the it by their Fourier
components in x-direction.
We consider two particular choices of source magnetization
ẑ M z δ(z − z ′ )δ(x),
(2.9)
x̂ M x δ(z − z ′ )δ(x),
(2.10)
where M x and M y will be used to construct the charged Bloch wall in the
next section. The respective solutions of Eq. (2.8) for M z and M z are Az
and Ax , respectively. As calculated in appendix A, these are
ik
Azk (z) = −µ0 M z
τ + |k|

τ z−|k|z ′

 e
′
e−|k|(z+z ) + (1 +
×

 e−|k|(z+z′ ) + (1 +
τ
|k| )
τ
|k| )
−|k|z ′
e
sinh(|k|z)
e−|k|z sinh(|k|z ′ )
, z < 0,
, z < z′,
, z > z′,
(2.11)
, z < 0,
, z < z ′,
, z > z ′.
(2.12)
and
|k|
Axk (z) = −µ0 M x
τ + |k|

τ z−|k|z ′

 e
′
e−|k|(z+z ) + (1 +
×

 e−|k|(z+z′ ) − (1 +
τ
|k| )
τ
|k| )
−|k|z ′
e
sinh(|k|z)
−|k|z
e
cosh(|k|z ′ )
√
where τ = λ−2 + k 2 . The full solution for Ak is found from Eqs. (2.11)
and (2.12) by superposition of solutions at various heights z ′ and distances
x′ .
11
Chapter 2. Domain wall and vortex matter
FGF
z
Block wall
+M
M
W
−M
F
F
111111111111111111111111111
000000000000000000000000000
000000000000000000000000000
111111111111111111111111111
Superconductor
vortex
000000000000000000000000000
111111111111111111111111111
000000000000000000000000000
111111111111111111111111111
h
a
x
Figure 2.5: Side view of Fig. 2.1, a charged Block wall above a superconductor. Wall width is 2W , thickness h, and gap to superconductor is a.
2.6
Modeling a charged Bloch wall
Let us now take the piecewise solutions from Sec. 2.5 and by superposition
find the full vector potential A = ŷA(x, y) for a charged Bloch wall above
a superconductor. The setup is as sketched in Fig. 2.1 and 2.5. The film
is magnetized in-plane and is split in two domains. Between the domains
there is a Block wall of finite thickness, 2W , where magnetization is in zdirection. As a simplification the Bloch wall is treated as bar magnet, while
in reality the magnetization flips continuously. Furthermore, it is assumed
that magnetization magnitude is everywhere constantly equals Ms . Let us
label
M ⊥ = Ms ,
M k = Ms sin(ϕ),
(2.13)
where ϕ is the projection angle. The angle ϕ gives the strength of the magnetic charges, and ϕ = 0 denotes the common model of an uncharged Bloch
wall. M ⊥ is nonzero for |x| < W and M k is nonzero for |x| > W . The
magnetization is anti-parallel for the two domains and we chose the sign
of M k to be −x/|x|. This choice assures that the z- and x-magnetizations
affect a single vortex with forces working in opposite directions. The magnetization is only nonzero within the film, a < z < a + h, where a is
superconductor-film gap and h is film thickness.
In order to find the vector potential we start with Eqs. (2.12) and (2.11)
for thin rods, and integrate over z ′ and x′ . Integration over z ′ give the
12
2.6 Modeling a charged Bloch wall
film
4
2
2
2
−2
−4
z
4
0
0
−2
0
x
2
0
−2
−4
4
−2
0
x
2
−2
−4
4
4
4
2
2
2
0
−2
−4
z
4
z
z
wall+film
4
z
z
wall
0
−2
0
x
2
0
x
2
4
−2
0
x
2
4
0
−2
−4
4
−2
−2
0
x
2
4
−2
−4
Figure 2.6: Magnetic field lines of a Bloch wall in free space (top) and above
half-space superconductor (bottom). The left is the conventional dipole
like field, Eq. (2.20). The middle comes from the in-plane magnetized film,
Eq. (2.21). The rightmost figure shows a superposition of the two. The
domain wall is for |x| < 1/2 and 1 < z < 2 and the superconductor is for
z < 0. All lengths are in units of λ.
following integrals
I1 =
Z
a+h
′
dz ′ e−|k|z ,
(2.14)
dz ′ sinh(|k|z ′ ),
(2.15)
dz ′ cosh(|k|z ′ ).
(2.16)
a
I2 =
Z
a+h
a
I3 =
Z
a+h
a
Correspondingly, the x′ integration
is over the Fourier components with the
R ′
substitution exp(ikx) → dx exp(ik(x − x′ ). These are
Z
W
−W
′
dx′ e−ikx =
2
sin(kW ),
k
(2.17)
13
Chapter 2. Domain wall and vortex matter
for M ⊥ and
Z
(
−W
−∞
−
Z
∞
W
′
)dx′ e−ikx =
−2
cos(kW ),
ik
(2.18)
for M k .
Thus the magnetic vector potential for a finite width Block wall above
a London superconductor is
Z
h
i
k
A(x, z) = dk A⊥
(z)
+
A
(z)
eikx
(2.19)
k
k
where
and
⊥ 2i sin(kW )
A⊥
I1
k (z) = − µ0 M
τ + |k|

τz

,z < 0
.
 e
τ
−|k|z
e
+ (1 + |k| ) sinh(|k|z) , z < a
×

τ
−|k|z
 e
+ II21 (1 + |k|
) e−|k|z
,z > a + h
k 2i cos(kW )
k
Ak (z) =µ0 M k
I1
|k| τ + |k|

τz

,z < 0
 e
τ
−|k|z
e
+ (1 + |k| ) sinh(|k|z) , z < a
×

 e−|k|z − I3 (1 + τ ) e−|k|z
,z > a+ h
I1
|k|
(2.20)
(2.21)
Note that Eq. (2.19) requires a choice of in-plane projection angle, ϕ, which
determines M k .
Fig. 2.6 plots the contour lines of Eq. (2.19) representing magnetic field
lines. The figure clearly shows many important features of the model. First,
the perpendicular magnetization creates a field which looks like a dipole
field. Second, the in-plane contribution has monopole-like features. Third,
the field lines only penetrates at depth λ in the superconductor. Fourth,
the set of field lines representing the full system is complicated. The reason
for this is that most sizes are of the same order of magnitude. Thus one
cannot apply scaling arguments to see which effect is dominant, but one
must rather perform the required calculations.
2.7
Bloch wall-vortex interaction force
Now I will discuss the interaction between the charged Bloch wall and a
superconducting vortex. Particular focus is on sign of interaction and how
14
0.15
0.15
0.1
0.1
0.05
0.05
F||/φ0Mx
F⊥/φ0Mz
2.7 Bloch wall-vortex interaction force
0
-0.05
0
-0.05
2W/λ=10
2W/λ= 5
2W/λ= 1
-0.1
-0.15
-20
2W/λ=10
2W/λ= 5
2W/λ= 1
-15
-10
-5
0
x/λ
5
10
-0.1
15
20
-0.15
-20
-15
-10
-5
0
x/λ
5
10
15
20
Figure 2.7: The Bloch wall-vortex interaction force as a function of vortex
position x, for various wall widths, 2W . Left: F ⊥ is the force from the
wall itself, Eq. (2.24). Right: F k comes from the in-plane magnetized film,
Eq. (2.25). The lengths are in units of λ and the forces in units of Ms φ0 .
For typical ferrite garnets Ms ∼ 100 kA/m, so that Ms φ0 ∼ 2 × 10−10 N.
the force scales with the Bloch wall width. Interactions between a magnet
and vortex has also been considered in Refs. [12, 13, 18, 19, 20].
Our system consists of four objects: the magnetic film, the magnetic
wall, the superconductor, and, finally, the vortex. In this case there are
actually four forces acting on the vortex: from the perpendicular wall,
from the in-plane film, and from the their respectively induced Meissner
currents. The calculations briefly outlined below show that the latter are
exactly equal to the direct contributions. We will hence just add them and
label F ⊥ as the force from the perpendicular magnetization and its induced
Meissner current, and similarly for F k , from the in-plane magnetization
and its induced Meissner current. Thus the sum of forces on a vortex in
x-direction is F ⊥ (x) + F k (x).
The direct force on the vortex from a magnet is most easily found from
the counter force on the magnet from the vortex. This is given by
F direct = −
∂U
,
∂x
(2.22)
R
where U is the free energy interaction term U (x) = dV Ms ·Bv [21]. Here
Ms is magnetization and Bv is the stray field from the vortex outside the
superconductor, calculated in Sec. 2.4.
The force on the vortex from the superconductor is, on the other hand,
most easily obtained through the superconducting Lorentz force, f = φ0 j×ẑ,
where j is Meissner current density. For thin rods, Eqs. (2.11) and (2.12),
R0
we find that F direct = − −∞ dzf (z), i.e., the two force contributions on the
vortex are exactly equal, in both magnitude and direction, as also found in
15
Chapter 2. Domain wall and vortex matter
Ref.[18].
Using the full vector potentials, Eqs. (2.20) and (2.21), we find the final
expression for the forces in x-direction on a vortex from a charged Bloch
wall,
Z
h
i
k
wv
⊥
F (x) = dk Fk + Fk eikx .
(2.23)
The Fourier components are
Fk⊥ = −4i
k
Fk = 4i
φ0 M ⊥ 1 − e−|k|h
e−|k|a sin W k ,
λ2 |k|τ (τ + |k|)
φ0 M k 1 − e−|k|h −|k|a
e
cos W k ,
λ2 kτ (τ + |k|)
(2.24)
(2.25)
√
where τ = λ−2 + k 2 , M ⊥ = Ms and M k = Ms sin(ϕ).
In the interesting situation where F ⊥ repels and F k attracts it is important to know how their respective magnitudes change with external parameters. Both expressions have the same scaling with regards to a and
h. The largest variation is the scaling with respect to W . Fig. 2.7 shows
that their scaling behaviors are opposite: F ⊥ increases with increasing wall
width while F k shrinks. Consequently, the wall width matters with regards
to the relative strengths. Fig. 2.7 also shows the scaling behaviors with
respect to distance x. For a vortex far from the wall F k (x) is strongest.
This is because of its monopole like origin which is stronger that the dipole
like field from F ⊥ (x) when x is large.
For the experimental numbers inserted in paper 1 it turned out that F ⊥
was strongest just below the domain wall, while F k dominated in a wide
region at both sides. Hence, it could explain why the domain wall of the
experiment could appear attractive when F ⊥ was repulsive.
2.8
Vortex matter response
How vortex matter responds to an external perturbation depends to high
extend on the pinning properties. I will here consider the limit of zero pinning. The main question is then how strongly the vortices interact. Strongly
interacting vortex matter is ’stiff’ and will resist the external perturbation.
Actually, extremely strongly interacting vortex matter will not change at all
and do, in that respect, appear similar to strongly pinned vortices. Weakly
interacting vortex matter will, on the other hand, be more adjustable and
rearrange willingly.
16
2.8 Vortex matter response
Let us now apply an external force, Fext , to the vortex matter. The
vortices rearrange, and reach a equilibrium when all forces balance,
X
Fext (ri ) =
Fvv (ri − rj ),
(2.26)
j6=i
for vortex positions ri . From this expression we are interested in how the
vortex matter rearranges, i.e., the vortex positions. An rough estimate is
achieved by treating the vortex matter as a continuum
Z
ext
F (r) = d2 r′ Fvv (r − r′ ) δN (r′ ),
(2.27)
R
where δN is the excess vortex density. It satisfies d2 r′ δN (r′ ) = 0.
Eq. (2.27) is easily expressed in Fourier space using the convolution
theorem. In our case, F ext δN are homogeneous in y-direction and this
leaves just a delta function, δ(ky ). Thus
k
F⊥ + F
δNk = k vv k
Fk
(2.28)
where k = kx .
Using Eq. (2.7) for vortex-vortex interaction and Eqs. (2.24) and (2.25)
for the domain wall-vortex interaction, the final expression for the excess
vortex density reads as
δNk = 4
µ0 Ms 1 − e−|k|h τ e−|k|a sin(ϕ) cos(W |k|) − sin(W |k|)
φ0
|k|
|k| + τ
|k|L + 1/(λ2 τ (|k| + τ ))
(2.29)
with vortex length L, film thickness h, superconductor-film gap a, wall
halfwidht W , and in-plane projection angle ϕ. All these parameters are
defined in Fig. 2.5.
Let us discuss the vortex length, L. There are two important things to
note about this quantity. First, Eq. (2.29) depends only weakly on London
penetration depth, λ, but the validity of the expression depends on λ in
a crucial way: if typical vortex nearest neighbor distance is larger than λ
the mutual vortex interaction is only through the surface term and the true
interaction gets independent of L even for a bulk superconductor. Second,
vortex lines may bend [6] so that the vortex matter lattice stays unperturbed in the bulk. These two arguments means that we cannot treat L as
the true vortex length, but merely as a measure of mutual vortex interaction
strength. In paper 1, L was treated as a fitting parameter which turned out
to be of the same order of magnitude as the true superconductor thickness.
17
Chapter 2. Domain wall and vortex matter
0.01
2W/λ=0
2W/λ=1
2W/λ=2
0.04
L/λ=5
L/λ=10
L/λ=100
φ0 δN/µ0M
φ0 δN/µ0Mx
0.005
0
0.02
0
-0.005
-0.02
-0.01
-30 -25 -20 -15 -10 -5
0
x/λ
5
10 15 20 25 30
-30 -25 -20 -15 -10 -5
0
x/λ
5
10 15 20 25 30
Figure 2.8: The excess vortex density δN , Eq. (2.29), as function of distances x to domain wall center. Left: for various W ; a = h = λ, L = 100λ,
and sin(ϕ) = 0.34. Right: for various L; a = h = W = λ, and sin(ϕ) = 0.34.
The interpretation as mutual vortex interaction strength is clearly seen in
Fig. 2.8. When L is short, the vortex matter rearranges willingly, while a
long L makes it stiff and less able to change.
The excess vortex density, Fig. 2.8, visualizes clearly the strong dependency on Bloch wall width, as also discussed in connection with forces in
Sec. 2.7. The plot tells that the attraction of vortices observed in Fig. 2.3
can only appear for relatively narrow domain walls. For wide domain walls
the conventional perpendicular contribution is dominant and the wall will
be repulsive. The domain wall width is further discussed in Sec. 2.9.
The mathematical treatment of this section is on the equilibrium of vortex matter influenced by an external force in absence of pinning. However,
a true vortex distribution is not in equilibrium, but rather a metastable
state. The reason is the pinning which provides a threshold for how weak
forces that can still perturb vortex matter. Eq. (2.29) is thus not reliable
for large distances, x, where forces are weak. Consequently, the experimental excess vortex density of paper 1 was only compared with Eq. (2.29) in
a narrow region just below the wall and, in fact, for large x the fit is no
longer good. As seen in Fig. 2.8, one must go to very large x before the
curves saturate. Hence, with no pinning the domain wall would perturb
vortex matter in a far wider region than what was observed experimentally.
A last thing to note about δN is its relation to the background vortex
density N0 . In the mathematical treatment these were unrelated while in
reality they are related. The reason is that we only consider rearrangement
of vortex matter, not creation of new vortices. In other words: Eq. (2.26)
only works when there is a supply of vortices to rearrange. Hence, N0 + δN
must always be positive or zero, it cannot be negative.
18
Intensity
2.9 Discussion of Bloch wall width
-4
-3
-2
-1
0
1
2
3
4
x [µm]
Figure 2.9: Image intensity across the Bloch wall of Fig. 2.3. The domain
wall width is of order 1 µm.
2.9
Discussion of Bloch wall width
The experiment discussed in paper 1 uses a Bloch wall to manipulate vortex
matter. However, one question was not fully discussed in the paper, namely,
what is the true Bloch wall width? The width is highly important since
it is the parameter that determines whether the model of a charged Bloch
wall can explain the experiment or not. As discussed in Secs. 2.7 and 2.8,
only narrow walls can actually attract vortices in a situation like Fig. 2.3.
When looking at the image of Fig. 2.3 the domain wall appears to have
width 2 µm. However, the image is deceptive about the true size of the wall
due to the image processing needed to expose single vortices. Based on raw
data, Fig 2.9 shows the image profile across the wall. We see that the wall
has nice continues shape and the width is not more that 1 µm. Actually,
the dominant part parts of the wall fits well with the 0.6 µm used for the
experimental numbers in paper 1.
Theoretical estimates for the Bloch wall’s width has been given by
L. E. Helseth in Ref. [12]. That work discusses how the width is influenced by the presence of the superconductor. The conclusion is that the
effect is rather weak, and the width changes of order 20% compared to a
Bloch wall in free space. Below I repeat the argument of Ref. [12]. The values for the required exchanged energy constant and anisotropy constant are
also from the same reference, except from the sign of anisotropy constant,
which is opposite.
Now let us use energy considerations to estimate the width of the wall.
We will bring in three surface energy terms [11]: the uniaxial anisotropy
energy σu , the exchange energy σex , and the magneto-static energy σm . All
of these energies depend on wall width w = 2W and the minimum of energy
19
Chapter 2. Domain wall and vortex matter
3
σ [nJ/m2]
2.5
2
Ms= 100 kA/m
Ms= 75 kA/m
Ms= 50 kA/m
1.5
1
0.5
0
0.1
0.2
0.3
0.4
0.5 0.6
w [µm]
0.7
0.8
0.9
1
Figure 2.10: The surface energy σ of the Bloch wall as a function of wall
width. The minima of the graphs give the equilibrium wall width. Reduced
magnetization gives wider walls and also less pronounced minima, which
means that the theoretical estimate becomes less accurate. The parameters
are realistic for the experimental situation, Aex ∼ 2 × 10−11 J/m, h =
0.8 µm, and Ku ∼ 103 J/m3 . The magnetization of the reported film is at
superconducting temperatures about 50 kA/m which gives w ≈ 0.6 µm.
with respect w determines equilibrium wall width.
The exchange energy is the energy cost of neighboring atomic spins
having slightly different angles. The energy is calculated in the Heisenberg
model and gives that the surface energy is inversely proportional to the
width,
1
(2.30)
σex = π 2 Aex ,
w
where Aex is the exchange energy constant. The exchange energy tries to
make the wall wider. The exchange energy constant used here is Aex ∼
2 × 10−11 J/m.
The uniaxial anisotropy energy of a Bloch wall tells the energy of switching away from the in-plane direction. A simple estimate gives an anisotropy
surface energy proportional to the wall width
σu =
1
wKu .
2
(2.31)
The energy is characterized by the uniaxial anisotropy constant which for
the reported film is Ku ∼ −103 J/m3 . The anisotropy of corresponding
in-plane magnetized films is also measured in Refs. [22, 23], but without
determining the uniaxial anisotropy constant. The effect of the anisotropy
energy depends on the sign of the constant. The preferred magnetization
direction for the material which the film is made of is out-of-plane. Hence,
20
2.9 Discussion of Bloch wall width
the anisotropy energy is negative and anisotropy tends to make the wall
wider.
The magnetostatic surface energy can be estimated by modeling the
wall as a magnetized elliptic cylinder
σm =
w2
1
µ0
M 2,
2 w+h s
(2.32)
where h is film thickness. This model was originally developed to find the
transition between Bloch walls and Néel walls depending on film thickness.
For the suggested charged Bloch wall, this expression should actually be
modified by taking into account also the magnetostatic energy of the magnetic charges.
The sum of the three surface energy terms σ = σm +σu +σex is plotted in
Fig. 2.10 for numbers relevant to the experiment. The curves shows a minimum for a specific wall width and that minimum shifts towards narrower
walls with increasing magnetization, Ms . The relatively weakly magnetized film of the experiment, Ms ≈ 50 kA/m, gives a minimum energy at
w ≈ 0.6 µm, a value that actually fits well with the experimental profile of
Fig. 2.9.
21
Chapter 2. Domain wall and vortex matter
22
Chapter 3
Thin film flux dynamics
3.1
Background
This chapter is connected to papers 2 and 3 and the topic is flux penetration
in type-II superconducting thin films. The length scales are much longer
than what was considered in chapter 2, so that the discrete vortex nature
is ignored and the magnetic field is treated as continuum. The samples
that are studied are all thin films where the applied field is transverse to
the films. The flux dynamics simulations follow the formalism developed
by E. H. Brandt [24, 25, 26, 27, 28, 29, 30, 31, 32, 33]. Paper 2 uses
this formalism to explore carefully how an indentation of the edge of a
strip affects flux penetration. Paper 3 extends the formalism to multiplyconnected samples and discusses samples with holes.
There are four main motivations for this chapter. First, to show the
importance of true magnetic dynamics, which gives results other that what
is expected from the conventional Bean model [34]. Second, to solve flux
penetration on films rather than bulk samples. Most experiments relevant
for superconducting devices are actually on films, so there is truly a need
for thin film results. Third, to find the importance of sample shape and
patterning, and how the magnetic properties are altered by e.g., the presence of non-conducting holes. Fourth, to see the interplay between the
above three points, which give highly interesting and often surprising dynamical properties. One examples is the ’lightning up’ of holes deep inside
the sample at small fields [35], seen in Figs. 3.3 and 3.4. The magnetic flux
appearing in the hole is caused by the non-local electrodynamics of thin
films, and together with creep dynamics it means that the flux does not
stay in the hole, but also creeps to the nearby region.
23
Chapter 3. Thin film flux dynamics
Ha
Ha
B=0
B=0
Figure 3.1: Sketch of magnetic field (left) and current (right) on a type-II
superconducting thin rectangular film. The interior of the sample is fluxfree, but not current-free. Note that the magnetic field is weak near the
corners where the current turns 90◦ .
3.2
Thin film flux dynamics
Magnetic field always enters a superconductor from the edges. Thus the
penetration follows tightly the sample shape and in this way it is possible to
guess approximate flux distributions. E.g., the Bean model [36, 37, 34] is an
invaluable tool to find approximate flux distributions of type-II superconductors. However, the original Bean model does not allow true dynamics
nor thin film geometry. There exists a thin film generalization of the Bean
model for strips [38, 25], but for general geometries and creep dynamics
the magnetic field distributions must be found by numerical simulations.
The simulation formalism applied in this work is carefully explained in
papers 2 and 3, and will not be repeated here. This section gives only
a short summary of the most significant quantities and equations. The
formalisms follows mainly Ref. [33], in the λ → 0 limit. The topic of
thin film flux penetration under the creep has been thoroughly explored by
E. H. Brandt [24, 25, 26, 27, 28, 29, 30, 31, 32, 33]. An improved performance version of this formalisms has been developed in Refs. [39, 40, 41].
The current distribution of a thin film superconductor is in reality very
complicated. The main current flow is in two layers of size λ, which itself
may be of the same order of magnitude as the sample thickness. A major
simplification is to consider the sheet current instead of the true current
density. The sheet current is
J(r) =
Z
d/2
dz j(r, z),
(3.1)
−d/2
where j = (jx , jy ) is current density, d is sample thickness, and r = (x, y)
are the in-plane coordinates.
The main quantity of the simulation is the local magnetization g(r)
which is defined by the relation
J = ∇ × ẑg.
24
(3.2)
3.2 Thin film flux dynamics
The main idea behind Brandt’s method for flux penetration simulations
is to invert the Biot-Savart law and get an equation for the time evolution
of g expressed by Ḃz . We write
(3.3)
ġ = Q̂−1 Ḃz − Ḣa ,
where Q̂−1 denotes the inversion of Biot-Savart law. The inversion is nonlocal and depends on sample shape. In the terminology of Brandt Q̂ is the
integral kernel of Biot-Savart law, or just kernel for short, and Q̂−1 is the
inverse kernel. All figures of this section are made with the same method
as described in Ref. [33] and paper 2 and 3. The key idea is to discretize
Biot-Savart law on a finite grid with grid points ri and weights w. Then,
the discrete kernel can be written
!
X
(3.4)
qil − qij ,
Qij = δij Ci /w +
l
where qij = 1/4π|ri − rj |3 for i 6= j and qii = 0. The function C depends
on the sample geometry
Z
dr′2
C(r) =
.
(3.5)
′ 3
outside 4π|r − r |
The Q̂−1 of Eq. (3.3) is in the discrete case just matrix inversion of Eq. (3.4).
The method works for connected thin films of any shape. With the addition
described in paper 3 it can also be used for multiply-connected samples.
The generality of the algorithm is illustrated in next section, which includes
simulation results for a large number of sample shapes, all created with the
same kernel and the same implementation. These should be compared with
the outcomes made by specially targeted kernels, for rings and disks [31],
rectangles [28], and strips [25].
Eq. (3.3) can only describe proper flux dynamics as long as Ḃz is a
known functional of g. For the described simulation the relation is in two
steps. The first step is Faraday’s law
Ḃz = −(∇ × E)z .
(3.6)
Second step is a material law, which binds E to g. Hence the thin film
flux dynamics is described with g as the only variable. The material law is
actually what characterizes the superconductor. The whole rest of the formalism is in fact valid for any thin conductor. The peculiar superconductor
dynamics is conventionally modeled by a highly nonlinear current voltage
25
Chapter 3. Thin film flux dynamics
y
a
a
FL
x
d−line
FL
Edge
Figure 3.2: Left: All samples are embedded in a square with side lengths
2a. Film thicknesses are d. Right: The sketch of a d-line from a 90◦ corner.
The direction of FL , the Lorentz force on vortices, is also shown.
curve. The flux creep regime that we are interested in is well described by
a power-law relation [42, 28, 43],
E = ρ0
j
jc
n−1
j,
(3.7)
where E is electric field, j is current density, jc is critical current density,
n is the exponent, and ρ0 is a resistivity constant. The most important
parameter here is n, where n = 1 gives ohmic conductor while n → ∞ gives
Bean’s critical state model. General flux creep is for n < ∞. The role of
the material law is discussed in papers 2 and 3 and Refs. [42] and [28].
Note that the simulations could equally well have been carried out with
Bz instead of g. The major complication about this is to enforce the boundary conditions on Bz , opposed to the simple g = 0 at the boundaries. For
sample with inner boundaries, switching to Bz in stead of g can be useful,
as described in paper 3.
3.3
Survey of sample geometries
The main point of this section is to show how important sample shape is
for the flux distribution. Shape in this context means both outer and inner
edges, i.e., holes. All samples are embedded in a square with half-width
a and of thickness d, as sketched in Fig. 3.2. The strips are modeled by
periodic boundary conditions. Furthermore, all simulations are with n = 19
and applied field is ramped with constant rate µ0 Ḣa = ρ0 Jc /ad, where n
and ρ0 come from the material law, Eq. (3.7). In this regime creep is low
but not negligible. All simulations start from completely flux free sample
and the second critical field is set to zero.
26
3.3 Survey of sample geometries
Since flux always penetrates from the edges, sample shape is crucially
important. In the same manner, non-conducting holes inside the samples
perturb flux distributions in large part of the sample, not just in the vicinity
of the defect. Figs. 3.3 and 3.4 shows flux distributions at medium and full
penetration and current stream lines at full penetration, for a selection of
samples. The samples are rectangle, square, square with a circular hole,
disk, ring, and strip with an array of holes. What is common for all of
them is that magnetic flux density is high at the edges and there is a certain
flux free region in the middle. The data is plotted to facilitate qualitative
comparison with magneto-optical images, like Ref. [10]. This means that
there are two things in particular one should notice in the figures. First, the
shape of the flux front, which is where the flux density drops to zero inside
the sample. The image intensity is set so that the flux free region is grayish,
not black, in order to use to the same scale also in images with negative flux.
For gradually increasing applied fields negative flux only appears in samples
with holes. Second, notice the shape of the d-lines at full penetration. The
d-lines are seen as dark lines in flux distributions and they coincide with
the places where current changes direction abruptly. E.g., from the square
and rectangular sample, Figs. 3.3 (a) and (b), the d-lines are 45◦ -starting
from the edges, while single holes of Figs. 3.3 (c) gives almost parabolic
shape. The array of holes in Figs. 3.4 (f) gives a very complicated set of
d-lines. For more discussion about d-lines, please see papers 2 and 3, and
book [34].
Fig. 3.5 and 3.6 contain profiles across some selected samples. The
profiles visualize better actual sizes, which are often lost in the intensity
plots of Fig. 3.3 and 3.4. Fig. 3.5 compares a disk with a ring. The disk has
current flowing in the whole sample at all times. The currents ensures that
magnetic field is shielded in the central parts of the sample. In the ring, the
current is excluded from the central parts, with the consequence that flux
shielding breaks down. Thus an inner flux front appears
Before the
R [31].
2
inner and main flux front meet, the simulations satisfies d rBz (r) = 0 for
integration from the ring center to the inner flux front. This condition is a
formulation of flux conservation and it is actually also a correctness check
for the simulation algorithm and the boundary condition implementation
of paper 3. Fig. 3.6 shows profiles for a strip with an array of holes. This
sample is chosen since guiding of magnetic flux from an array of holes might
be utilized for applications [35, 44]. Currently there are not so many results
for large arrays of holes in thin films. Exceptions are Refs. [45] and [41].
27
Chapter 3. Thin film flux dynamics
Magnetic field
Magnetic field
Current stream lines
Ha /Jc = 0.3
Ha /Jc = 0.9
Ha /Jc = 0.9
(a)
(b)
(c)
Figure 3.3: Survey of geometries: (a) rectangle, (b) square, (c) square with
a hole; cf. Refs. [28] and [46]. The rectangle (a) has proportions 1:2. The
hole in (c) has radius 0.1a at distance 0.25a from the edge.
28
3.3 Survey of sample geometries
Magnetic field
Magnetic field
Current stream lines
Ha /Jc = 0.3
Ha /Jc = 0.9
Ha /Jc = 0.9
(d)
(e)
(f)
Figure 3.4: Survey of geometries: (d) disk, (e) ring, (f) strip with array of
holes; cf. Refs. [31], [35] and [45]. The disk (d) and the ring (e) have outer
radii R=a and ring inner radius is 0.5a. The strip (f) has holes of radii
0.05a at y/a = ±0.8, ±0.6, ±0.4 and ±0.2.
29
Chapter 3. Thin film flux dynamics
Ring
1
1
0.8
0.8
0.6
0.6
Bz/µ0Jc
(a)
Bz/µ0Jc
Disk
0.4
0.2
0
0.4
0.2
0
-1
-0.5
0
-1
1
1
0.8
0.8
0.6
0.4
0.2
0.2
0
-1
-0.5
x/a
0
-1
-0.5
x/a
0
-1
-0.5
x/a
0
0.6
J/RJc
0.6
J/RJc
0.6
0.4
0
(c)
0
x/a
J/Jc
(b)
J/Jc
x/a
-0.5
0.4
0.2
0.4
0.2
0
0
-1
-0.5
x/a
0
Figure 3.5: Comparison of disk and ring. Same as Figs. 3.4 (d) and (f). Profiles of (a) Bz , (b) J, and (c) g. Applied fields are Ha /Jc = 0.1, 0.2, 0, 3, 0.4;
cf. Refs. [31, 47]. The exclusion of current in the ring means that shielding
breaks down, giving nonzero flux density (but still zero total flux) in the
ring.
30
3.3 Survey of sample geometries
Array of holes
1
1
0.8
0.8
Bz/µ0Jc
(a)
Bz/µ0Jc
Plain strip
0.6
0.4
0.2
0.6
0.4
0.2
0
0
-1
0
1
-1
0
x/a
1.2
1
1
0.8
0.8
J/Jc
J/Jc
1.2
(b)
0.6
0.4
0.2
0.2
0
-1
-0.8 -0.6 -0.4 -0.2
0
x/a
0.2
0.4
0.6
0.8
1
0.8
0.8
0.6
0.6
g/aJc
g/aJc
0.6
0.4
0
(c)
1
x/a
0.4
0.2
-1
-0.8 -0.6 -0.4 -0.2
0
x/a
0.2
0.4
0.6
0.8
1
-1
-0.8 -0.6 -0.4 -0.2
0
x/a
0.2
0.4
0.6
0.8
1
0.4
0.2
0
0
-1
-0.8 -0.6 -0.4 -0.2
0
x/a
0.2
0.4
0.6
0.8
1
Figure 3.6: Illustration of how an array of holes affects flux penetration of
a strip. Profiles of (a) Bx , (b) J, and (c) g of a strip with eight holes, same
sample as Fig. 3.4 (f). Left: away from the holes. Right: through hole
centers; Ha /Jc =0.3, 0.6 and 0.9.
31
Chapter 3. Thin film flux dynamics
3.4
Electric field and polarization
In this section I will discuss briefly the electric field distributions and electrical polarizations of samples with holes. The presentation refers to thesis [46], which is on this subject in connection with magneto-optical imaging.
The electric field distribution is important since it tells where and how
fast flux moves. The flux motion is especially complicated and interesting
near defects, holes and edges. Fig. 3.7 shows electric field distributions on
a square with a hole and a strip with an array of holes. The main electric
field distribution is as expected [28], i.e., E is high at the edges and low in
the corners. The hole and hole arrays create a channel of greatly enhanced
E, as also discussed in paper 3. The high E is accompanied by an enhanced
flux transport.
What is not obvious is that the flux penetration polarizes the sample,
also leading to a nonzero charge density, ρin = ǫ0 ∇ · E, induced by the moving
[48]. The simulations satisfy the necessary charge conservation
R 2vortices
d ρin = 0. The excess charge density given by ∇ · D, where D is electric
displacement, is of course everywhere zero.
Fig. 3.7 shows ρin , and the distribution has many interesting features.
The values of ρin are highest at the boundaries, near d-lines, and in connection with the hole. The signs of ρin are always opposite on each side of
d-lines. Moreover, the d-lines divide the sample in segments, where integrated ρin is zero within each segment. Holes and defects create d-lines so
ρin -distribution is particularly complicated there.
3.5
Inversion of Biot-Savart law in Fourier
space
The essential point in the flux penetration simulation formalism of E. H.
Brandt, is inversion of the thin-film Biot-Savart law [33]. For discrete formulation of the law, the inversion was carried out as a matrix inversion. In
this section I will present a formulation of the Biot-Savart law in Fourier
space following Refs. [49, 10]. This formulation is appealing, since it is fast
and has low memory demands on a computer, contrary to the matrix formulation. On the other hand, it cannot be utilized directly in flux penetration
simulations since it assumes an infinite superconductor and consequently it
does not deal with boundaries in the correct way. However, in next section
I will sketch how the boundary condition can be handled. The method is
also useful in itself to find currents from a magnetic field distribution known
32
3.5 Inversion of Biot-Savart law in Fourier space
Square with hole
Strip with hole array
Ha /Jc = 0.9
Ha /Jc = 0.6
E
ρin
Figure 3.7: Electric field, E, and induced charge density ρin for a square
with a hole (left) and strip with eight holes (right). The samples are the
same as Fig. 3.3 (c) and 3.4 (f) and parameters are in Sec. 3.3. cf. Ref. [46].
33
Chapter 3. Thin film flux dynamics
e.g. by magneto-optical imaging.
Now let us assume an infinite thin film. The assumed infinite space is
necessary since the translation invariance makes the relation local in Fourier
space. For finite samples, e.g., rectangles [28], the current-field relation is
non-local in Fourier spaces and has the same complexity as the real space
expression.
In a thin film where current can only flow in-plane, the Biot-Savart law
reads as
Z
Z
r − r′
µ0
2 ′
′
′
′ ′
d r
dz ∇ g(r , z ) ·
Bz (r, z) =
, (3.8)
4π
((r − r′ )2 + (z − z ′2 )2 )3/2
where g, defined in Eq. (3.2), is the local magnetization and r = (x, y).
Since the Fourier transform of the first term is ikg̃k and the two terms are
dotted, only the Fourier transform of the last term in k̂ direction needs to
be found. It is
Z
k̂ · r
e−ik·r = 2πie−kz .
(3.9)
d2 r 2
2
3/2
(r + z )
Convolution of Eq. (3.8) gives the Biot-Savart law for thin films in Fourier
space
Z
′
µ0
dz ′ g̃(k, z ′ ) e−k|z−z | .
(3.10)
B̃z (k, z) =
2
For thin films we ignore the z ′ dependency over the thickness d so that
B̃z (k, z) =
µ0
d k g̃(k) e−k|z| .
2
(3.11)
The ignored z ′ -dependency does not imply that g̃(k, z ′ ) is independent of
z ′ , but it merely defines the effective local magnetization g̃(k).
Inversion of Biot-Savart law means to determine g from Bz in the plane
z = 0,
2 1
g̃(k) =
B̃z (k, 0).
(3.12)
µ0 d k
The inversion experience problems at k = 0. This means that the magnetic
field does not depend on the background level of g, so that if a constant
is added to g, Bz is left unchanged. This is physically correct since the
currents density comes from the derivative of g and the absolute level of g
is insignificant.
Numerical implementations of the Fourier transforms are likely to use
the discrete Fourier transform, which is periodic in real space. The formulas
of this section works well also in this case as long as one is careful to
rearrange the Brillouin zones in the correct manner.
34
3.6 An alternative simulation formalism
Figure 3.8: Flux penetration on a square with a slit, run with the alternative
simulation formalism on a 256×256 grid. The slit details are meant to
reassemble the slit in Ref. [50]. Applied fields are Ha /Jc = 0.2 and 0.5;
n = 9.
3.6
An alternative simulation formalism
This section sketches how to base flux penetration simulations on the analytical inversion of Biot-Savart law in Fourier space, Eq. (3.12), presented in
Sec. 3.5. This inversion is preferable to the discrete matrix inversion used
in paper 2 and 3 due to of speed, scalability and memory consumption.
For N grid points the matrix inversion requires a N × N matrix, while the
Fourier space inversion only requires matrices of dimension N .
Schematically, the equation governing macroscopic thin film flux dynamics can be written as [33]
(3.13)
ġ = Q̂−1 Ḃz − µ0 Ḣa
where g is the local magnetization defined in Eq. (3.2) and Ha is the applied
field. The Ḃz is a known functional of g by a material law inside the sample.
The operator Q̂−1 is an inversion of the Biot-Savart law. However, the
inversion, Eq. (3.12), cannot be utilized here directly since it assumes an
infinite space, and as clearly visualized in Figs. 3.3 and 3.4: boundaries are
of most crucial importance.
The inversion Eq. (3.12) does yield correct ġ when Ḃz is given in the
whole plane z = 0. However, from the material law Ḃz is just known inside
the sample. Only by reconstructing Ḃz outside the sample one can apply
Eq. (3.12) to find ġ. But this is indeed the same problem as was solved
35
Chapter 3. Thin film flux dynamics
in paper 3 for non-conducting holes. In order to apply the formulas from
paper 3 one must hence provide additional free space on all sides of the
sample for the reconstructed Ḃz . For an exact inversion by Eq. (3.12) this
area must be infinite. However, for squares and disks quite small areas
provide good results. For elongated samples, like rectangles and strips, the
magnetic field drops off slowly, as 1/r (at least for a while), which means
that larger areas must be supplied outside the sample, and the method does
not work so well.
A simulation result example is shown in Fig. 3.8. The samples is a square
with a slit and the slit has very fine details. The slit details are meant to
reassemble the slit in Ref. [50] which has been cut out with a laser. The
non-smooth edges gives a rib-like shape of the flux distribution around the
slit. The pattern is somewhat similar to the pattern from concave corners,
e.g., crosses [30]. The resolution of Fig. 3.8 is currently not manageable on
a desktop computer with the matrix inversion method utilized for the other
simulation results of this chapter.
36
Chapter 4
Landau-Zener transitions
in superconducting qubits
4.1
Background
This chapter is connected to paper 4 and the topic is Landau-Zener transitions of superconducting qubits in presence of noise. A successful qubit
must remain in a coherent state for a large number of operations. In addition, it must be possible to prepare it in exact initial states and read out results. Hence, there are conflicting demands: the need for coherence suggest
the system should be extremely weakly coupled to the surroundings while
the need for manipulation means that it cannot be isolated completely. In
other words, contact with environment is inevitable and a central practical
and theoretical problem is how to describe and eventually reduce noise influence on the quantum state. Good models for qubits should hence both
take into account environment coupling as well as the non-trivial dynamics
of qubit operations.
Paper 4 considers an interesting single qubit operation, the LandauZener transition [51, 52, 53]. A Landau-Zener transition can prepare a
qubit in a desired state by dynamically interchanging the two qubit energy
levels. Simply by changing the driving rate one can put the qubit in ground
state, excited state, or superposition of the two. Paper 4 discusses two
modifications of the traditional Landau-Zener problem. First, the energy
level splitting is driven as a general power of time,
∆(t) ∝ |t|ν sign(t),
(4.1)
for an arbitrary exponent ν. Conventional Landau-Zener transitions cor37
Chapter 4. Landau-Zener transitions in superconducting qubits
Energy
+E
g
−E
time
Figure 4.1: Time-dependent energy eigenvalues with an avoided levelcrossing of energy gap g.
responds to ν = 1. Second, it considers the transition due to noise, not
transition caused by an avoided level crossing as in the traditional LandauZener dynamics. The noise is in form of random telegraph noise, a noise
model which is both physically relevant for qubits and capable of giving
analytical results for some selected problems, as discussed in detail in paper 4.
The rest of this chapter focus on the conventional Landau-Zener problem
and derives the exact solution of this, following C. Zener [51]. Also it goes
through some of most relevant noise sources for superconducting qubit and
nanoscale devices. The noise problem is also relevant for e.g. single vortex
devices. Appendix B contains an adiabatic basis formulation for the system
of a qubit coupled to telegraph noise. This basis enables other kinds of
approximations than the conventional diabatic formulation and can also be
utilized for stable numerical solutions.
4.2
The Landau-Zener Hamiltonian
The Landau-Zener transition is a fundamental problem in non-stationary
quantum mechanics. The idea behind it is as follows. Let us initially
prepare a system in its ground state and then change the Hamiltonian
so that the energy levels switch; the ground state becomes the excited
state and visa versa. In nature, energy levels try to avoid direct crossing
which means there is a finite minimum gap like sketched in Fig. 4.1. For a
time, when the energy levels are close, the tunneling between the levels is
significant, and the outcome of the transition depends strongly on the rate
at which the system is driven. Fast rate drives the system to the excited
state (diabatic dynamics) while slow rate means it ends in the ground state
(adiabatic dynamics). For finite driving rate the end state is a superposition
of the two states. Such transitions are called Landau-Zener transitions.
38
4.3 Zener’s solution
Many variants of the problem exists. The simplest variant includes just
two levels and diagonal energy splitting that increases linearly with time.
This problem was solved in 1932 by C. Zener [51], L. D. Landau [52],
and E. C. G. Stueckelberg [53]. At that time, the system in mind was
slow molecular collisions, which in fact is dynamics of chemistry. Zener
solved the problem exactly, expressed by special functions. Landau did a
semiclassical approximation, but was able to extract the exact transition
probability. His solution relied on matching of functions through analytical
continuations, a method which requires careful treatment. I will not go
through Landau’s approach here, but refer to Refs. [54],[55], and [56].
Consider the qubit Hamiltonian
1
1
1
∆
g
,
(4.2)
HLZ (t) = ∆σz + gσx =
g −∆
2
2
2
where σ = (σx , σz ) are Pauli matrices and g is the avoided level-crossing
gap. Dynamics is given by the corresponding Schrödinger equation
1
∆
g
φ̇1
φ1
,
(4.3)
i~
=
φ2
g −∆
2
φ̇2
with state vector (φ1 , φ2 ) formulated in the diabatic basis.
The Landau-Zener Hamiltonian is Eq. (4.2) with a diagonal splitting
driven linearly with time
∆(t) = at,
(4.4)
where a is driving rate.
The outcome of a Landau-Zener transition depends on the ratio g 2 /a.
Fig. 4.2 plots the staying probability PLZ (t) = |φ1 (t)|2 as a function of time.
We see that fast driving, g 2 /~a ≪ 1, makes the system stay in the same
diabatic state, while slow driving, g 2 /~a ≫ 1, makes the system change
diabatic state.
The explicit time-dependency of the Hamiltonian also manifests itself
in the energy eigenvalues of Eq. (4.2),
p
(4.5)
±2E(t) = ± ∆2 (t) + g 2 ,
which are shown in Fig. 4.1. The time-dependency of E means that the
ordinary dynamic phase factors do no give the proper time evolution.
4.3
Zener’s solution
In 1932 Zener [51] found the time evolution of the Hamiltonian Eq. (4.2).
The solution was expressed by non-elementary functions and I will repeat
39
Chapter 4. Landau-Zener transitions in superconducting qubits
1
PLZ
0.8
0.6
0.4
0.2
√
g/√~a = 0.1
g/√~a = 0.5
g/√~a = 1
g/ ~a = 2
0
-15
-10
-5
0
p
a/~ t
5
10
15
Figure 4.2: The Landau-Zener
diabatic staying probability as a function of
√
time, for various g/ ~a.
the derivation here. The time evolutions of the two elements of the state
vector (φ1 , φ2 ) are given by the Schrödinger equation Eq. (4.3), which on
component form is
2i~φ̇1 = +∆φ1 + gφ2 ,
2i~φ̇2 = −∆φ2 + gφ1 ,
(4.6)
where ∆ = at. Here φ1 and φ2 are complex functions of time that satisfy |φ1 |2 + |φ2 |2 = 1. Isolating the φ1 component gives one second order
equation
1
(4.7)
φ̈1 + 2 (∆2 + g 2 + 2i~a)φ1 = 0.
4~
Let us define
r
a
ξ= i t
(4.8)
~
and we get the equation
∂ 2 φ1
−
∂ξ 2
1 2 1
ξ − − n φ1 = 0,
4
2
where
n = −i
40
g2
.
4a~
(4.9)
(4.10)
4.4 Charge qubit
The solutions of Eq. (4.9) are Parabolic Cylinder Functions U (−1/2 −
n, ±ξ) = Dn (±ξ), where the ± denotes two independent solutions1 [57, 58].
The Parabolic Cylinder Functions can be expressed by Confluent Hypergeometric Functions, and the relation to the functions 1 F1 are in appendix C.2.
The asymptotic values of Dn (ξ) for large arguments and | arg z| < 3π/4 are
|Dn (±ξ) | → exp (in arg(±ξ)) ,
(4.11)
√
√
where arg( i) = π/4 and arg(− i) = −3π/4. The physical solution must
satisfy |a(t)| ≤ 1 and |a(−∞)| = 1. The above function with negative
sign satisfies the condition that |a(−∞)| ≥ |a(∞)| so that the solution of
Eq. (4.9) with correct boundary conditions is
r
g2
a
−π
φ1 (t) = e 4 4~a Di g2 − i t .
(4.12)
~
4~a
A equivalent way to write Eq. (4.12) is by D−n−1 (±iξ), which is how Zener
expresses the solution in his original paper.
The expansion Eq. (4.11) also gives the transition probability
π g2
2
.
(4.13)
PLZ = |φ1 (∞)| = exp −
2 ~a
The states φ1 and φ2 are the diabatic states, so that PLZ (∞) is the probability for a system starting in the ground state to end in the excited state.
The limit of fast driving, ~a/g 2 → ∞, the system ends in the excited state.
For the adiabatic limit, ~a/g 2 → 0, the system ends in the ground state.
4.4
Charge qubit
A qubit is a two-level system with a tunable Hamiltonian. Evidently, this
can be implemented in numerous ways, and even when restricting the discussion to superconducting qubits there are several fundamentally different
designs. Most superconducting qubits base their quantum state either on a
quantized flux line or on an isolated Cooper pair, and they are respectively
called flux and charge qubits. For an overview of superconducting qubits
and noise therein, please see Ref. [59].
Paper 4 uses the noise model of a single random telegraph noise process.
This noise source is only relevant for solid state qubits of very small spatial
dimensions and at very low temperatures. Here I will give a simplified
1 The
functions Dn (x) are called Whittaker functions in “Abromowitz and Stegun”
and Weber functions in “Whittaker and Watson”.
41
Chapter 4. Landau-Zener transitions in superconducting qubits
JJ
Vg
Φ
Cg Box
JJ
Figure 4.3: Sketch of a Josephson charge qubit. The qubit is controlled by
the gate voltage, Vg , and the trapped flux Φ. The spots in the Josephson
junctions indicate trapped charges as a noise source.
presentation of one such device which can be relevant, namely the Josephson
charge qubit [60, 61, 62, 63, 64]. A charge qubit is single Cooper pair box
with a capacitive coupling and a Josephson junction, as shown in Fig. 4.3.
In the figure the Josephson coupling is substituted with a SQUID so that
the effective Josephson energy can be tuned by varying the trapped flux.
The two states of the system are zero and one excess Cooper pair in the
box, respectively. In this basis, the Hamiltonian is
H=
1
1
Ec (1 − Ng )σz + EJ (Φ)σx ,
2
2
(4.14)
where Ec is the charging energy, Ng is the dimensionless gate charge and
Ej is the Josephson energy. The point is that both these energies can be
externally tuned. The diagonal term is tuned by the gate voltage Vg which
controls the dimensionless gate charge
Ng = Cg Vg /2e,
(4.15)
where Cg is the gate capacitance. The off-diagonal term is tuned by the
changing the external flux Φ in the SQUID,
π Φ
0
,
(4.16)
Ej (Φ) = 2Ej cos
2 Φ0
where Ej0 is the energy of the Josephson junctions and Φ0 is the flux quantum.
Hence, a Landau-Zener transition of a Josephson charge qubit means to
sweep the gate voltage. An avoided level crossing happens when Ej (Φ) 6= 0.
It must be noted that the two-level system is only an approximation and
when applying a too high gate voltage other energy levels will be relevant.
42
4.5 Noise in solids
Paper 4 considers transverse noise. In terms of Josephson charge qubits
this is noise in the Josephson energy. The characteristics of telegraph noise
in Josephson devices has been measured in several work, for charges trapped
in the isolating junction [65], or, if the SQUID is a type-II superconductor,
because of pinned vortices in the material of the SQUID [66, 67].
4.5
Noise in solids
Noise normally denotes spontaneous fluctuations with undesired consequences for a devices. Undesired consequences includes things like inaccuracy of measurement and decoherence of quantum devices. The common
way to describe noisy problems is to split the full system in two unequal
parts, called system and environment . The system is the device of interest
and the environment is everything else. The description of the two parts
is asymmetric: the environment affects the system, while the the system
affects the environment only weakly. Usually it is a goal to formulate the
problem in quantities involving the system only.
Superconducting qubits are always operating at very low temperatures.
Then the dominating noise sources are discrete fluctuations of localized
states appearing at impurity traps in the solid [68, 63, 59, 69, 70]. Quantum
noise means that the qubit and impurity are strongly coupled and form an
entangled pair. Thermal noise, on the other hand, means that the dynamics
of the impurity is mainly unaffected by the qubit and follows the thermal
fluctuations of the solid in stead.
I will now go through formalism describing classical noise in solids and
also classify the noise sources most relevant for superconducting qubits.
Please consult the book [71] for a thorough description of noise sources and
basic mathematical formalism on the topic.
The most important quantity in noise characterization is the noise autocorrelator
S(t) = hχ(t)χ(0)i,
(4.17)
where the brackets denote assemble average over all realization of noise
random process χ(t). The cosine transform of Eq. (4.17) is the noise power
spectrum
Z
∞
Ŝ(ω) =
dt S(t) cos(ωt).
(4.18)
0
Gaussian white noise
At high temperatures the dominating noise source is typically Gaussian
white noise. The Gaussian noise is typically created by large number of
43
Chapter 4. Landau-Zener transitions in superconducting qubits
fluctuating quantities and it has infinitely short correlation time, S(t) =
Aδ(t), with A a constant. The power spectrum is constant Ŝ(ω) = A.
1/f-noise
At low temperatures the frequency dependent, 1/f noise is a major source
of decoherence of qubits [59]. The 1/f noise has its name from the form of
the power spectrum, Ŝ(ω) ∼ 1/ω. The 1/f noise is universal in the sense
it is almost always present in solid state devices. It is typically created by
localized fluctuators that sit in defects distributed at random position of
the material. It must be noted that the effect of the noise does not just
depend on the power spectrum, but also on the nature of each fluctuator.
1/f noise from Gaussian fluctuators behaves different from e.g., 1/f noise
from non-Gaussian telegraph processes [72, 68].
Random telegraph noise
Telegraph noise comes from bistable fluctuators, i.e., fluctuators randomly
switching between two levels. It can be used as building block for other
kinds of noise, like Gaussian white noise or 1/f noise. If one works with
very tiny devices one can even consider only the most prominent single
fluctuator. Random telegraph noise is used as noise source in paper 4 and
it is described in book [71] and Refs. [62, 73, 68].
The probability Pk for the fluctuator to switch k times in time interval
t is given by the Poisson distribution,
Pk =
(γt)k −γt
e
k!
(4.19)
with switching rate γ.
The random telegraph noise is randomly fluctuating between two levels,
i.e., χ(t)χ(0) = ±v 2 where v is the fluctuator strength and the + and −
signs are from even and odd number of switches, respectively. Hence the
autocorrelator for random telegraph noise is
S(t) = hχ(t)χ(0)i =
=v
2
∞
X
χk (t)χk (0)Pk ,
k=0
k
−γt
k (γt)
(−1)
k=0
∞
X
k!
e
It gives a Lorentzian power spectrum
Z ∞
dt S(t) cos(ωt) = v 2
Ŝ(ω) =
0
44
(4.20)
2 −2γt
=v e
.
2γ
.
(2γ)2 + ω 2
(4.21)
4.6 Bloch notation
When the switching rate is fast, γ → ∞, the autocorrelator has infinitely
short correlation time. Thus, in this limit a single telegraph process behaves
like Gaussian noise. Furthermore, when it also satisfies v 2 ∼ γ the power
spectrum is a constant and we get Gaussian white noise.
4.6
Bloch notation
Bloch notation describes the dynamics of a quantum two level system as
precession of a spin in a magnetic field. The precessing spin vector represents the quantum state while the magnetic field represents the Hamiltonian. This notation is common for description of qubits and I will here
show the relation between the Bloch state and the density matrix. The
density matrix of a two level system is
ρ11 ρ12
φ1
∗ ∗
,
(4.22)
(φ1 φ2 ) =
ρ=
ρ21 ρ22
φ2
where φ1 and φ2 are the components of the state vector. The components
of the density matrix satisfies that ρ11 and ρ22 are real and ρ21 = ρ∗12 . From
the density matrix, the Bloch vector r = (x, y, z) is defined as
x = 2ℜρ12 ,
y = 2ℑρ12 ,
(4.23)
z = ρ11 − ρ22 .
For a pure quantum state the length of the vector is constant, r = 1, and
the vector moves on the Bloch sphere. Dynamics is given by the Bloch
equation, which is a reformulation of Heisenberg’s equation of motion
~ṙ = −r × B.
(4.24)
This equation describes precession around the ’magnetic field’ B. The magnetic field comes from the Hamiltonian written like
1
H = B · σ,
(4.25)
2
where σ = (σx , σy , σz ) is the vector of the Pauli matrices.
The Bloch formulation comes from the density matrix and hence it is
also valid when the quantum state is no longer pure, i.e., when the two level
system is in contact with environment. In that respect it is interesting to
look at the von Neumann entropy
X
pi log(pi ),
(4.26)
S=
i
45
Chapter 4. Landau-Zener transitions in superconducting qubits
where pi are the eigenvalues of the density matrix. For the two-level system,
the eigenvalues of Eq. (4.22) are
p± =
1
(1 ± r) ,
2
(4.27)
where r = |r| is the length of the Bloch vector. Then the von Neumann
entropy is
1
1
1
1
S(r) = − (1 − r) ln (1 − r) − (1 + r) ln (1 + r).
2
2
2
2
(4.28)
When the system is coupled to environment r is in general less than unity.
Maximum entropy is for r = 0 and minimum is for r = 1.
46
Appendix A
Domain wall, calculations
The London equation
Laplace transforms are convenient to solve problems with plane interfaces.
In our case the interface is between a superconductor and vacuum. The
Laplace transforms in z-direction is combined with a Fourier transform
in the xy-plane and the solutions for the various parts are put together by
requiring continuity for the function and its derivative at z = 0. The mostly
used Laplace and inverse Laplace transforms are in appendix C.1.
We use the London equation expressed by the vector potential A and
with a magnetic source term M
λ−2 A − ∇2 A = µ0 (∇ × M).
(A.1)
This form of the London equation assumes the London gauge ∇ · A = 0.
When λ → ∞ the equation is the Poisson equation and describes vacuum
in stead of a superconductor.
System: Superconductor
I will now solve the London equation for a superconductor in half space
z<0
−∇2 A + λ−2 A = 0.
(A.2)
Fourier transform in the xy-plane and Laplace transform in z-direction gives
L{Ak } =
k
sAk |0 + ∂A
∂z |0
,
s2 − τ 2
(A.3)
47
Chapter A. Domain wall, calculations
where τ =
√
λ−2 + k 2 . Inverse Laplace transform gives the relation
Ak = Ak |0 cosh(τ z) +
1 ∂Ak
|0 sinh(τ z).
τ ∂z
(A.4)
Consistency as z → −∞ gives the equation
1 ∂Ak
|0 = Ak |0 ,
τ ∂z
(A.5)
Ak (z < 0) = Ak |0 eτ z .
(A.6)
where the solution is
This result also applies vacuum when λ → ∞, so that τ → k. For a solution
in the upper half space let z → −z.
System: z-magnetized rod
The system is vacuum with a thin rod pointing in y-direction and magnetized in z-direction. The rod is at x = 0 and z = a,
M = M δ(z − a)δ(x)ẑ.
(A.7)
The curl of the magnetization is ∇ × M = −∂x M ŷ. Solution for the vector
potential is
Ak (z) = Ak |0 cosh(|k|z) +
1 ∂Ak
|0 sinh(|k|z)
|k| ∂z
ik
+ µ0 M
sinh(|k|(z − a)) θ(z − a).
|k|
(A.8)
Consistency for z → ∞ gives the relation
Ak |0 +
1 ∂Ak
ik −|k|a
|0 + µ0 M
e
= 0.
|k| ∂z
|k|
(A.9)
Put together with a the superconductor for z < 0 gives
Ak |0 = −µ0 M
ik
e−|k|a
τ + |k|
(A.10)
and
Ak (z) = Ak |0 e
48
−|k|z
k
− µ0 M i
|k|
e−|k|a sinh(|k|z), z < a
.
e−|k|z sinh(|k|a), z > a
(A.11)
System: x-magnetized rod
The system is vacuum with a thin rod pointing in y-direction and magnetized in x-direction. The rod is at x = 0 and z = a,
M = M δ(z − a) δ(x) x̂.
(A.12)
The curl of the magnetization is ∇ × M = ∂z M ŷ. Solution for the vector
potential is
Ak (z) = Ak |0 cosh(|k|z) +
1 ∂Ak
|0 sinh(|k|z)
|k| ∂z
1
cosh(|k|(z − a)) θ(z − a).
+ µ0 M
|k|
(A.13)
Consistency z → ∞ gives
Ak |0 +
1 ∂Ak
|0 + µ0 M e−|k|a = 0.
|k| ∂z
(A.14)
Put together with a superconductor at z < 0 gives
Ak |0 = −
µ0 M |k| −|k|a
e
τ + |k|
(A.15)
and
Ak (z) = Ak |0 e
−|k|z
+ µ0 M
−e−|k|a sinh(|k|z), z < a
+e−|k|z cosh(|k|a), z > a
(A.16)
System: Superconductor with vortex
Now solve the London equation with one vortex in the half space z < 0.
The London equation is
−∇2 A + λ−2 A =
φ0 1
eϕ .
2πλ2 r
(A.17)
Define
1
(ẑ × ik) .
(A.18)
k2
Fourier transform in xy-plane and Laplace transform in z-direction gives
Φk = −φ0
(s2 − τ 2 )Aks = −Φk
∂A
1
+ sAk |z=0 +
|z=0 .
s
∂z
(A.19)
49
Chapter A. Domain wall, calculations
√
where τ = k 2 + λ−2 . Inverse Laplace transform gives and consistency for
z → −∞ gives
1
1
Ak (z) = Φk 2 2 + Ak |z=0 − Φk 2 2 eτ z .
(A.20)
λ τ
λ τ
50
Appendix B
Qubit, calculations
This appendix has two motivations. First, Sec. B.1 provides an integral
form for the full problem of Landau-Zener transitions coupled to one telegraph noise process. This integral form is numerically stable and is used
for the plots in paper 4. The integral form is also useful for further approximate fast noise, beyond what is considered in paper 4. Second motivation
is the need for equations in the adiabatic basis, given in Sec. B.2. The
adiabatic basis was used by Landau in his semiclassical treatment and it
might be used to obtain approximate solutions for large minimum energy
gaps g. The formulations in the adiabatic basis are also attractive numerically, since they do not experience oscillations at long times. It must also
be noted that the transitions in the adiabatic basis are very sharp, which
makes it easy to identify a transition time. This is interesting with regards
to identification of the the true transition time, as discussed in Ref. [74].
This appendix covers a quantum two-level system with Hamiltonian
H=
1
1
∆σz + gσx ,
2
2
(B.1)
where σx and σz are Pauli matrices and ∆ = ∆(t) is an arbitrary function
of time.
The two-level system is coupled to one transverse telegraph noise process, i.e., a fluctuating addition
v
± σx
2
(B.2)
to the Hamiltonian. In this appendix we use units where ~ = 1.
The master equations for one qubit coupled to one transverse telegraph
process is discussed in paper 4 and Refs. [62, 73]. On component form the
51
Chapter B. Qubit, calculations
equations are
ẋp = −∆yp
ẏp = +∆xp − gzp − vzq
żp = gyp + vyq
ẋq = −2γxq − ∆yq
(B.3)
ẏq = −2γyq + ∆xq − gzq − vzp
żq = −2γzq + gyq + vyp
where rp = (xp , yp , zp ) is the average Bloch state and rq = (xq , yq , zq ) are
auxiliary quantities. When v = γ = 0 the equations are exactly the Bloch
equations of Sec. 4.6.
The rest of this appendix is mathematical massage of Eqs. (B.3).
B.1
Integral equations for zp and zq
Let us now find the integral equations for zp and zq for transverse noise and
g 6= 0. Furthermore, the boundary conditions are so that xp , yp , xq , and yq
all start in zero at t = −∞. Isolating zp and zq from the master equations,
Eq. (B.3), yield
Z
t
żp (t) = −
dt1 cos(θ(t) − θ(t1 ))
−∞
h
i
2
2 −2γ(t−t1 )
−2γ(t−t1 )
× (g + v e
)zp (t1 ) + gv(1 + e
)zq (t1 ) ,
Z t
żq (t) = −2γzq −
dt1 cos(θ(t) − θ(t1 ))
−∞
h
i
× (v 2 + g 2 e−2γ(t−t1 ) )zq (t1 ) + gv(1 + e−2γ(t−t1 ) )zp (t1 ) .
(B.4)
Rt
where θ(t) = 0 dt′ ∆(t′ ). These equations are exact for one telegraph process, and generalizations of the integral equation of paper 4.
B.2
Adiabatic transform
The master equations, Eq. (B.3) are formulated in the diabatic basis. This
is convenient when doing driving of energy levels so that the system mainly
stays in the same diabatic level. This section contains reformulation of the
master equations to a basis following the rotating frame.
52
B.2 Adiabatic transform
No noise
For system with no noise we write the Bloch equation, Eq. (4.24), as a
“Schrödinger equation”
iṙ = Mg r,
(B.5)
where r = (x, y, z) and


0 −i∆
0
0
−ig  ,
Mg =  i∆
0
ig
0
(B.6)
is Hermitian. The subscript g is added explicitly since we will later use the
matrix with other values than g on the transverse term. Digonalization of
Mg yields
(B.7)
Mg = Ug Λg Ug† ,
where Λg is the diagonal matrix of eigenvalues and Ug is the unitary matrix
of eigenvectors. Now we do a change of variables from the vector r to the
vector d,
(B.8)
d = Dg−1 Ug† r,
where the diagonal matrix Dg is a solution of
iḊg = Λg Dg .
Thus Eq. (B.5) can be transformed to
d˙ = D−1 U̇ −1 Ug Dg d,
g
g
(B.9)
(B.10)
which is an equation in d only, while Dg , Ug , and U̇g−1 are all known, at
least in principle. The eigenvalues of Eq. (B.6) are
p
(B.11)
λg = ∆2 + g 2 .
These correspond to the time-dependent energy eigenvalues, E = λ/2, of
the two-level Hamiltonian, Eq. (B.1). Spelling out the results from above
we get


λg
0
0
Λg =  0 −λg 0  ,
(B.12)
0
0
0


exp(−iϕg )
0
0
0
exp(iϕg ) 0  ,
Dg = 
(B.13)
0
0
1
Z t
λg (t′ )dt′ ,
(B.14)
ϕg (t) =
(B.15)
53
Chapter B. Qubit, calculations
and
Ug =
Ug† =
U̇g† =
U̇g† Ug
√

2g
∆ −∆
1 
√
iλg iλg √0  ,
2λg
2∆
−g
g


∆
−iλg
−g
1 
−∆ −iλg √g  ,
√
√
2λg
2g
0
2∆


g
0
∆
˙
∆g 
−g
0 √
−∆  ,
√
√
2λ3g
2g
− 2∆ 0


0 0 1
˙
1 ∆g
= √ 2  0 0 −1  .
2 λg
−1 1 0

(B.16)
(B.17)
(B.18)
(B.19)
From the above formulas we get the following equation for d = (a, b, c),



ȧ
c
exp(−iϕ
)
g
˙
.
 ḃ  = √∆g 
−c exp(iϕg )
2
2λ
g
−a exp(iϕg ) + b exp(−iϕg )
ċ

(B.20)
The two functions a and b are complex, while c is real. Now we will use the
above formulas and construct an integral equation similar to Eq. (B.10). If
we isolate c we get
Z t
˙
˙ ′ )g
∆(t)g
∆(t
ċ(t) = −2 √
cos(ϕ(t) − ϕ(t′ )) c(t′ ).
dt′ √
2λ2g (t) −∞
2λ2g (t′ )
(B.21)
The denominator contains the factor λ2g = ∆2 + g 2 , which makes the transitions sharp. For linear driving, ∆ = at, this implies that the time of the
transition is approximately g/a.
With noise
Let us first formulate Eq. (B.3) expressed with the quantities x+ = (xp +
xq )/2 and x+ = (xp − xq )/2,
iṙ+ = −iγr+ + iγr− + Mg+v r+ ,
iṙ− = −iγr− + iγr+ + Mg−v r− .
54
(B.22)
(B.23)
(B.24)
B.2 Adiabatic transform
The matrices d+ and d− transform from r+ and r− in the following way
†
−1
d+ = Dg+v
Ug+v
r+ ,
(B.25)
†
−1
d− = Dg−v
Ug−v
r− .
(B.26)
Thus in the transformed bases we get the following equations in d+ and d− ,
†
†
−1
−1
d˙+ = −γd+ + γDg+v
Ug+v
Ug−v d− + Dg+v
U̇g+v
Ug+v Dg+v d+ ,
†
†
−1
−1
d˙− = −γd− + γDg−v
Ug−v
Ug+v d+ + Dg+v
U̇g−v
Ug−v Dg−v d− .
(B.27)
(B.28)
These equations are easily solved by time-integration as long as g 6= 0. The
big advantage compared to the original equation, Eq. (B.3), is that the all
oscillations are damped out at long times. This makes it easier to get a
numerical solutions. The cross term of (B.27) and (B.28) is
√

 ′2
λ + λ′′2 λ′2 − λ′′2 −2√2∆v
1
†
(B.29)
Ug+v
Ug−v = ′2  λ′2 √
− λ′′2 λ′2 √
+ λ′′2 +2 2∆v  ,
2λ
′′2
+2 2∆v −2 2∆v
2λ
which for v = 0 is just the identity. The λ′ and λ′′ are defined by
λ′2 = λg+v λg−v ,
′′2
λ
2
= ∆ + (g + v)(g − v).
(B.30)
(B.31)
†
To get Ug−v
Ug+v let v → −v.
55
Chapter B. Qubit, calculations
56
Appendix C
Formulas
C.1
Forward and inverse Laplace transforms
L {f ′ (x)} = −f |0 + sL {f (x)}
∂f
L {f ′′ (x)} = − |0 − sf |0 + s2 L {f (x)}
∂x
1
1
=
sinh(|k|z)
L
2
2
s −k
|k|
s
−1
L
= cosh(|k|z)
s2 − k 2
−sa e
1
sinh(|k|(z − a))Θ(z − a)
L−1
=
2
2
s −k
|k|
−sa se
−1
= cosh(|k|(z − a))Θ(z − a)
L
s2 − k 2
1
1
1
−1
L
= 2 (cosh(kz) − 1)
2
2
ss −k
k
−1
C.2
(C.1)
(C.2)
(C.3)
(C.4)
(C.5)
(C.6)
(C.7)
Parabolic Cylinder Functions
The exact solution of the Landau-Zener problem, Sec. 4.3, is expressed
by parabolic cylinder functions, U (a, x) = D−n−1/2 (x). However, the
parabolic cylinder functions are rarely included in software packages, so
in order to plot the solution the functions must be related to the Confluent
57
Chapter C. Formulas
Hypergeometric Functions 1 F1 . Following “Abramowitz and Stegun” we
start with Weber’s equation
1 2
∂2y
−
x + a y = 0.
(C.8)
∂x2
4
Solutions of this equation are
1
1 1 1 2
y1 = e
a+ ; ; x ,
1 F1
2
4 2 2
1
3 3 1 2
− 41 x2
y2 = xe
a+ ; ; x .
1 F1
2
4 2 2
− 14 x2
(C.9)
(C.10)
Define
1 Γ 14 − 21 a
y1 ,
Y1 = √
π 2 21 a+ 14
1 Γ 34 − 21 a
y2 .
Y2 = √
π 2 12 a− 14
Then, the Parabolic Cylinder Functions are
1 1
1 1
U (a, x) = D−a− 12 = cos π
+ a Y1 − sin π
+ a Y2 .
4 2
4 2
58
(C.11)
(C.12)
(C.13)
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64
PAPER I
PRL 98, 117002 (2007)
PHYSICAL REVIEW LETTERS
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16 MARCH 2007
Interaction between Superconducting Vortices and a Bloch Wall in Ferrite Garnet Films
J. I. Vestgården, D. V. Shantsev, Å. A. F. Olsen, Y. M. Galperin, V. V. Yurchenko, P. E. Goa, and T. H. Johansen
Department of Physics and Center for Materials Science and Nanotechnology, University of Oslo,
P.O. Box 1048, Blindern, 0316 Oslo, Norway
(Received 3 July 2006; published 13 March 2007)
A theoretical model for how Bloch walls occurring in in-plane magnetized ferrite garnet films can serve
as efficient magnetic micromanipulators is presented. As an example, the walls’ interaction with
Abrikosov vortices in a superconductor in close contact with a garnet film is analyzed within the
London approximation. The model explains how vortices are attracted to such walls, and excellent
quantitative agreement is obtained for the resulting peaked flux profile determined experimentally in
NbSe2 using high-resolution magneto-optical imaging of vortices. In particular, this model, when
generalized to include charged magnetic walls, explains the counterintuitive attraction observed between
vortices and a Bloch wall of opposite polarity.
DOI: 10.1103/PhysRevLett.98.117002
PACS numbers: 74.25.Ha, 74.25.Qt, 75.60.Ch, 78.20.Ls
Microscopic magnetic potentials offer an efficient and
often indispensable way to manipulate various tiny objects,
e.g., to trap and guide Bose-Einstein condensates and
degenerate Fermi gases of ultracold atoms [1–3] or to
stretch and twist DNA strands attached to paramagnetic
beads [4,5]. How precisely one can position the trapped
object and how large forces can be applied are determined
by the steepness of the magnetic potential produced by the
manipulating device. Typically, an assembly of microfabricated wires or coils can generate fields with gradients up
to 102 –103 T=m [1,2,6,7]. Recently, magnetic manipulators were created using ferrimagnetic films, where domain
walls create locally even stronger field gradients. In particular, ferrite garnet films (FGFs) were used to trap ultracold neutral atoms [8], assemble and guide colloidal
particles [9,10], and manipulate vortices in superconductors [11]. Using in-plane magnetized FGFs, where domains
are separated by Bloch walls, has two strong advantages:
(i) it is easy to move the domain wall and thereby tune the
magnetic potential, and (ii) one can simultaneously observe the motion of the manipulated objects. This ‘‘see
what you do’’ ability stems from the large Faraday effect in
the FGFs, which today are extensively used as magnetooptical imaging (MOI) sensors [12]. In an optical polarizing microscope configuration the FGF allows direct visualization of both the stray field from the manipulated
magnetic objects and the Faraday rotation in the wall itself.
In this work we present a theoretical model for how
Bloch walls can function as magnetic manipulators, using
as an example their interaction with a lattice of Abrikosov
vortices in a type-II superconductor. The configuration
considered is that of a FGF located near, but a finite
distance from, the surface of a flat superconductor. It is
shown that the model, generalized to include charged
walls, explains how the vortices are attracted to such walls,
and predicts quantitatively the nonuniform flux density
distribution we find experimentally using MOI.
Figure 1 shows an image of the magnetic field distribution near two linear Bloch wall segments, which them0031-9007=07=98(11)=117002(4)
selves appear dark in the image. The superconductor is a
single crystal of NbSe2 [13] cooled to T 4 K in a 0.3 mT
perpendicular magnetic field, which transformed into
quantized vortices when entering the superconducting
state. Each vortex is here seen as a bright dot. Evidently,
the magnetic wall has a considerable attraction on the
vortices, since their number density increases near the
wall. This observation forms the experimental basis for
the theoretical modeling presented below. Interestingly, a
most surprising feature visible from Fig. 1 makes the
problem even more challenging. There is opposite contrast
between the dark wall and the bright vortices, which means
they are of opposite magnetic polarity. In this case one
would expect from the models presented previously in the
literature [14 –17] that a Bloch wall should repel the vortices. As will be shown, by accounting for additional
FIG. 1 (color online). Magneto-optical image of the vortex
distribution near a Bloch wall in a Bi-substituted lutetium iron
garnet film placed on top of a superconducting NbSe2 crystal
with transition temperature of 7.2 K. The slightly uncrossed
analyzer and polarizer setting used here implies that the dark
wall and the bright vortices have opposite field polarities. Image
dimensions are 70 70 m2 .
117002-1
© 2007 The American Physical Society
PRL 98, 117002 (2007)
PHYSICAL REVIEW LETTERS
magnetic charges due to misalignment between the wall
direction and the in-plane magnetization vector within the
domains, the sign of the interaction can become inverted.
The two Bloch wall segments seen in Fig. 1 are actually
part of a larger zigzag pattern. Extended zigzag domain
walls are commonly present in FGFs with strong in-plane
anisotropy [18]. An example is shown in Fig. 2 (top),
where the zigzag line separates two domains with antiparallel magnetizations that meet head-on. By folding into a
zigzag pattern, the domain boundary reduces the density of
magnetic charges at the wall, which helps to minimize the
energy [19]. To describe the interaction between one segment of such a zigzag wall and a superconducting vortex,
we introduce the model illustrated in Fig. 2 (bottom). The
superconductor occupies the half-space, z > 0, and the
wall is directed along the y axis, which forms an angle ’
with the magnetization direction of the two domains.
Experimentally, we detect only tiny changes of the wall
width, 2W, during cooling through Tc , in agreement with
[14]. Thus, in calculations the wall is approximated as a
fixed, uniform, out-of-plane magnetization Mz jxj < W Ms . Inside the domains there is an in-plane magnetization with a component normal to the wall given by
Mx jxj > W Ms x=jxj sin’, where Ms is the saturation magnetization. For ’ 0 the present model reduces
to the noncharged wall case. The My component is omitted
in the analysis since the wall is assumed to be infinitely
long.
Stray fields from the wall induce shielding currents in
the superconductor, which we determine using the London
theory. The equations valid inside and outside the superconductor then read
FIG. 2 (color online). Top: MO image showing a zigzag Bloch
wall in a FGF separating two domains with antiparallel in-plane
magnetization. Bottom: Sketch of an in-plane magnetized FGF
with a Bloch wall placed above a superconductor. The magnetic
charges along the vertical sides of the wall can lead to a net
attraction between a wall and vortices, as seen in Fig. 1.
2 A r2 A 0
r2 A 0 r My
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16 MARCH 2007
z 0;
z 0;
(1)
where is the London penetration depth and the vector
^ The shielding currents flow in the y
potential is A Ay.
direction and their density equals Jy A=0 2 . A vortex present in the superconductor then feels two forces.
First, the direct force from the FGF, which
can be calcuR
lated from the free energy term, 0 M Hv dV, where
Hv is the stray field from the vortex. Second, the Lorentz
force from the shielding currents in the superconductor,
FL Jr 0 integrated over the length of the vortex.
0 is the magnetic flux quantum, and we will simplify the
treatment by assuming the vortex to be straight and aligned
with the z axis. Interestingly, the two forces turn out to have
exactly the same magnitude and direction, as was noted
also in Ref. [15], where a similar configuration was
analyzed.
It is convenient to express the total force on the vortex in
the x direction as
Fvw F? Fk ;
(2)
where F? and Fk are the contributions from the perpendicular and in-plane components of M, respectively. Their
Fourier transforms are obtained as
Fk? 4i
Ms 0 1 ejkjh jkja
e
sinWk;
2 jkj jkj
(3)
Ms 0 1 ejkjh jkja
e
cosWk sin’; (4)
2 k jkj
p
where 2 k2 , a is the gap between superconductor and FGF, and h is the FGF thickness. For the configuration illustrated in Fig. 2, the force Fk is always attractive,
whereas F? is repulsive. This qualitative result can be
easily understood by considering the interaction between
the magnetic charges involved. The stray field from a
vortex is closely approximated by that of a magnetic
monopole located at z and with strength 20 [20].
Thus, the vortex is attracted to the positive charges along
the vertical sides of the wall and repelled by the perpendicular dipole charges. The charge representation also
yields the correct magnitude of forces given by Eqs. (3)
and (4) in the limit ! 0. The superconductor then perfectly screens the magnetic field created by M and its
presence should be accounted for by adding the corresponding mirror charges.
From the inverse transform of (3) and (4), we obtain the
spatial variation of the two force contributions, which are
plotted in Fig. 3 together with the total force on the vortex.
At sufficiently large x the magnitude of Fk is larger than
that of F? , which is expected since the monopolemonopole interaction should dominate at long distances.
However, at short distances F? becomes dominant, and the
total force changes sign at some distance x . For our set of
parameters, the repulsive region jxj < x is very small, less
117002-2
Fkk 4i
vw
20
10
x*
0
0.4
Φ0 δN [mT]
F
⊥
F
||
F
30
F [pN]
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PHYSICAL REVIEW LETTERS
PRL 98, 117002 (2007)
-10
-20
2W
2W
2W
2W
0.2
= 0.0 µm
= 0.5 µm
= 1.0 µm
= 2.0 µm
0.0
-0.2
-30
-10
-3
-2
-1
0
1
2
3
x [µm]
FIG. 3 (color online). The calculated forces on the vortex from
the Bloch wall: F? is repulsive and Fk is attractive. Their sum
Fvw changes sign at x 1 m. The parameters used here are:
sin’ 0:34, 2W 0:6 m, h 0:8 m, a 140 nm, 200 nm, and Ms 50 kA=m.
than 1 m. This is why the repulsive region under the wall
is not visible in the image of Fig. 1. This also explains why
we observe the counterintuitive attraction between the
Bloch wall and the vortices of opposite polarity. A related
phenomenon when ferromagnetic domain wall stimulates
superconductivity due to its stray fields was considered in
Ref. [21].
We consider next an initial state with a uniform distribution of vortices in the superconductor, and a subsequent
introduction of a Bloch wall. This results in a perturbation
of the vortex density, Nx, which creates an additional
force acting on every vortex. In the equilibrium, the additional force everywhere balances the force from the Bloch
wall, i.e.,
Z
Fvw x Fvv x x0 ; y0 Nx0 dx0 dy0 ;
(5)
where Fvv is the x component of the repulsive vortexvortex interaction, and the vortex matter is considered as
a continuum. This equation represents a perfect shielding
of the domain wall by the vortex matter. In Fourier space,
the perturbed vortex density becomes Nk Fkvw =Fkvv .
The vortex-vortex interaction can be obtained from the
currents around a flux line in a half-space, first calculated
by Pearl [22], and gives
20 ik
1 1
1
Fkvv ;
(6)
jkjL
2 jkj 0 2 jkj2
where L is the flux line length. The term proportional to L
is the conventional (Abrikosov) bulk contribution, while
the other term is the surface contribution.
The resulting perturbation Nx of the vortex density
induced by a Bloch wall is shown in Fig. 4. Its profile is
strongly dependent on the wall width 2W. In the small W
limit the total force on a vortex is everywhere attractive;
hence, the vortex density increases monotonically as one
approaches the wall. For increasing W the density profile
-5
0
x [µm]
5
10
FIG. 4 (color online). The excess vortex density Nx near
the Bloch wall for various wall width 2W calculated using L 160 m and the other parameter values as indicated in the
caption of Fig. 3.
develops a minimum below the center of the wall, and the
resulting Nx becomes nonmonotonic with a minimum
at the center and maxima near the two wall edges. For
sufficiently large W the vortices right below the wall are
expelled creating a narrow depleted area.
The theoretical vortex density profile Nx can be
compared to our MOI observations of the vortex distribution near the Bloch wall. The FGF had a thickness h 0:8 m, and saturation magnetization Ms 50 kA=m.
The film, with no additional layers (contrary to standard
MOI indicators), was placed directly on top of the 0.3 mm
thick NbSe2 crystal with a gap of a 140 nm, as determined from the optical interference pattern [23]. This gap
equals one quarter of a wavelength, and gives optimal
transmission.
The vortices were formed slightly below Tc where flux
pinning is negligible. Thus, one expects that the vortex
positions seen in Fig. 1 represent a frozen picture of an
arrangement where the vortices adjust only to balance the
interaction with the wall. To obtain a better view, we made
use of the fact that further cooling to 4 K increased the
vortex pinning considerably, and removed the Bloch wall
without creating noticeable change in the vortex positions,
see and Fig. 5 (top). From this image, the positions of all
the vortices inside the marked rectangular area were identified, and the Wigner-Seitz cell of each vortex was determined using standard triangulation [24]. The local vortex
density was obtained by inverting the cell area, and is
shown in Fig. 5 (bottom) for every vortex versus its coordinate x. The vortex density near the wall clearly exceeds
the background density of 0 N0 0:3 mT. The theoretical curve N0 Nx, plotted in Fig. 5 (bottom), was
calculated using ’ 20 determined from Fig. 1, and a
wall width of 2W 0:6 m. The calculated curve reproduces very well not only the sign, but also the magnitude of the excess vortex density. This agreement was
achieved using the penetration depth 200 nm as an
adjustable parameter. This value of corresponds to the
temperature of 6.8 K (slightly below Tc 7:2 K) which is
thus the temperature when the vortices got frozen. The
117002-3
PRL 98, 117002 (2007)
PHYSICAL REVIEW LETTERS
week ending
16 MARCH 2007
the image of objects of the order of the light wavelength,
0:55 m.
In conclusion, mobile domain walls found in in-plane
magnetized ferrite garnet films were investigated for possible use as magnetic micromanipulators. It was shown,
choosing superconducting vortices as a case example, that
such films can serve to both apply forces and simultaneously monitor the results of the action. A theoretical
model for the interaction was developed, with the vortices
described within the London approximation, and the domain wall represented by a charged magnetic wall. The
charged wall model, which includes magnetic poles on all
the sides of the wall’s rectangular cross-section, is shown
to give a very good quantitative description of the attraction of vortices to such a wall. The comparison was made
by direct observation of individual vortices using the
magneto-optical imaging technique.
This work was supported financially by The Norwegian
Research Council, Grant No. 158518/431 (NANOMAT)
and by FUNMAT@UIO. We gratefully acknowledge
discussions with V. Vlasko-Vlasov, L. Uspenskaya, and
E. Il’yashenko.
FIG. 5 (color online). Top: Distribution of vortices formed in
the presence of a Bloch wall. The image was taken after the wall,
seen in Fig. 1, was removed by an in-plane field of the order of a
few T applied perpendicular to the indicated x axis. Bottom:
Vortex density obtained from the image (each symbol represents
one vortex) together with the theoretical curve calculated for
L 160 m, 0 N0 0:3 mT and remaining parameters as
listed in the caption of Fig. 3.
vortex length was set to L 160 m. It is smaller than the
crystal thickness 300 m to compensate for the overestimation of the Abrikosov interaction term in the continuum
approximation.
An open question remains regarding the large apparent
width of the Bloch wall, approximately 3 m as seen from
the image in Fig. 1. The theoretical estimate obtained by
minimizing the sum of exchange, anisotropy and magnetostatic energies is given implicitly by the equation [14]
1 w2 w2 1 Ku =0 Ms2 , where w 2W=h
is the normalized wall width and 22 A=0 Ms2 h2 .
Substituting the effective exchange constant A 2 1011 J=m and the uniaxial anisotropy constant Ku 103 J=m3 [14] we obtain 2W 0:6 m. The discrepancy
between the observed and estimated wall width is probably
due to the optical diffraction which significantly distorts
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PAPER II
Flux Penetration in Superconducting Strip with Edge-Indentation
J. I. Vestgården, D. V. Shantsev, Y. M. Galperin and T. H. Johansen1
1
Department of Physics and Center for Advanced Materials and Nanotechnology,
University of Oslo, P. O. Box 1048 Blindern, 0316 Oslo, Norway
The flux penetration near a semicircular indentation at the edge of a thin superconducting strip
placed in a transverse magnetic field is investigated. The flux front distortion due to the indentation
is calculated numerically by solving the Maxwell equations with a highly nonlinear E(j) law. We
find that the excess penetration, ∆, can be significantly (∼ 50 %) larger than the indentation radius
r0 , in contrast to a bulk superconductor in the critical state where ∆ = r0 . It is also shown that the
flux creep tends to smoothen the flux front, i.e. reduce ∆. The results are in very good agreement
with magneto-optical studies of flux penetration into an YBa2 Cu3 Ox film having an edge defect.
PACS numbers: 74.25.Ha,74.78.Bz,74.25.Qt
I.
INTRODUCTION
Magnetic field penetrates type-II superconductors as a
set of quantized flux lines – vortices. An important feature of this vortex matter is pinning, which leads to zero
electrical resistance at zero temperature. The pinning
results in a non-uniform distribution of magnetic flux
forming a critical state. The critical state determines
the macroscopic properties, e.g. the maximum current
density and magnetic susceptibility, that are important
for applications. According to the critical state model,1
at any point of the sample the local value of the electrical current density is equal to its critical value, jc , for a
given magnetic field and temperature.
An interesting property of the critical state is that local material defects affect the field and current distributions on a global scale. For example, even a small
non-superconducting cavity or an edge indentation create sample-spanning discontinuity lines where the current
flow direction changes abruptly.2 At a non-zero temperature, the critical state is relaxed due to flux creep that is
conventionally described by a highly nonlinear currentvoltage curve, E ∝ j n where n ≫ 1 and E is electric
field. Nevertheless, the same tendency persists: a small
cavity of size ℓ in a bulk superconductor perturbs the
field distributions on a much larger scale of ∼ nℓ.3 Many
applications of superconductors are based on thin films
where this tendency must be even stronger since the relation between the magnetic field and current is nonlocal.4
Usually this leads to poorer performance of superconducting devices whose global properties are deteriorated
by numerous natural defects blocking the current flow.
However, the same tendency can help control the flux
motion on a global scale by patterning the superconductor with arrays of small holes designed, e.g., to guide the
flux in a particular direction.5,6
Surprisingly, a quantitative understanding on how a
single local defect affects the flux penetration into a superconducting film is still rather poor. Even the simple
case of an infinitely long thin strip with a semicircular
edge-indentation is not solved. For a bulk superconductor in the critical state such an indentation creates
an excess flux penetration exactly equal to the indentation radius.2 However, the nonlocal electrodynamics
in thin films and hence the presence of Meissner currents in the flux-free regions make the picture much more
complicated.7 It has been observed using magneto-optical
imaging that the excess flux penetration in films can significantly exceed the size of the indentation it originates
from.8 However, for a particular sample it is not always
transparent which mechanism is responsible for the enhancement. It could be related to the effect of thin-film
geometry, to the flux creep or to thermal instabilities nucleated at the indentation.
This work aims to clarify this question by presenting
a detailed study of flux penetration into a strip with a
semicircular edge-indentation in the flux creep regime.
We determine how the excess penetration depends on the
size of the indentation, the applied magnetic field and the
creep exponent n.
II.
MODEL
Consider a thin superconducting strip of thickness d
placed in a transverse magnetic field. The strip is infinite in the y-direction, has the width 2w ≫ d in the
x-direction and a semicircular indentation with radius r0
at the edge, see Fig. 1. The flux dynamics in the creep
regime is conventionally described using a local relation
between electric field E and current density j,9–11
E = ρj ,
(1)
with a highly nonlinear resistivity
n−1
ρ = ρ0 (j/jc )
,
(2)
which does not explicitly depend on the magnetic induction B. Here ρ0 is a constant, jc is the critical current
density, while n is the creep exponent, n ≫ 1. This exponent can be related to the activation energy U for thermal depinning as n ∼ U/kT . Hence, large n means small
creep, and the Bean critical state model1 is regained in
the limit n → ∞.
2
Strip center
a
y
w
w
∆
r0
Ha
x
Edge
FIG. 1: A superconducting strip of width 2w with a semicircular indentation of radius r0 in transverse field Ha .
For numerical simulations of flux penetration into the
strip we use the formalism developed by Brandt4,9,11–17
that can be applied to thin type-II superconductors of
various shapes. For a thin superconductor, it is appropriate to look at length scales larger than the thickness
d, and introduce a sheet current
J(r) =
Z
d/2
dz j(r, z) ,
where r = (x, y) are in-plane coordinates. Due to the
current conservation, ∇ · J = 0, the sheet current can be
expressed through a scalar function g(r) as
A
Here Ha is the applied field and A is the sample area.
The integral kernel is equal to the field of a dipole of unit
strength,
µ0 2z 2 − (r − r′ )2
.
4π [z 2 + (r − r′ )2 ]5/2
(5)
The integral Eq. (4) with kernel Eq. (5) is divergent at
r → r′ and z → 0. In a numerical procedure, the divergence can be handled in three ways: (i) by keeping a finite
z during the calculation;18 (ii) by working in the Fourier
space;11 (iii) by converting the integral to a matrix form
and using the flux conservation to determine the diagonal
elements.12,16,17 Here we use the third method. Since, for
Ha = 0, the total flux through the zR = 0 plane is zero,
the kernel should have the property d2 r Q(r, r′ , 0) = 0.
This yields
1
Bz (r) = Ha + g(r)C(r) −
µ0
Z
A
Cstrip (x) =
w
1
.
2
π w − x2
(8)
In addition, the indentation gives a contribution from the
semicircle, which is calculated numerically from Eq. (7).
(3)
where g has the interpretation of the local
magnetization.9 Substituting the current from Eq. (3)
into the Biot-Savart law one arrives at a non-local
relation between Bz and g,
Z
d2 r′ Q(r, r′ , z) g(r′ ) .
(4)
Bz (r, z) = µ0 Ha +
Q(r, r′ , z) =
where the scalar function C is an integral over the area
outside the superconductor
Z
dr′2
C(r) =
.
(7)
′ 3
outside 4π|r − r |
For a uniform strip of width 2w it yields
−d/2
J = ∇ × ẑg
FIG. 2: The simulated flux density map in a strip of width
2w, with a semicircular indentation of radius r0 = 0.2w, in
applied field Ha = 0.3Jc , n = 19, and ramped with a rate
µ0 Ḣa = ρ0 Jc /wd. Note that ∆ is not equal to r0 .
d2 r′ g(r′ ) − g(r)
, (6)
4π |r − r′ |3
In the following we use an equidistant square grid and
ascribe the same area s to each grid point. The discrete
version of the kernel then acquires the form17
!
Qij
Ci X
(9)
= δij
+
qil − qij ,
µ0
s
l
3
where qij = 1/4π|ri − rj | for i 6= j and qii = 0. All elements of the discrete kernel
Eq. (9) are nondivergent and
R
the flux conservation, d2 r B(r) = 0, is guaranteed. Relating the magnetic field and the current by the Faraday’s
law, and using the inverted Biot-Savart law one obtains
the dynamic equation for the local magnetization:
Z
h
i
ġ(r, t) =
d2 r′ Q−1 (r, r′ ) fˆg(r′ , t) − Ḣa (t) , (10)
A
where
fˆg ≡ ∇ · (ρ∇g)/dµ0 .
For discrete formulation of the problem the inverse kernel Q−1 is just the inverse of the matrix Eq. (9), hence
the matrix must be calculated and inverted only once.
Mathematical details are in appendix A.
III.
RESULTS AND DISCUSSION
a. Magnetic field and current The simulations were
performed by ramping the applied field at a constant
3
0.3
1
n=101
n=19
n=9
n=5
0.8
∆/w
0.6
0.1
Bean
n=101
n=19
n=9
n=5
0.4
0.2
0
0
0
0
0.1
0.2
Ha/Jc
0.3
0.1
0.4
FIG. 3: Width of Meissner state, a, as the applied field is
ramped with a constant rate µ0 Ḣa = ρ0 Jc /wd. Stronger flux
creep, i.e. smaller n, leads to deeper penetration. The Bean
model curve is given as aBean = w/ cosh(πHa /Jc ).
0.3
0.4
0.3
0.2
0.1
rate µ0 Ḣa = ρ0 Jc /wd, starting at zero field and a fluxfree strip. The flux penetrates from the edges forming
well-defined flux fronts that move towards the strip center
as the applied field increases. Shown in Fig. 2 is a typical
result of the flux density distribution presented as seen
in a magneto-optical image, i.e., the image brightness
represents the magnitude of the perpendicular magnetic
field. The sample edge is seen as a bright line, i.e., the
flux density is highest at the edge.
Far from the indentation the flux penetration front is
straight, and leaves a fraction a/w of the strip in the
flux-free Meissner state, seen here as a black region. The
penetration of this straight front versus applied field is
shown in Fig. 3 for different values of the creep exponent.
For large n the simulations approach the film Bean model
result,4 aBean = w/ cosh(πHa /Jc ), while for smaller n,
i.e., stronger flux creep, the penetration is deeper, all as
expected for a strip with straight edges.
Near the indentation the flux penetration largely follows the circular shape. At both sides of the indentation there are dark regions of reduced flux density. As
penetration gets deeper these will become narrow lines.
They are called discontinuity lines, or d-lines, since they
appear where current turns discontinuously in the Bean
limit n → ∞. The d-lines of semicircular indentations
have parabolic shape.2 With finite n the parabolic shape
is only approximated. However, the main effect of the
indentation is that it pushes magnetic field deeper into
the sample. In order to quantify this we define the excess
penetration ∆ as the difference between the deepest penetration and the penetration far away from the indentation. Fig. 4 shows how ∆ evolves with increasing Ha . Evidently, the excess penetration is not equal to the indentation radius, r0 , as in the case of the bulk Bean model.2,3,19
Moreover, ∆ turns out to be field-dependent. Initially,
∆ increases, then reaches a maximum followed by a de-
0.2
Ha/Jc
∆/w
a/w
0.2
r0 / w = 0.1
n=101
n=19
n=9
n=5
0
0
r0 / w = 0.2
0.1
0.2
Ha/Jc
0.3
0.4
FIG. 4: Evolution of the indentation-induced excess penetration, ∆, as a function of applied field. The two panels
correspond to different indentation radii, r0 /w =0.1 and 0.2,
respectively; µ0 Ḣa = ρ0 Jc /wd.
crease at larger Ha . This surprising non-monotonous behaviour is supported by magneto-optical measurements
of the flux penetration in a uniform YBa2 Cu3 Ox film containing an edge defect, see Fig. 5. The film was shaped
as a strip of half-width w = 0.4 mm, and the figure shows
the flux distribution at 25 K for 3 different applied fields.
In (a) the field was very small, µ0 Ha = 3 mT, creating
negligible penetration so that the actual shape of the defect appears in the image as the bright ”bay area” inside
the strip. In this state the excess penetration is equal
to the depth of the defect, and measures ∆ = 80 µm.
In (b) and (c) the applied field is 17 mT and 36 mT,
respectively, and the corresponding excess penetration is
∆ = 115 µm and 100 µm. This gives for ∆/w = 0.20, 0.29
and 0.25, demonstrating an excess penetration that exceeds the depth of the indentation by nearly 40 %, in
very good agreement with the high n results plotted in
Fig. 4.
The Fig. 4 includes the behaviour of ∆/w for two different r0 /w. Comparing the two panels we see that larger
indentations produce a larger ∆. However, the relative
excess penetration, ∆/r0 is larger for the smaller indentation. The excess penetration can exceed the indentation
4
a
0.1 mm
b
∆
∆
∆
c
FIG. 5: Magneto-optical images of flux penetration into an
YBa2 Cu3 Ox strip with a defect at the edge. Only the lower
half of the strip is shown. In (a), (b) and (c) the applied fields
was 3, 17 and 36 mT, respectively. The excess flux penetration, ∆, is maximal at the intermediate field, in agreement
with simulations.
depth by almost 50 % for r0 = 0.1w and large values of
n. For smaller values of the creep exponent one always
finds smaller ∆, implying that creep tends to smoothen
perturbations in the flux front.3
Our results demonstrate that an indentation in a thin
film affects the flux distribution in a stronger and more
complex way than it does in bulk superconductors. This
must be due to the non-local electrodynamics of thin
films, and in particular due to the presence of Meissner
currents in the flux free regions. These Meissner currents do not make the same sharp turns as the critical
currents in the flux penetrated region, see Fig. 6 and also
Refs. 9,11. As a result, the Meissner currents concentrate
in front of the indentation where their density reaches jc
and hence lead to even deeper flux penetration. This is
why the flux front near the indentation advances faster
than in the rest of the film. This accelerated advancement eventually terminates when the penetration depth
becomes comparable to the strip halfwidth.
b. Electric field The Lorentz force pushing magnetic
flux is directed perpendicular to the local current density.
Even a small indentation distorts the current stream lines
over a large area, and hence significantly modifies the
trajectories of flux motion. In particular, all the flux arriving to the fan-shaped region rooted at the indentation
must have entered the sample through this indentation,
see Fig. 5. It creates a dramatic local enhancement of
electric field since E is a direct measure of the intensity
of flux traffic.
Analytical solution for the electric field distribution
around an indentation in thin films is not available.
Therefore the results obtained for the case of a slab are
often utilized as approximations also for films.3,14,20 We
will now analyze to what extent such estimates are valid
by comparing them with our simulation results for a strip.
In the fan-shaped region that originates from the semicircular indentation, the electric field can be found by
solving the Maxwell equation ∇ × E = −Ḃ in cylindri-
FIG. 6: Simulated flux distributions (top) and current stream
lines (bottom) in an increasing applied field, where r0 = 0.1w
and the other parameters the same as in Fig. 2. From left, the
values of Ha are 0.05Jc , 0.2Jc , and 0.4Jc with corresponding
values of a 0.98w, 0.8w, and 0.5w.
cal coordinates. Since the evolution of B-distribution is
usually not very far from the bulk Bean model, one can
assume Ḃ = µ0 Ḣa , which leads to the solution14
"
#
2
µ0 Ḣa (w − a + r0 )
E1 (x) =
− (w − |x|)
(11)
2
w − |x|
for |x| > a − r0 and zero for |x| < a − r0 . Far away
from the indentation the solution of the same equation
in Cartesian coordinates, ∂x E = −µ0 Ḣa , is
E0 (x) = µ0 Ḣa (|x| − a)
(12)
for |x| > a and zero for |x| < a. Note that the width
w enters Eq. (11) only because of the specific choice of
the x-coordinate, where the edge is located at x = w.
Replacement x → x + w removes the w-dependence.
Figure 7 compares E0 (x) and E1 (x) with the simulated electric field profiles. The quantitative agreement
is poor, though the shape of profiles (both across the
indentation and away from it) is fairly well reproduced,
in agreement with Ref. 8. The expected enhancement
of E due to indentation is also obvious. The formulas
above predict the relative enhancement for the peak val(max)
(max)
ues E1
/E0
= (w − a)/2r0 + 1 for a bulk sample.
One can see from the plot that the effect of indentation
(max)
(max)
is
/E0
is even stronger for thin films: the ratio E1
slightly higher and the excess penetration is larger (the
flux front here corresponds to the point where E(x) = 0).
A locally enhanced electric field near edge indentations
and hence enhanced Joule heating is predicted to facilitate nucleation of a thermal instability.14,20 The instability in thin superconductors is usually observed in form of
macroscopic dendritic flux avalanches21 or macroscopic
uniform flux jumps22 . However, a third scenario is also
possible when a series of microscopic flux avalanches repeatedly take place in the same region, each leading to
a small advancement of the flux front.23 It creates an
5
2
1.5
r0 / w
1
E/Ec
E(x,0)
E1(x)
E(x,∞)
E0(x)
0.5
0
0.5
0.6
0.7
0.8
0.9
1
x/w
FIG. 7: Electric field profiles across the indentation, E(x, 0),
and far away from it, E(x, ∞), computed numerically from
Eq. (1). The corresponding analytical bulk Bean-model profiles E1 (x) and E0 (x) are given in Eqs. (11) and (12), respectively. The simulation parameters are the same as for Fig. 2
except that r0 = 0.1w and Ha = 0.25Jc . Ec = ρ0 Jc /d. A
strong field enhancement near the indentation is clearly seen.
additional front distortion since the avalanches are expected to be larger and occur more frequently at the
indentation, where the local E is maximal. Experimentally the individual avalanches can be very small, and
hence it is not easy to determine whether the thermal
effects contribute to an observed front distortion. To
identify the penetration mechanism one can compare the
observed flux profiles with the simulations. The maximal
excess penetration due to non-thermal effects is found to
be 150 % of the indentation radius for our parameters.
Consequently, when the observed excess penetration is
larger, the flux penetration probably occurs via thermal
micro-avalanches.
IV.
CONCLUSIONS
We have numerically solved the Maxwell equations to
describe flux penetration into a thin superconducting
strip with an edge-indentation and analyzed the time
evolution of flux front in an increasing applied field,
Ha . The excess penetration, ∆, due to the indentation
is not equal to the indentation radius, r0 , in contrast
to the well-known case of a bulk superconductor in the
1
2
3
4
5
C. P. Bean, Rev. Mod. Phys. 36, 31 (1964).
A. M. Campbell and J. Evetts, Critical Currents in Superconductors (Taylor and Francis LTD, London, 1972).
A. Gurevich and M. Friesen, Phys. Rev. B 62, 4004 (2000).
E. H. Brandt and M. Indenbom, Phys. Rev. B 48, 12893
(1993).
R. Wördenweber, P. Dymashevski, and V. R. Misko, Phys.
Rev. B 69, 184504 (2004).
Bean model. Three different mechanisms that influence
the excess penetration were analyzed. (i) The nonlocal
electrodynamics in films leads to a characteristic ∆(Ha )
dependence with a smooth peak. The ratio ∆/r0 at the
peak equals 1.5 when r0 is 0.1 of the strip half-width and
becomes even larger for smaller r0 . (ii) The flux creep
always tends to smoothen the flux front and decrease
the excess penetration. (iii) Thermal flux avalanches
are more likely to occur at the indentation, which can
increase the apparent front distortion. Our results
can be very helpful in order to identify which of these
three mechanisms is the dominant one in a concrete
experiment.
Acknowledgments
We thank C. Romero-Salazar and Ch. Jooss for fruitful discussions. This work was supported financially by
The Norwegian Research Council, Grant No. 158518/431
(NANOMAT) and by FUNMAT@UIO.
APPENDIX A: NUMERICAL DETAILS
The simulations are carried out on an equidistant
square grid with N × N points, xm = w(2m + 1)/N − w
and yn = w(2n + 1)/N − w, for 0 ≤ m, n < N . The
system has two symmetries that must be incorporated
in the kernel: first, the periodic boundary, which means
that we must add mirror strips at x < −w and x > w.
Second, the symmetry around x = 0. The latter means
that we can work with half the kernel.11 The simulations use a grid size of N = 100, which means that a
5000 × 5000 matrix must be put in memory and inverted.
The memory consumption is the main limiting factor of
the simulations. The kernel is stable, so there is no need
for additional smoothening. For most exponents a pure
power law is used, but for the Bean limit, n = 101, a
cutoff on the resistivity ρ < ρmax was necessary to ensure stability. The flux front position was determined at
every time step and then smoothened as a function of
time. It allows the front position to be determined with
an accuracy much better than the distance between two
grid points.
6
7
8
9
V. V. Yurchenko, R. Wördenweber, Y. M. Galperin, D. V.
Shantsev, J. I. Vestgården, and T. H. Johansen, Physica
C 437-438, 357 (2006).
J. Eisenmenger, P. Leiderer, M. Wallenhorst, and
H. Dötsch, Phys. Rev. B 64, 104503 (2001).
T. Schuster, H. Kuhn, and E. H. Brandt, Phys. Rev. B 54,
3514 (1996).
E. H. Brandt, Phys. Rev. Lett. 74, 3025 (1995).
6
10
11
12
13
14
15
16
17
18
E. Zeldov, N. M. Amer, G. Koren, A. Gupta, and M. W.
McElfresh, Appl. Phys. Lett. 56, 680 (1990).
E. H. Brandt, Phys. Rev. B 52, 15442 (1995).
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T. Schuster, H. Kuhn, E. H. Brandt, M. V. Indenbom, M. Kläser, G. Müller-Vogt, H.-U. Habermeier,
H. Kronmüller, and A. Forkl, Phys. Rev. B 52, 10375
(1995).
R. G. Mints and E. H. Brandt, Phys. Rev. B 54, 12421
(1996).
E. H. Brandt, Phys. Rev. B 55, 14513 (1997).
E. H. Brandt, Phys. Rev. B 64, 024505 (2001).
E. H. Brandt, Phys. Rev. B 72, 024529 (2005).
K. A. Lörincz, M. S. Welling, J. H. Rector, and R. J. Wi-
19
20
21
22
23
jngaarden, Physica C 411, 1 (2004).
T. Schuster, M. V. Indenbom, M. R. Koblischka, H. Kuhn,
and H. Kronmüller, Phys. Rev. B 49, 3443 (1994).
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D. V. Denisov, D. V. Shantsev, Y. M. Galperin, E.-M.
Choi, H.-S. Lee, S.-I. Lee, A. V. Bobyl, P. E. Goa, A. A. F.
Olsen, and T. H. Johansen, Phys. Rev. Lett. 97, 077002
(2006).
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B 74, 220511(R) (2006).
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PAPER III
Flux Distribution in Superconducting Films with Holes
J. I. Vestgården, D. V. Shantsev, Y. M. Galperin and T. H. Johansen1
1
Department of Physics and Center for Advanced Materials and Nanotechnology,
University of Oslo, P. O. Box 1048 Blindern, 0316 Oslo, Norway
Flux penetration into type-II superconducing films is simulated for transverse applied magnetic
field and flux creep dynamics. The films contain macroscopic, non-conducting holes and we suggest a
new method to introduce the holes in the simulation formalism. The method implies reconstruction
of the magnetic field change inside the hole. We find that in the region between the hole and the
edge the current density is compressed so that the flux density is slightly reduced, but the traffic
of flux is significantly increased. The results are in good agreement with magneto-optical studies of
flux distributions in YBa2 Cu3 Ox films.
PACS numbers: 74.25.Ha,74.78.Bz,74.25.Qt
I.
INTRODUCTION
The behavior of vortex matter in superconductors can
to a large degree be controlled by introducing artificial
defects. It has been known for a long time that randomly
distributed defects, created e.g. by neutron irradiation,
allow a dramatic enhancement of the critical current density, jc . One may reach more specific goals by tuning
the arrangement of artificial defects. In particular, experiments on superconducting thin films have revealed
a large number of interesting effects, including matching effects,1 noise reduction in SQUIDs,2 rectified vortex
motion,3,4 anisotropy of jc ,5 and vortex guidance.6
In parallel with the experimental progress, the theoretical understanding of how artificially created patterns
interact with vortex matter is also developing. Interaction between a single vortex and a cylindrical cavity in a
bulk superconductor was considered within the London
approximation in Ref. 7. This work extends the classical paper, Ref. 8, predicting the maximal number of flux
quanta that can be trapped by a single hole. Current
distribution around a 1D array of holes was calculated
within the Ginsburg-Landau theory in Ref. 6. However,
these theoretical works consider a bulk superconductor,
while most experiments are on patterned thin films.1–6
Moreover, a realistic model should take into account the
strong pinning of vortices in the superconducting areas
around the artificial defects. When the defect size is
much larger than the London penetration depth, one can
consider the average vortex density B rather than individual vortices. Such an approach was used in Ref. 9 to
simulate flux penetration into a thin film with a 2D array of holes. It allowed to explain an asymmetrical flux
penetration due to asymmetry in the hole shape. At the
same time, the case of an individual hole in a thin film
has not yet been carefully analysed. A main purpose of
the present work is to acquire details of flux and current
distributions in a superconducting strip with one individual hole.
An approximate picture of the current distribution
around a non-conducting hole can be obtained within
Bean’s critical state model.10 In the Bean model current
stream lines are added from the edge with equal spacing representing the critical current density. The presence of a hole forces the current to flow around it and
hence pushes the flux front deeper into the sample. Both
holes and sample corners give rise to so-called d-lines
where the current changes direction discontinuously.11
They are seen as dark lines12 in images showing magnetic flux distributions.13,14 For example, 90◦ corners
give 45◦ straight d-lines15 while semicircular indentations
of the edge give parabolic d-lines.16 The magneto-optical
image of Fig. 1 shows d-lines spreading out from a circular hole towards the flux-free region. The same hole
also introduces another pattern: a darkened region starting from the hole and extending towards the edge. This
pattern is similar to the one observed by Eisenmenger
et al., Ref. 17. The pattern does not fit with the common interpretation of the Bean model, which leaves the
the currents between the hole and the edge unperturbed.
Ref. 17 discusses how to reinterpret the Bean model and
explain the observed pattern as a second parabolic d-line.
In this work, we will go further and do full dynamical simulations of flux penetration taking into account the nonlocal electrodynamics of films as well as flux creep. Our
results provide details of flux and current distributions
in the vicinity of a hole and suggest a new interpretation
for the observed anomaly.
II.
A.
MODEL
Single-connected superconductors
Consider a type-II superconducting thin film placed
in an increasing transverse magnetic field. The superconductor responds by generating screening currents to
shield its interior. The current density is highest at the
edges where the Lorentz force eventually overcomes the
pinning force, leading to penetration of flux. According
to the Bean model, the vortices move only when the local current density exceeds the critical value, jc . A more
realistic model for flux penetration also allows for flux
creep at j < jc . Macroscopically, flux creep is introduced
2
FIG. 1: Left: Magneto-optical image of Bz near a hole. Note
the parabolic d-lines going upwards and a dark area going
downwards from the hole. Right: a sketch of the strip with
a circular hole indicating how peculiarities in the flux distribution are related to bending of the current stream-lines.
Notations: film half-width is a, distance from the edge to the
hole center is s, and the hole radius is r0 .
FIG. 2: Simulated magnetic field distribution in a long strip,
plotted in the style of magneto-optical images, where the intensity represents Bz . At small applied field (left) the hole
produces a field dipole and at large field (right) one can see
the parabolic d-lines and a dark region between the hole and
the edge, cf. the experimental image, Fig. 1; r0 /a = 0.1,
s/a = 0.5, n = 19, Ha /Jc = 0.2 (left) and 1 (right), and
µ0 Ḣa = ρ0 Jc /ad.
by a dipole of unit strength,15
Q(r, r′ , z) =
through a highly non-linear current voltage relation15,18
n−1
j
E = ρ0
j,
(1)
jc
where E is electric field, ρ0 a resistivity constant, j is
current density, and n is the creep exponent. For thin
films of YBa2 Cu3 Ox , n is typically in the range from 10
to 70 depending on temperature and pinning strength.19
Flux dynamics of single-connected type-II superconductors in transverse geometry has been described thoroughly by E. H. Brandt. This work uses the same formalism and hence we only give a short summary of the
simulation basics, mainly following Refs. 15,20 and 21.
The next section will be devoted to additional changes
for multiply-connected samples.
For films, it is a great simplification to work with the
R d/2
sheet current J(r) = −d/2 dz j(r, z), r = (x, y), in stead
of the current density j. This is justified as long as thickness, d, is small compared to the in-plane dimensions but
much larger than the London penetration depth, λ. Finite λ can be handled with a small modification of the
algorithm.21 Since the current is conserved, ∇ · J = 0, it
can be expressed as J = ∇ × ẑg, where g = g(r) is the
local magnetization.20
For single-connected thin films the Biot-Savart law can
be formulated as
Z
d2 r′ Q(r, r′ , z) g(r′ ), (2)
Bz (r, z)/µ0 = Ha +
A
where Ha is the applied magnetic field, and A is the
sample area. The kernel Q represents the field generated
1 2z 2 − (r − r′ )2
.
4π [z 2 + (r − r′ )2 ]5/2
(3)
We discretize the kernel on an equidistant grid with grid
points ri and weights w and obtain21
Qij = δij
Ci /w +
X
l
qil
!
− qij ,
(4)
where qij = 1/4π|ri − rj |3 for i 6= j and qii = 0. The
function C depends on the sample geometry. It is given
as
Z
dr′2
.
(5)
C(r) =
′ 3
outside 4π|r − r |
The time evolution of g comes from the inverse of
Eq. (2),
ġ(r) =
Z
d2 r′ Q−1 (r, r′ ) [Ḃz (r′ ) − µ0 Ḣa ],
(6)
A
where Q−1 for discrete problem is the matrix inverse of
Eq. (4). Ḃz is given from Faraday’s law as
Ḃz (r) = − (∇ × E)z = ∇ · (
ρ
∇g),
dµ0
(7)
with ρ = ρ0 |∇g/Jc |n−1 obtained from Eq. 1. The righthand side of Eq. (6) is expressed only via g and Ha so
that time evolution of g can be found by integrating the
equation numerically.
3
FIG. 3: The current stream lines for the bulk Bean model
(left) and for film with finite n (right).
B.
Superconductors with holes
For macroscopic, arbitrarily shaped, single-connected,
type-II superconducting films flux dynamics is fully described by Eq. (6). This basic equation can also be used
for multiply connected samples, but in this case one needs
to specify the dynamically changing value of g at the hole
boundary. In Refs. 9 and 22 this value was set to the
lowest value of g along the hole perimeter. This method
turned out to be quite feasible but unfortunately it cannot reproduce the discussed pattern of Fig. 1. Moreover,
it also introduces unphysical net flux into the hole before
the flux front has reached it.
A completely different approach is to consider the holes
as part of the sample, but ascribe to them a large Ohmic
resistance or a strongly reduced Jc .23 Then, Eq. (6) applies to the whole sample including the holes, while the
material law, Eqs. (1) and (7), is spatially non-uniform.
This approach is physically justified but numerically challenging due to huge electric field gradients. In addition,
there still remain small but non-zero currents flowing
within the holes.
In this work we propose a new approach that does not
require any additional assumptions, though requires a
larger computational time. In this approach the integration in Eq. (6) is extended over the whole sample area
including the holes. Then the dynamics of g is described
by the equation
Z
d2 r′ Q−1 (r, r′ ) [Ḃz(s) (r′ ) + Ḃz(h) (r′ ) − µ0 Ḣa ],
ġ(r) =
A
(8)
where A is the sample area including the hole. Here we
(h)
(s)
(h)
presented Ḃz as a sum Ḃz + Ḃz where Ḃz is nonzero
(s)
only in the hole and Ḃz is nonzero within the supercon(s)
ducting areas. Ḃz is calculated in the straightforward
(h)
way using Eq. (7). The other term, Ḃz , is defined by
two conditions. The first condition is that current does
not flow beyond the superconducting areas, i.e., ġ is constant within the hole. This constant is determined by
the second condition, that the total change of magnetic
flux inside the hole is related to the electric field at its
FIG. 4: Simulation results for a strip with a hole: the current
stream lines (top), Bz contour lines (middle), and E contour
lines (bottom). Note that the electric field is greatly enhanced
in the channel between the hole and the edge;24 Ha /Jc =
0.3 and 0.9. The remaining parameters are the same as for
Fig. 2.
boundary through Faraday’s law,
Z
Z
2
d r Ḃz = −
hole
dl · E.
(9)
hole edge
(h)
In order to find a Ḃz that satisfies the two conditions
(h,0)
we use an iteration scheme. An initial guess, Ḃz , is
substituted into Eq. (8) to find ġ (h,0) inside the hole. The
next approximation is found as
Z
d2 r′ Q(r, r′ )ġ (h,0) (r′ ) + K,
Ḃz(h,1) (r) = Ḃz(h,0) (r) −
hole
(10)
where the constant K is chosen so that Eq. (9) is satis(h,1)
fied. Ḃz
is then inserted into Eq. (8) to find ġ (h,1) .
(h,1)
This ġ
is in general non-uniform, but when the procedure is repeated ġ (h,n) becomes more uniform with every new iteration. A smart choice of the initial guess
(h,0)
of Ḃz
is the final value at the previous time step,
(h,0)
(h,n)
Ḃz (r, t) = Ḃz
(r, t − ∆t). With this choice only
a couple of iterations are sufficient.
4
R
FIG. 6: Total flux inside the hole, Φh = hole d2 r Bz , as a
function of Ha , for various distances s from the edge. For low
fields Φh is zero since the flux front has not reached the hole
yet. For high fields Φh grows linearly with Ha since the strip
is saturated with J ≈ Jc . Hole radii are r0 /a = 0.1. The
remaining parameters are the same as for Fig. 2.
FIG. 5: Profiles of Bz , J, g and E through y = 0 for a strip
with a hole. The curves correspond to applied fields Ha /Jc =
0.1, 0.3, 0.5, 0.7, and 0.9, and Ec = ρ0 Jc /d. The remaining
parameters are the same as for Fig. 2.
Note that the scheme presented here is in no way
bound to the discrete formulation of the kernel, Eq. (4).
It can be used for any formulation as long as both the forward and inverse relations between ġ and Ḃz are known.
Further mathematical details are in appendix A.
III.
STRIP WITH A CIRCULAR HOLE
In this section, Eq. (8) is solved for an infinite superconducting strip in linearly increasing magnetic field.
The strip is modeled using periodic boundary conditions
in the y-direction, and examples of magnetic field distributions are given in Fig. 2. In the upper part one observes regular flux penetration with maximum of Bz at
the edges. Flux penetration in the lower half is strongly
affected by the presence a small, circular, non-conducting
hole. Note that the flux distribution is perturbed in a region that significantly exceeds the hole dimensions.
The left image of Fig. 2 corresponds to a small field
for which the flux front has not reached the hole yet. In
this case the hole shows up as a field dipole, in agreement
with magneto-optical observations; cf. Refs. 17 and 25.
Namely, there is positive field at the farther side of the
hole and a negative field at the side closer to the film
edge. The negative fields shrinks when the flux front
reaches the hole, but the asymmetry of the flux distribution inside the hole remains, as seen in the right image.
As expected, the front becomes distorted so that the penetration is significantly deeper in the vicinity of the hole.
For the full penetration image of Fig. 2 one also clearly
see the d-lines as dark line originating at the hole and
directed towards the middle of the strip. Such d-lines
were first described in Ref. 11 within the Bean model
framework and they are called d-lines because current
changes direction discontinuously there. The discontinuity is most clearly seen in current stream line plot of
Fig. 3 (left). For the Bean model, d-lines from circular
holes are parabolic and by convention d-lines from small
holes inside superconductors are often called parabolas.
In the presence of flux creep the change of current direction is smeared as follows from Fig. 3 (right). However,
the d-lines are still clearly visible, at least for n ≫ 1.
Comparing the two panels of Fig. 3 we notice a qualitative difference between the current flow in the bulk
Bean model and for films under the creep. In the Bean
model the current density is everywhere constant and all
the current that is blocked by the hole turns towards the
strip center. The region between the hole and the edge
is hence unaffected by the presence of the hole. For film
creep dynamics this is no longer true and a certain fraction of the current will force its way here. As a result,
Simulation
Experiment
Edge
Geometry
Slit
the current density is enhanced which is seen as denser
stream lines in Fig. 3 (right). Since the stream lines bend
they create the feature visible in the flux distribution of
Fig. 2: a slightly darkened region starting at the hole
and widening towards the edge. This feature can also be
observed experimentally; cf. Fig. 1. It was analysed in
detail in Ref. 17 and interpreted in terms of the Bean
model as additional parabolic d-lines. Our experiment
and simulations suggest a different interpretation. We
believe that one should speak about an area of reduced
flux density rather than new d-lines. Moreover, the appearance of this area is due to locally enhanced current
density, hence it cannot be explained within the Bean
model, postulating J = Jc . An enhanced current density also implies a strongly enhanced electric field. This
is clearly seen in Fig. 4 showing the contour lines of E.
A locally enhanced E means that there is an exceptionally intensive traffic of magnetic flux through the channel
between the edge and the hole. The channel width is approximately given by the hole diameter, but increases
slightly towards the edge. The width depends in general on the distance to the edge and the creep exponent
n. Both larger distance and smaller n tend to make the
channel wider.
After arrival to the hole, the flux is further directed in
the fan-shaped region between the d-lines. Electric field
within this region is also relatively high, again implying
an intensive flux traffic. This situation is similar to the
case of a semicircular indentation at the sample edge considered in Refs. 16, 26, and 27. The hole thus strongly
rearranges trajectories of flux flow.
The above discussion is further confirmed by profiles
of Bz , J, g, and E through the line y = 0 shown in
Fig. 5. The J profiles show features commonly observed
in strips,28 i.e., plateaus with values ∼ Jc in the penetrated regions and shielding currents with J < Jc in the
Meissner regions. The profiles show clearly the enhanced
J and E between the edge and the hole. It is also interesting to see the negative Bz for low values and how
the negative values gradually vanish when the main flux
front gets in contact.
R Fig.2 6 shows the total flux in a circular hole, Φh =
hole d r Bz , as a function of the applied field Ha for
various distances between the hole and the edge. In the
beginning, Φh ≈ 0, until the main flux front is in contact
with the hole. Then it starts to increase. For high fields
Φh grows almost linearly with Ha at a universal growth
rate determined by the hole area. The linear rate is not
just the case for small holes in strips, but has also been
found for e.g. ring geometry.29 Note that for small fields
Φh is close to, but not exactly zero. The reason is the creation of two additional flux fronts: one positive towards
the flux-free region and one negative towards the edge,
as also seen in Fig. 5. Only when integrating Bz over a
larger area that includes this additional penetration one
finds that the total flux is exactly zero.30 This integral
is also a good consistency check of the boundary condition implementation, since a wrong value of g (h) tends to
Slit
5
FIG. 7: A square with two slits (only the lower half is shown).
Top: Sample sketch, experimental magneto-optical image of
YBa2 Cu3 Ox film, and simulated magnetic field distribution.
Bottom: current stream lines, Bz and E contour lines at
Ha /Jc = 0.3 and 0.9, with n = 19 and µ0 Ḣa = ρ0 Jc /ad.
Note the strongly enhanced J and E between the slit and the
edge and the complicated set of d-lines at full penetration.
introduce a net, unphysical flux in the hole.
IV.
SQUARE WITH TWO SLITS
This section presents results of simulations of a square
superconducting film with rectangular slits. The sample geometry is chosen to reassemble one particular
YBa2 Cu3 Ox film and comparison of the simulation with
a magneto-optical image of the sample is shown in Fig. 7.
The experimental film thickness is 250 nm, and side
lengths are 2.5 mm. The two slits have been cut out
6
with a laser. Details of the film preparation can be found
elsewhere.31
The experiment and simulations show a great similarity both in large and in the details. The flux density is
considerably enhanced everywhere along the slit edges,
and reaches the maximal values at the upper corners.
Our main result found for circular holes holds true also
for rectangular slits. Namely, we again find a distinct
dark region starting at the slit and widening towards the
edge. It can be attributed to the over-critical current
density in that region, which is clearly seen in the current stream-line plot. A new result for slits is a slightly
brightened regions near the upper corners that appear
due to concave current turns. A similar situation arises
in superconductors of some other shapes having concave
corners, e.g. in crosses.26
There also exist a few minor discrepancies between flux
distributions obtained in the simulations and in the experiment of Fig. 7. The most notable is the details of the
region of reduced Bz at the side of slits close to the edge.
The values of Bz appear to be less in the simulation than
in experiment. This might be caused by simplifications,
like the disregarded B-dependency of the material law or
the simplification of using the sheet current in stead of
the true current density.
V.
SUMMARY
We have proposed a new method for treating boundary
conditions of non-conducting holes inside macroscopic,
type-II superconducting films. The key point is to reconstruct the at first unknown Ḃz inside the holes, at each
time step of the simulation. The method is capable of
handling any number of holes of arbitrary shape.
The simulations of flux dynamics assuming a material
law E ∼ j n reproduce very well flux distributions observed by magneto-optical imaging in YBa2 Cu3 Ox films,
for circular holes as well as rectangular slits. In particular, they demonstrate a significant enhancement of
current density in the region between a hole (slit) and
the edge leading to a more intensive traffic of flux. This
region appears darker in magneto-optical images due to
a slight bending of current stream lines.
We thank C. Romero-Salazar and Ch. Jooss for fruitful discussions and M. Baziljevich for experimental data
1
2
3
4
V. V. Moshchalkov, M. Baert, V. V. Metlushko, E. Rosseel,
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on Fig 7. This work was supported financially by The
Norwegian Research Council, Grant No. 158518/431
(NANOMAT) and by FUNMAT@UIO.
APPENDIX A: NUMERICAL DETAILS
The simulations are run on a N × N square grid.
The creep exponent and the ramp rate are n = 19 and
µ0 Ḣa = ρ0 Jc /ad, a regime in which creep is low, but
not negligible. Changing n would only do quantitative
changes to the results. For small exponents the plateaus
of current profiles, like Fig. 5, would be less flat and there
would also be more current compressed between the holes
and the edge.
The main limiting factor of the simulations is memory consumption since the kernel matrix Q, Eq. (4),
has dimension N 2 × N 2 . The simulations are run with
N = 100 grid points, which yields a kernel matrix of
dimension 5000 × 5000, when the sample symmetry has
been exploited.15
The kernel Q in Eq. (4) depends explicitly on the sample shape. Since the strip is infinite in the y-direction,
Q should be computed via an infinite sum over strip segments. However a good approximation is achieved with
only one segment on each side of the “main” strip. The
strip segments further away contain zero net current and
the dipole like character means that they have a negligible effect. A good accuracy of this approximation
was checked by comparing the Meissner state width, b,
obtained for very high n with the analytical film Beanmodel result,32 b = a/ cosh(πHa /Jc ).
The reconstruction of Ḃz inside the hole, Eq. (10),
need not use the full Q from Eq. (4). The best is to
use a smaller kernel, Q̃, also generated with Eq. (4), but
only including points inside the hole. Fast convergence
of Eq. (10) is achieve by ignoring currents at the hole
perimeter, which means that Q̃ should use C(r) = 0.
The most difficult numerical problem in our method
is the calculation of the electric field at the boundary
in Eq. (9). The electric field is given by the power law,
Eq. (1), and is largely fluctuating between neighboring
grid points. A stable way to handle this is to take the
average of only the most significant values of E and use
2πr and πr2 for the hole circumference and area.
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6
7
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E inside the hole cannot be found from the material law
and is simply put to zero in the plots. The correct E inside
the hole must be found from Faraday’s law.33 .
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Shantsev, J. I. Vestgården, and T. H. Johansen, Physica
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PAPER IV
Nonlinearly driven Landau-Zener transition with telegraph noise
J. I. Vestgården,1 J. Bergli,1 and Y. M. Galperin1, 2, 3
1
Department of Physics and Center for Advanced Materials and Nanotechnology,
University of Oslo, P. O. Box 1048 Blindern, 0316 Oslo, Norway
2
A. F. Ioffe Physico-Technical Institute of Russian Academy of Sciences, 194021 St. Petersburg, Russia
3
Argonne National Laboratory, 9700 S. Cass Av., Argonne, IL 60439, USA
We study Landau-Zener like dynamics of a qubit influenced by transverse random telegraph noise.
The telegraph noise is characterized by its coupling strength, v and switching rate, γ. The qubit
energy levels are driven nonlinearly in time, ∝ sign(t)|t|ν , and we derive the transition probability
in the limit of sufficiently fast noise, for arbitrary exponent ν. The longitudinal coherence after
transition depends strongly on ν , and there exists a critical νc with qualitative difference between
ν < νc and ν > νc . When ν < νc the end state is always fully incoherent with equal population of
both quantum levels, even for arbitrarily weak noise. For ν > νc the system keeps some coherence
depending on the strength of the noise, and in the limit of weak noise no transition takes place. For
fast noise νc = 1/2, while for slow noise νc < 1/2 and it depends on γ. We also discuss transverse
coherence, which is relevant when the qubit has a nonzero minimum energy gap. The qualitative
dependency on ν is the same for transverse as for longitudinal coherence. The state after transition
does in general depend on γ. For fixed v, increasing γ decreases the final state coherence when
ν < 1 and increase the final state coherence when ν > 1. Only the conventional linear driving is
independent of γ.
PACS numbers: 03.65.Yz,85.25.Cp,05.40.Ca
I.
INTRODUCTION
Driven quantum systems are exceedingly more complicated to study than stationary systems, and only few
such problems have been solved exactly. An important
exception is the Landau-Zener transitions.1–3 In the conventional Landau-Zener problem, a two-level system is
driven by changing an external parameter in such a way
that the level separation ∆ is a linear function of time,
∆(t) = at. Close to the crossing point of the two levels
an inter-level tunneling matrix element g lifts the degeneracy in an avoided level crossing. When the system is
initially in the ground state the probability to find it in
the excited state after the transition is exp(−πg 2 /2a).
Hence, fast rate drives the system to the excited state,
while the system ends in ground state when driven slowly.
The Landau-Zener formalism was originally developed
for molecular and atomic physics, but has since then been
applied to various systems and many generalizations of
the linearly driven two-level system exists, like avoided
level crossing of multiple levels,4,5 repeated crossings,6
non-linear model,7 and non-linear driving functions.8
In connection with decoherence of qubits there has recently been increased interest in Landau-Zener transitions in systems coupled to an environment. This problem is both of theoretical interest and of practical importance for qubit experiments.9 The noise affects the
qubit in two ways. First, it destroys coherence by random additions to the phase difference of the two states
(dephasing). Second, it causes transitions and alters the
level occupation (relaxation). The noisy Landau-Zener
problem has been discussed by several authors10–14 both
for quantum and classical environments. In this work we
will study classical noise processes. In particular, we will
use a random telegraph process as the noise source. This
allows us to study the effect of noise with long correlation
time (slow, or non-Gaussian noise). In the limit of short
correlation times we will recover the results of Pokrovsky
and Sinitsyn15 who have considered this problem in the
limit of fast noise. An important result of their analysis
was that there is a characteristic time scale, tnoise , during
which the noise is active. If this time scale is long compared to the time of the Landau-Zener transitions, tLZ ,
dynamics can be separated in a noise-dominated regime
for long times and a pure, noiseless Landau-Zener transition for short times. This allows one to study separately
transitions driven purely by noise and the usual LandauZener transitions driven by the tunneling amplitude g.
We will follow this approach, which simplifies the problem considerably.
Most works on noisy Landau-Zener transitions are
mainly concerned with transition probabilities. However, in the case of an open system it is also interesting to study the amount of decoherence, or purity, of
the state after the transition is passed. In terms of the
Bloch vector, the transition probability is given by the
z-component of the vector whereas the purity is given by
its length. By generalization from the stationary case, it
is clear that longitudinal noise (noise in the level spacing ∆) will cause dephasing at all times, and the final
state will always be on the axis of the Bloch sphere, i.e.,
the x- and y-components of the Bloch vector decay to
zero. For transverse noise (noise in the anticrossing energy g) the situation is less evident since the effect of the
noise is reduced by the factor g/∆. When ∆ increases
sufficiently fast as function of time one can in a sense
‘run away’ from the noise, and the final state will not
decohere maximally. This motivates us to study the ef-
2
fect of nonlinear time dependences for the level splitting,
similar to those considered in Ref. 8 for Landau-Zener
transitions without noise. In particular, we will study
power-law driving functions, ∆ ∼ sign(t)|t|ν , and we will
find that there exists a critical νc such that the system is
completely decohered for ν < νc even for arbitrarily weak
noise coupling. For ν > νc some coherence is retained.
The critical νc will depend on the correlation time of the
noise.
II.
A.
MODEL
Hamiltonian
Consider a solid state qubit, e.g., a Josephson charge
qubit.9,16,17 The qubit is modeled as a two-level system
and it couples to environment through a randomly fluctuating addition χ(t) on its off-diagonal terms. Let us
here only consider dynamics for one realization of χ(t),
while in next section we will use the particular model of
random telegraph noise to derive master equations for
the noise averaged quantities.
The Hamiltonian is
H=
1
1
∆ν (t) σz + [g + χ(t)] σx
2
2
(1)
where σx and σz are Pauli matrices, ∆ν (t) is the diagonal
splitting, g is the minimal energy gap at the avoided level
crossing.
The interesting dynamics comes from a power-law
time-dependency
∆ν (t) = αν |αν t|ν sign(t)
(2)
with sweep rate αν and exponent ν. Linear sweep and no
noise give exactly the Landau-Zener dynamics. However,
our focus will be on entirely noise-driven transition for
any exponent.
From here and throughout this work the quantum state
is described by the Bloch vector r ≡ (x, y, z). The Bloch
vector is is given from the density matrix ρ as
x = 2 Re ρ12 ,
y = 2 Im ρ12 ,
z = ρ11 − ρ22 .
B.
Telegraph noise
The noise model applied in this work is random telegraph noise. Such noise occurs when defects create
bistable traps, atomic or electronic, in solids, and is
assumed18 to be a basic source for various kinds of high
and low-frequency noise.19 For example, a large number
of fast fluctuators with a narrow distribution of switching rates give Gaussian white noise. A broad distributions of switching rates can, on the contrary, give rise
to non-Gaussian, 1/f noise.20,21 In experiments on solid
state qubits, the low-frequency 1/f noise is often the
dominant source of decoherence.22 For tiny devices, a
small number, or even single fluctuators, can be important. Relevant for transverse noise on Josephson
charge qubits telegraph noise characteristics has been
measured for electrons trapped in Josephson junctions,23
for intrinsic Josephson junctions in granular high-Tc
superconductors,24 and for trapped single flux quanta.25
If the bistable system, or fluctuator, is more strongly
coupled to its surroundings than to the qubit we can consider its dynamics to be independent of the qubit, and
it will act as a classical noise source, driven by its environment. With this approximation, the effect of the fluctuator on the qubit appears through a randomly switching addition ±v to the tunneling energy. The constant,
v, represents fluctuator-qubit coupling strength, which
will be called noise strength for short. We assume the
switchings between the two fluctuator states to be independent, random events. The rates of random switching is assumed to be the same between both fluctuator
levels, γ+− = γ−+ = γ. This holds when the fluctuator level-spacing is small compared to the temperature.
Our fluctuator model is thus a stochastic process and the
probability Pk to switch k times in a time interval t is
given by the Poisson distribution,
Pk =
(γt)k −γt
e
.
k!
The telegraph process has the property χ(t)χ(0) = ±v 2 ,
where the + and − sign are for a even and odd number
of switches, respectively. Hence, the autocorrelator is
S(t) = hχ(t)χ(0)i =
∞
X
χk (t)χk (0)Pk ,
k=0
(3)
=v
For a pure quantum system the vector r is a unit vector.
Under the influence of noise its average value is in general
less than unity.
The dynamics of r is given by the Bloch equation,
(5)
2
∞
X
k=0
k
−γt
k (γt)
(−1)
k!
e
(6)
2 −2γt
=v e
,
(4)
for t > 0. Correspondingly, the cosine transform of
Eq. (6) (the noise power spectrum) is a Lorentzian
Z ∞
2γ
. (7)
Ŝ(ω) =
dt S(t) cos(ωt) = v 2
(2γ)2 + ω 2
0
analogous to a spin precessing in magnetic field B(t) =
(g +χ(t), 0, ∆ν (t)). We use units where ~ = 1 throughout
this work.
The noise power spectrum is important since all results
for fast noise can be expressed by this function.
It must be noted that for many qubit experiments
the environment cannot be considered as classical and
ṙ = −r × B,
3
1
0.8
0.6
zp
a quantum description of noise is necessary.26 The SpinBoson model was discussed in Ref. 17 for stationary system and in Ref. 13 in connection with Landau-Zener transitions. Ref. 27 has developed a model for fluctuating
charges at finite temperature. Random telegraph noise
is the high temperature limit of this model.
0.4
C.
Master equations
0.2
We will now average Eq. (4) over the noise and derive
master equations for a qubit coupled to one random telegraph process. The quantum state is now only known
with a certain probability and we need to operate with
averaged quantities rather than the pure quantum states.
The average value of r is
Z
rp = hri = d3 r p(r, t) r.
(8)
where p = p(r, t) is the probability of being in Bloch state
r at time t.
For the particular model of one random telegraph process there are two possible values of the effective magnetic
field acting upon the qubit according to Eq. (4):
B± = B0 ± v .
(9)
where v is a constant vector. Here B0 (t) = (g, 0, ∆ν (t))
controls the time evolution of the quantum mechanical
system. We will now derive the set of master equations.
The derivation is in fact valid for any two-level system
coupled to one fluctuator in arbitrary direction, not just
Landau-Zener like dynamics and transverse noise. The
derivation follows Refs. 28 and 29.
Let p = p(r, t) be the probability to be in r at time t.
Now split p(r, t) = p+ (r, t) + p− (r, t) where p+ (r, t) and
p− (r, t) are the probabilities to be in state r at time t
under rotation around B+ and B− , respectively.
The master equations for p+ and p− are
p+ (r, t + ǫ) = αp+ (r − δr+ , t) + βp− (r − δr− , t),
p− (r, t + ǫ) = αp− (r − δr− , t) + βp+ (r − δr+ , t),
where ǫ is a small time change and α and β are the staying
and switching probabilities, respectively. When ǫ ≪ γ
we can neglect multiple switchings, and Eq. (5) can be
expanded to give α ≈ P0 ≈ 1 − γǫ and β ≈ P1 ≈ γǫ. The
spatial changes δr± represent the vector’s displacements
during the time interval ǫ. This is given from the Bloch
equation, Eq. (4), as δr± = −r × B± ǫ. Expanding to
first order in ǫ gives
ṗ+ = −γp+ + γp− + (r × B+ ) · ∇p+ ,
ṗ− = −γp− + γp+ + (r × B− ) · ∇p− .
The probabilities enableR us to define equations for the
averaged quantities r± = d3 r rp± ,
ṙ+ = −γr+ + γr− − (r+ × B+ ),
ṙ− = −γr− + γr+ − (r− × B− ).
(10)
ν=2
ν=1
ν=0.7
0
-400 -300 -200 -100
0 100 200 300 400
ανt
FIG. 1: The zp (t) as a function of time for fast noise, γ/αν = 2
and v/αν = 0.5. The transition time extends significantly
with decreasing ν.
The quantities r+ and r− are just auxiliary quantities
and the final master equations are expressed by rp =
r+ + r− and rq = r+ − r− . The quantities normally measured in experiment are those quantities averaged over p,
and rp are the averaged components of the Bloch vector.
Isolating rp and rq yields
ṙp = −rp × B0 − rq × v,
ṙq = −2γrq − rq × B0 − rp × v.
(11)
The above equations are exact for one telegraph process.
Compared to the noiseless case, the number of equations
rise from two (i.e., three equation and constraint of |r| =
1) to six equations. Adding more fluctuators, the number
of equations will grow exponentially.28
D.
Master equations for simplified problem
Let us now study the simplified problem of entirely
noise-driven transition, i.e., g = 0. In this case the set of
six equations, Eq. (11), decouple in two sets of equations
in (xp , yp , zq ) and (xq , yq , zp ), respectively. A system initially prepared in one energy eigenstate has xp = yp = 0.
Assuming also the initial state of the fluctuator to be
random we have zq (−∞) = 0, which means that xp and
yp remain zero as long as g = 0. Thus coherence only
relays on zp and we will for the following concentrate on
the set (xq , yq , zp ). The master equations are

 


ẋq
−2γ −∆ν 0
xq
 ẏq  =  ∆ν −2γ −v   yq  .
żp
0
v
0
zp
(12)
4
1
0.8
zp(∞)
intervals, the relaxation rate in each interval being given
by the usual expression for the static case. This can only
be done in the limit of fast noise.
For the particular model of random telegraph noise Ŝ
is given by Eq. (7) and Eq. (15) reads as
v=0.01
v=0.1
v=0.5
v=1
0.6
2γ
żp
= −v 2
.
zp
(2γ)2 + ∆2ν (t)
0.4
0.2
0
0.5
1
ν
1.5
FIG. 2: The zp (∞), Eq. (18), as function of ν, for fast noise;
γ/αν = 10. The weak noise has a strong ν-dependency near
ν = 1/2.
Isolating zp yields the integral equation
żp = −
=−
t
Z
dt1 cos(θ(t) − θ(t1 )) S(t − t1 ) zp (t1 )
−∞
Z ∞
dt2 cos(θ(t) − θ(t − t2 )) S(t2 ) zp (t − t2 ),
0
(13)
where
θ(t) =
Z
t
dt′ ∆ν (t′ ) =
0
1
|αν t|ν+1
ν+1
(14)
and S(t) given by Eq. (6). The integral equation,
Eq. (13), is exact for one telegraph process, and valid for
all transition rates. The equation is the same as found in
Ref. 15 for any fast noise source. Hence, all conclusions
drawn from Eq. (13) in the limit γ → ∞ are also valid
for any Gaussian noise source.
III.
FAST NOISE
With fast noise we mean finite but large γ, γ ≫ αν .
Then the relevant contributions in the integral of Eq. (13)
are for small t2 . Series expansions in t2 yields
Z ∞
żp
≈−
dt2 cos(∆ν (t)t2 ) S(t2 ) = −Ŝ(∆ν (t)). (15)
zp
0
The solution is
Z
zp (t) = exp −
t
−∞
dt Ŝ(∆ν (t )) ,
′
′
(16)
with noise power spectrum Ŝ. Recalling17 that the relaxation rate of a qubit without driving is Γrelax = Ŝ(E) at
the qubit level spacing E we can understand the above
expression as the total relaxation over many short time
(17)
The full integrated Eq. (17) is expressed through hypergeometric functions, which will not be written here. A
numerical solution is plotted in Fig. 1 for various exponents. It illustrates that the fast noise curves are smooth
and all fluctuations are averaged out. Also, it shows that
transitions times get longer for decreasing ν.
The most interesting quantity, however, is the value at
infinity which for ν > 1/2 is
#
"
1/ν−1
π/2ν
v 2 2γ
.
(18)
zp (∞) = exp −2 2
αν αν
sin(π/2ν)
This equation makes it possible to explore how the final
state depends on v, γ, and ν.
For ν < 1/2 the integral of Eq. (17) diverges and
we get zp (∞) = 0, independently of v and γ. When
zp (∞) = 0 both levels are occupied with same probability
and this represents a fully incoherent state. The fact
that the result is independent of v means that arbitrarily
weak noise destroys coherence completely. This is similar
to a stationary system where noise always dominates at
long times. The result is actually a bit surprising. It is
obvious that a static system finally looses all coherence.
However, in this case the energy levels split by up to
square root of time and even this is not enough to avoid
total decoherence. For ν > 1/2 the results are no longer
independent of v and γ. In this sense one can say that
the regimes for ν < 1/2 and ν > 1/2 are qualitatively
different. Thus we identify the critical νc = 1/2 in the
limit of fast noise.
Fig. 2 shows zp (∞) as a function of ν. For decreasing
v the change near ν = 1/2 get sharper and in the limit
v → 0 it approaches a step function of ν.
Another interesting feature of Eq. (18) is how zp (∞)
changes with increasing γ. For ν < 1 increasing γ means
that zp (∞) decreases and goes to zero in the extremely
fast noise limit, γ/αν → ∞. In other words, faster noise
reduces end state coherence. The opposite is the case for
ν > 1. Then faster noise increases the end state coherence and in fact zp (∞) → 1 when γ/αν → ∞. This behavior is to some extent counterintuitive since one could
initially expect faster noise would always decrease coherence. The linear driving is truly a special case since
zp (∞) is independent of γ for ν = 1. Note that in Ref. 15
where the case ν = 1 was considered, the limit γ → ∞
was taken together with the limit v → ∞ in such a way
that v 2 /γ remained constant. In their case, zp (∞) depends on γ and goes to 0 when γ → ∞.
From the denominator of Eq. (17) one can identify
a time scale characteristic for the action of the noise,
5
1
0.8
0.5
0.6
zp
zp(∞)
1
v=0.5
v=0.1
v=0.01
0
0.4
-0.5
0.2
-1
-30
0
0.2
0.3
0.4
ν=0.7
ν=1
ν=2
0.5
-20
-10
ν
FIG. 3: The zp (∞) as function of ν for weak and slow noise;
γ/αν = 0.1. Obtained by numerical integration of Eq. (13).
The plot shows a critical value of νc ≈ 0.2 which is less than
the value for fast noise seen in Fig. 2.
1/ν
tnoise = α−1
. Thus tnoise increases with inν (2γ/αν )
creasing γ and decreasing ν. For very large times,
t ≫ tnoise , the z(t) will approach its end value as power
of time. Integration of Eq. (17) in this limit yields the
asymptotic solution
!
2
v
1 2γ
1−2ν
,
(αν t)
zp (t) = zp (∞) 1 +
2ν − 1 αν αν
(19)
with zp (∞) given by Eq. (18). Eq. (19) illustrates again
the message of Fig. 1, namely that convergence gets
slower for decreasing ν and near the critical value of
ν = 1/2 the transition is very slow. For ν < 1/2, the
expansion, Eq. (19), is not valid.
For the important linear case there is also a nice explicit solution of Eq. (17) for all times,
2 2 v π
α1
zp (t) = exp − 2
+ arctan
t
,
(20)
α1 2
2γ
in which the end state simplifies to
2
zp (∞) = e−π(v/α1 ) .
IV.
(21)
SLOW AND WEAK NOISE
Now we will study the influence of one slowly varying
telegraph process, γ . αν in the limit of weak noise,
v ≪ αν . We start with Eq. (13), which is exact for both
fast and slow telegraph noise. A series expansion in v/αν
yields
Z ∞ Z t
zp (∞) ≈ 1 − v 2
dt
dt1 cos[θ(t) − θ(t1 )] e−2γ(t−t1 ) ,
−∞
−∞
(22)
with θ defined in Eq. (14).
0
ανt
10
20
30
FIG. 4: The zp (t) as a function of time for slow and strong
noise; γ = 0 and v/αν = 1. This case is mathematically
equivalent to a Landau-Zener transition and rapid oscillations
are observed, unlike for fast noise, cf. Fig. 1.
In the extreme limit γ = 0 the equations are the same
as for the nonlinear Landau-Zener system without noise.
In this limit the integral Eq. (22) can be solved exactly,
recovering the results of Ref. 8:
zp (∞) = 1−2
v
αν
2 ν
− ν+1
(1 + ν)
Γ
1
ν+1
2
. (23)
where Γ is the gamma function. Eq. (23) shows only
weak ν-dependency. Thus the ν-dependency for a finite
and small γ will also be weak. The reason is that the
first order in γ will also be proportional to the a power of
the small factor (v/αν ). The expression Eq. (23) is only
approximately valid for small, but finite, γ, provided that
ν > νc .
Let γ be small but nonzero. As for fast noise we define
the critical νc by zp (∞) = 0 for all ν < νc independently
of v. Hence, νc can be identified by studying the convergence of Eq. (22). The integral diverges for ν < νc
and converges for ν > νc .30 We have not been able to
analyze the convergence of this integral analytically. Instead, Eq. (13) is solved numerically for a selected small
value of γ. This value gives a hint of how νc depends on
γ. Practically, νc is found by plotting zp (∞) as a function of ν for fixed γ/αν and decreasing values of v/αν .
The plot in Fig. 3 shows the expected behavior: zp (∞)
decreases when ν decreases, and goes to zero at finite ν,
even for very small values of v/αν . The behavior is analogous to the fast noise plot of Fig. 2, but the critical value
is significantly lower. For γ/αν = 0.1 we find νc ≈ 0.2.
This lowering is expected since νc = 0 for γ = 0. It must
be noted that there is large numerical inaccuracy for the
low ν in Fig. 3, since the integral is close to divergency.
6
1
γ=0.5
γ=0.1
γ=0.01
γ=0
zp(∞)
0.5
0
-0.5
-1
0
0.5
1
1.5
2
2.5
v /αν
FIG. 5: The zp (∞) as function of v/αν , for slow noise; γ/αν =
0.1 and ν = 1. Obtained by numerical integration of Eq. (13).
For γ/αν ≪ 1 the value zp (∞) can take any value between -1
and 1, not just those in the upper half of the Bloch sphere.
V.
equations, Eq. (11), with g 6= 0 is difficult, and in the
spirit of Pokrovsky and Sinitsyn15 we consider the case
where the characteristic time tLZ of the Landau-Zener
transition is much shorter than the time over which the
noise is effective, tnoise . In principle, this would mean
that we should study Eq. (11) in the case g = 0, starting at t = t0 , where tLZ ≪ t0 ≪ tnoise . As long as we
are only interested in determining the critical νc and not
in the precise value of the transition probability we can
therefore consider a Bloch vector starting in the equatorial plane of the Bloch sphere, r⊥ (0) = r0 and zq (0) = 0
at time t = 0. With this starting point we now assume
g = 0 and ∆ν again as an arbitrary power of time. From
Eq. (11) we have
 



ẋp
0 −∆ν 0
xp
 ẏp  =  ∆ν
0
−v   yp  ,
(24)
żq
0
v −2γ
zq
which should be compared with Eq. (12) for (xq , yq , zp ).
Isolating zq gives
SLOW AND STRONG NOISE
zq (t) = v
Let us again look at slow noise, γ . αν , but without
restrictions on v/αν . In particular we are interested in
the regime in which v is of same order of magnitude as
αν . In this regime the results depend strongly on the
actual values of αν , v, γ, and ν. The transitions are
quite sharp and give rapid oscillations after the transition, as seen in Fig. 4, contrary to the smoothened transitions of the fast noise, exemplified in Fig. 1. Unlike fast
noise the results depends strongly on γ also for ν = 1.
Fig. 5 shows how zp (∞) depends on v/αν for slow noise
and linear driving. One first thing to notice is that slow
noise, contrary to fast noise, can drive the system to the
other diabatic level. This is seen as zp (∞) < 0 in the
plot. Second, some curves for zp (∞) go through the center of the Bloch-sphere when v increases. The center of
the Bloch sphere represents maximum decoherence since
both states are occupied with equal probability. Consequently, under some conditions increasing noise strength
will also increase the system purity after transition.
VI.
TRANSVERSE COHERENCE
We will now discuss transverse
q coherence (phase coherence). It is given by r⊥ = x2p + yp2 , where xp and
yp are the transverse components of the Bloch vector. In
particular we are interested in the behavior for fast noise
and long times and see if we can identify a critical νc , as
we did for the longitudinal coherence.
Transverse coherence becomes relevant when there is a
nonzero anticrossing energy g in the Hamiltonian Eq. (1).
In that case the Bloch vector makes a rotation away
from the z-axis, acquiring nonzero r⊥ . This rotation is a
Landau-Zener transition. A full solutions of the master
Z
t
dt2 yp (t − t2 )e−2γt2 .
(25)
0
For fast noise and long times the important contributions
again come from small t2 . However, we must be careful
when doing expansions of yp (t) since the product ∆ν (t)t2
is not necessarily small. Let us define A(t) = xp (t) +
iyp (t) and explicitly take out the problematic, long-time
Rt
phase factor θ(t) = 0 dt′ ∆ν (t′ ):
A(t) = r⊥ (t) eiθ(t)+iϕ(t) ,
(26)
where ϕ(t) is a phase factor that varies less rapidly than
θ(t). Now expanding at long times t ≫ t2 ,
A(t − t2 ) ≈ r⊥ (t) eiθ(t)+iϕ(t)−i∆ν (t)t2 .
(27)
Inserting this into Eq. (25) and isolating r⊥ yields
v2
ṙ⊥
=−
sin(θ + ϕ)
2
r⊥
(2γ) + ∆2ν (t)
× {2γ sin(θ + ϕ) − ∆ν (t) cos(θ + ϕ)} .
(28)
At long times the sine and cosine functions oscillate
rapidly and we substitute these terms with their respective average values, giving the final equation for r⊥ ,
2γ
v2
1
ṙ⊥
=−
= − Ŝ(∆ν (t)).
r⊥
2 (2γ)2 + ∆2ν (t)
2
(29)
where Ŝ is the noise power spectrum.
Eq. (29) has the same form as Eq. (15) for zp , so
the whole discussion of Eq. (15) is in fact valid also for
Eq. (29). In particular this means they share the same
critical value. Thus νc = 1/2 for both transverse and
longitudinal coherence; when ν < νc = 1/2, the system
end state is fully incoherent no matter the value of v and
7
γ. We have not searched for the critical exponent of the
transverse coherence for slow noise, γ . αν . However, if
it exists it need not have the same numerical value as for
the longitudinal coherence.
The right hand side of Eq. (29) can be interpreted
as the instantaneous dephasing rate. In that case one
recovers17 the result from transverse noise without driving, Γϕ = Ŝ(E)/2, where E is qubit level spacing. The
relation to the instantaneous relaxation rate is Γϕ =
Γrelax /2; exactly the same as for the weak coupling limit
of a Gaussian noise source.
There is one more thing to note about Eq. (29). The
approximations needed to get to this expressions are
coarser than those for zp . In fact, the fast noise regime
of zp start at αν t & 1, while for r⊥ it must be truly large,
αν t ≫ 1.
VII.
SUMMARY
We have considered Landau-Zener like dynamics of a
qubit in noisy environment. The environment is modeled as transverse, classical, telegraph noise. The qubit
diagonal splitting is driven as a power law, ∆ν (t) =
αν |αν t|ν sign(t), with driving rate αν , where particular
attention has been on the role of ν.
An expression, Eq. (18), for the state after transition,
zp (∞), has been derived in the limit of fast noise, γ ≫ αν .
From this expression we have found that there exists a
critical νc = 1/2 such that the system looses all coherence
when ν < νc , even for very weak noise, v ≪ αν . When
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|z(∞)|, representing increased coherence.
Acknowledgments
This work was supported financially by The Norwegian
Research Council, Grant No. 158518/431 (NANOMAT).
The work of YG was partly supported by the U. S. Department of Energy Office of Science through contract
No. DE-AC02-06CH11357.
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In principle higher order terms in v/αν can diverge, even
if this term converges, so the convergence of this integral
gives only a lower bound on νc .