#  Hysteresis in Two Dimensional Arrays of Magnetic Nanoparticles

Manish Anand<EMAIL_ADDRESS>Department of Physics, Bihar National
College, Patna University, Patna-800004, India.

###### Abstract

We perform computer simulations to probe the magnetic hysteresis in a two-
dimensional ($L_{x}\times L_{y}$) assembly of magnetic nanoparticles as a
function of dipolar interaction strength $h_{d}$, temperature $T$, aspect
ratio $A_{r}=L_{y}/L_{x}$, and the applied alternating magnetic field’s
direction. In the absence of magnetic interaction ($h_{d}\approx 0$) and
thermal fluctuations ($T=0$ K), the hysteresis follows the Stoner and
Wohlfarth model, as expected. For weak dipolar interaction and substantial
temperature, the hysteresis has the dominance of superparamagnetic behaviour,
irrespective of the applied magnetic field’s direction and $A_{r}$.
Interestingly, the hysteresis curve has all the characteristics of
antiferromagnetic interaction dominance for $A_{r}\leq 6$ and considerable
dipolar interaction strength ($h_{d}>0.2)$, which is independent of applied
magnetic direction. When the magnetic field is applied along the system’s
shorter axis ($x$-direction), a non-hysteresis straight line is observed with
large $h_{d}$. In the case of the magnetic field applied along the long axis
of the sample ($y$-direction), ferromagnetic interaction dominates the
hysteresis for large $h_{d}$ and $A_{r}>6$. Irrespective of $h_{d}$ and
applied magnetic field’s direction, the coercive field $\mu_{o}H_{c}$ and
remanence $M_{r}$ are minimal for $A_{r}\leq 6$ and significant temperature.
They are found to increase with $h_{d}$ when $A_{r}$ is enormous. Remarkably,
the variation of hysteresis loop area $E_{H}$ as a function of these
parameters is the same as that of the coercive field variation. We believe
that the concepts presented in this work are relevant in various technological
applications such as spintronics and magnetic hyperthermia, in which such
self-assembled nanoparticle arrays are ubiquitous.

##  I Introduction

In recent years, two-dimensional arrays of magnetic nanoparticles (MNPs) have
received significant attention due to their unique magnetic properties and
potential technological applications Gallina and Pastor (2020); Alkadour  _et
al._ (2019); Bupathy  _et al._ (2019); Wang and Luijten (2019); Leo  _et al._
(2018); Wang  _et al._ (2008); Takagaki  _et al._ (2008); Kechrakos and
Trohidou (2002). From theoretical perspectives, they pose an exciting research
topic in which competing interactions, reduced dimensionality, shape
anisotropy, etc., lead to novel collective behaviours Puntes  _et al._ (2004);
Fraerman and Sapozhnikov (2002); Farhan  _et al._ (2013). On the other hand,
such systems are ideal candidates for diverse technological applications such
as spintronics, high-density data storages, magnetic hyperthermia, etc. Valdés
_et al._ (2021); Frey and Sun (2010); Sarella  _et al._ (2014); Karmakar  _et
al._ (2011); Mohammadpour and Majidzadeh-A (2020); Takagaki and Ploog (2005).

The magnetic properties of non-interacting MNPs are well understood Usov
(2010); Fu  _et al._ (2018). However, they are found to interact in an
assembly due to dipolar interaction Ovejero  _et al._ (2016); Zhao and Rinaldi
(2018). The dipolar interaction is long-ranged and anisotropic Barrera  _et
al._ (2021). Further, it can allow for both ferromagnetic and
antiferromagnetic couplings depending on the relative position of MNPs. As a
consequence, it has varied effects on systemic properties Kachkachi  _et al._
(2000); Anand (2021a); Plumer  _et al._ (2010). In some cases, the dipolar
interaction causes spins’s frustration, modifies energy barriers, and imparts
spin-glass-like properties. When the system size is smaller than the range of
dipolar interaction, it can create unconventional spins morphologies.
Depending on the lattice structure, there have been observations of magnetic
vortices, antiferromagnetic states, checkerboard patterns, etc. Löw  _et al._
(1994); Edlund and Jacobi (2010). Consequently, they exhibit unusual ground-
state and relaxation characteristics. For example, the dipolar interaction
elevates the blocking temperature even with weak interaction strength Luo  _et
al._ (1991). The dipolar interaction also affects the ground state spins
configuration depending on the geometry of the system. In the face-centred
cubic arrangement of MNPs, Luttinger et al. have found the ground state to be
ferromagnetic while it is antiferromagnetic in a simple cubic lattice
Luttinger and Tisza (1946). The dipolar interaction promotes antiferromagnetic
coupling with the square arrangement of MNPs MacIsaac  _et al._ (1996);
De’Bell  _et al._ (1997). On the contrary, the dipolar interaction promotes
ferromagnetic coupling in the assembly of magnetic moments arranged in
triangular lattice Politi  _et al._ (2006).

The dipolar interaction also drastically modifies the magnetic properties in
such cases Anand  _et al._ (2016); Landi (2014); Anand (2021b); Russier
(2001); Xue and Yan (2006); Figueiredo and Schwarzacher (2007). For instance,
V. Russier studied the magnetic properties, and orientational structure for
spherical assembly of MNPs arranged in square and hexagonal lattices at
temperature $T=0$ K Russier (2001). The dipolar interaction is found to induce
in-plane orientation of magnetic moments. Xue et al. studied the magnetic
hysteresis behaviour in a hexagonal array of MNPs with perpendicular
anisotropy Xue and Yan (2006). The shape of the hysteresis curve changes from
rectangular to a non-hysteresis straight line, increasing dipolar interaction
strength. For the triangular arrangement of MNPs, Figueiredo et al. observed
an increase in the blocking temperature and relaxation time with increased
dipolar interaction strength Figueiredo and Schwarzacher (2007). The dipolar
interaction also affects the hysteresis loop area in an assembly of MNPs, one
of the most valuable parameters for magnetic hyperthermia applications. Using
the molecular dynamics anisotropic diffusion model, Myrovali et al. have found
the dipolar interaction increases the hysteresis loop area in an assembly of
interacting MNPs Myrovali  _et al._ (2016). Using kinetic Monte Carlo
simulation, we have shown that dipolar interaction increases the hysteresis
loop area in a one-dimensional chain of MNPs Anand (2020). Using micromagnetic
simulations, Haase and Nowak reported that magnetic hyperthermia efficiency
decreases with an increase in dipolar interaction strength Haase and Nowak
(2012). The dipolar interaction also has a drastic effect on the tunnel
magnetoresistance, a useful quantifier for spintronics based applications
Black  _et al._ (2000); Tan  _et al._ (2009, 2010); Kechrakos and Trohidou
(2008); Yang  _et al._ (2006). For example, Tan et al. found the detrimental
effect of dipolar interaction on the tunnel magnetoresistance in square arrays
of MNPs Tan  _et al._ (2010). Kechrakos et al. and Yang et al. observed an
enhancement in the tunnel magnetoresistance because of dipolar interaction
Kechrakos and Trohidou (2008); Yang  _et al._ (2006).

The above studies clearly suggest that dipolar interaction plays an essential
role in determining the magnetic properties of self-assembled MNPs, which are
of profound technological importance. However, a systematic study of the
effect of dipolar interaction strength, temperature, the shape of the system,
and external magnetic field’s direction on the underlying system’s magnetic
properties have received little attention in the literature. These studies
could provide a concrete theoretical basis in choosing suitable system
parameters for various technological applications of interest. In the present
work, we perform kinetic Monte Carlo (kMC) simulations to study the magnetic
hysteresis in the two-dimensional assembly of MNPs as a function of various
parameters of interest. In particular, we perform a detailed study of the
dependence of magnetic hysteresis on dipolar interaction strength,
temperature, aspect ratio and direction of the applied magnetic field.

This paper is organized as follows. In Sec. II, we present the model for an
assembly of MNPs and discuss the kMC simulations briefly. The numerical
results will be presented in Sec. III. Finally, we summarize and discuss the
main results in Sec. IV.

##  II Theoretical Framework

We consider an assembly of spherical magnetic nanoparticles arranged in a two-
dimensional lattice in the $xy$-plane with system dimension $L_{x}\times
L_{y}$. The particle has a diameter $D$, and the lattice spacing is $a$, as
shown in Fig. 1(a). Each particle has a magnetic moment $\mu=M_{s}V$, $M_{s}$
being the saturation magnetization , and $V=\pi D^{3}/6$ is the volume of the
nanoparticle. Further, each nanoparticle is assumed to have uniaxial
anisotropy $\vec{K}=K_{\mathrm{eff}}\hat{k}$, where $K_{\mathrm{eff}}$ is the
uniaxial anisotropy constant, and $\hat{k}$ is the direction of anisotropy or
easy axis. We have assumed anisotropy axes to be randomly oriented. The energy
associated with such nanoparticle due to magnetocrystalline anisotropy is
given by Carrey  _et al._ (2011); Anand  _et al._ (2018)

$E_{K}=K_{\mathrm{eff}}V\sin^{2}\theta$ (1)

Here $\theta$ is the angle between anisotropy axis and magnetic moment.

MNPs also interact due to the long-ranged dipolar interaction Pei  _et al._
(2019). We can calculate the dipolar interaction energy $E_{\mathrm{dip}}$ in
an assembly of MNPs as Odenbach and Thurm (2002); Rosensweig (2013); Usov  _et
al._ (2017)

$E_{\mathrm{dip}}=\frac{\mu_{o}}{4\pi a^{3}}\sum_{j,\ j\neq
i}\left[\frac{\vec{\mu_{i}}\cdot\vec{\mu_{j}}}{(r_{ij}/a)^{3}}-\frac{3\left(\vec{\mu_{i}}\cdot\hat{r}_{ij}\right)\left(\vec{\mu_{j}}\cdot\hat{r}_{ij}\right)}{(r_{ij}/a)^{3}}\right].$
(2)

Here $\mu_{o}$ is the permeability of free space; $\vec{\mu}_{i}$ and
$\vec{\mu}_{j}$ are the magnetic moment vectors of $i^{th}$ and $j^{th}$
nanoparticle respectively, and $r_{ij}$ is the center-to-center separation
between $\mu_{i}$ and $\mu_{j}$. $\hat{r}_{ij}$ is the unit vector
corresponding to $\vec{r}_{ij}$. The corresponding dipolar field
$\mu_{o}\vec{H}_{\mathrm{dip}}$ is given by the following expression
Rosensweig (2013); Tan  _et al._ (2014)

$\mu_{o}\vec{H}_{\mathrm{dip}}=\frac{\mu\mu_{o}}{4\pi a^{3}}\sum_{j,j\neq
i}\frac{3(\hat{\mu}_{j}\cdot\hat{r}_{ij})\hat{r}_{ij}-\hat{\mu_{j}}}{(r_{ij}/a)^{3}}.$
(3)

As it is clearly evident from Eq. (2) and Eq. (3) that dipolar interaction
varies as $1/r^{3}_{ij}$, we can define the strength of dipolar interaction
$h_{d}=D^{3}/a^{3}$ Tan  _et al._ (2010). So $h_{d}=1.0$ is the largest
dipolar interaction strength, and $h_{d}=0$ can be termed as the non-
interacting case.

We apply an oscillating magnetic field to probe the hysteresis behaviour of
dipolar interacting MNPs. It is given by Anand (2020)

$\mu_{o}H=\mu_{o}H_{\mathrm{o}}\cos\omega t,$ (4)

where $\mu_{o}H_{\mathrm{o}}$ and $\omega=2\pi\nu$ are the amplitude and
angular frequency of the applied magnetic field respectively, $\nu$ is the
linear frequency, and $t$ is the time. Therefore, the total energy of the
underlying system under the influence of dipolar and external magnetic field
is given by Tan  _et al._ (2014); Anand  _et al._ (2019)

$E=K_{\mathrm{eff}}V\sum_{i}\sin^{2}\theta_{i}+\frac{\mu_{o}\mu^{2}}{4\pi
a^{3}}\sum_{j,\ j\neq
i}\left[\frac{\hat{\mu_{i}}\cdot\hat{\mu_{j}}-{3\left(\hat{\mu_{i}}\cdot\hat{r}_{ij}\right)\left(\hat{\mu_{j}}\cdot\hat{r}_{ij}\right)}}{(r_{ij}/a)^{3}}\right]-\mu_{o}\sum_{i}\vec{\mu}_{i}\cdot\vec{H}$
(5)

Here $\theta_{i}$ is the angle between the anisotropy axis and the $i^{th}$
magnetic moment of the system.

We implement kMC simulations to probe the hysteresis response as a function of
dipolar interaction strength $h_{d}$, temperature $T$, aspect ratio
$A_{r}=L_{y}/L_{x}$, and the applied magnetic field’s direction. In the kMC
algorithm, dynamics is captured more accurately, which is essential to study
the dynamic hysteresis response of dipolar interacting MNPs Chantrell  _et
al._ (2000). Using this technique, we can accurately describe the dynamical
properties of MNPs in the superparamagnetic or ferromagnetic regime without
any artificial or abrupt separation between them Tan  _et al._ (2014). We have
used the same algorithm, which is described in greater detail in the work of
Tan et al. and Anand et al. Tan  _et al._ (2014); Anand  _et al._ (2019).
Therefore, we do not reiterate it here to avoid repetition.

The amount of heat dissipated by the MNPs assembly during one complete cycle
of the external magnetic field equals the hysteresis loop area $E_{H}$. It can
be evaluated using the following formula Anand  _et al._ (2016)

$E_{H}=\oint M(H)dH,$ (6)

The above intergral is evaluated over the entire period of the external
magnetic field change. $M(H)$ is the magnetization of system at magnetic field
$H$.

##  III Simulations Results

We consider spherically shaped nanoparticles of Fe3O4 arranged in a two-
dimensional lattice ($L_{x}\times L_{y}$) with $D=8$ nm,
$K_{\mathrm{eff}}=13\times 10^{3}$ Jm-3, and $M_{s}=4.77\times 10^{5}$ Am-1.
The total number of nanoparticles in the assembly is considered as $n=400$. We
have considered seven values of system sizes viz. $L_{x}\times L_{y}=20\times
20$, $16\times 25$, $10\times 40$, $8\times 50$, $4\times 100$, $2\times 200$
and $1\times 400$. The corresponding aspect ratio $A_{r}(=L_{y}/L_{x})$ of the
underlying system is $1.0$, 1.56, 4.0, 6.25, 25, 100 and 400, respectively.
The dipolar interaction strength $h_{d}$ is varied from 0 to 1.0. The
temperature $T$ is changed between 0 and 300 K. The anisotropy axes are
assumed to have random orientations. The alternating magnetic field is applied
along $x$ and $y$-direction. The amplitude of the magnetic field strength
$\mu_{o}H_{\mathrm{o}}$ and its frequency $\nu$ is taken as $0.10$ T and
$10^{5}$ Hz, respectively.

First, we study the magnetic hysteresis in a square lattice with $L_{x}\times
L_{y}=20\times 20$. we have considered six representative values of dipolar
interaction strength $h_{d}=0$, 0.2, 0.4, 0.6, 0.8 and 1.0. The temperature is
taken as $300$ K, and the magnetic field is applied along $x$-direction in
Fig. 1(b) and along $y$-direction in Fig. 1(c). In all the hysteresis curves
shown in the present article, the $x$-axis (applied magnetic field) and
$y$-axis (magnetization) have been scaled by single-particle anisotropy field
$H_{K}=2K_{\mathrm{eff}}/M_{s}$ Carrey  _et al._ (2011) and $M_{s}$,
respectively. The hysteresis curve has the negligible value of coercive field
and remanence for weakly interacting MNPs ($h_{d}\leq 0.2$) indicating the
dominance of superparamagnetic character. Interestingly, the dipolar
interaction induces antiferromagnetic coupling for large magnetic interaction
strength ($h_{d}>0.2$). As a consequence, the corresponding hysteresis curve
has all the signatures of the dominance of antiferromagnetic interaction,
i.e., zero value of remanence. These observations are in perfect agreement
with the work of Chen et al. Chen  _et al._ (2017). It is also in qualitative
agreement with the work of Ewerlin et al. Ewerlin  _et al._ (2013). These
observations are also independent of the direction of the applied magnetic
field as along it is applied in a plane of the system.

To see the effect of aspect ratio $A_{r}$ and direction of the applied
magnetic field on the hysteresis, we then study magnetic hysteresis for six
values of $A_{r}$ in Fig. (2). The magnetic field is applied along the
$x$-direction, and temperature $T$ is taken as 300 K. Irrespective of $A_{r}$,
the hysteresis loop area is negligibly tiny for very weak dipolar interaction
strength ($h_{d}\leq 0.2$), indicating the dominance of superparamagnetic
behaviour. For large dipolar interaction strength and relatively small aspect
ratio $A_{r}\leq 6.0$, the dipolar interaction induces antiferromagnetic
coupling, which is reflected in the negligible value of remanence and coercive
field. When the aspect ratio is large ($A_{r}>6.0$), we observe non-hysteresis
for considerable dipolar interaction strength. We can explain it using the
fact that the magnetic moment’s natural tendency is to get aligned along the
sample’s longer axis (along $y$). The applied magnetic field forces the
magnetic moment to aligned along the shorter side of the sample. As a
consequence, non-hysteresis is observed for sizeable dipolar interaction
strength.

In Fig. (3), we apply the magnetic field along the $y$-direction. All other
parameters are the same as that of Fig. (2). There is a dominance of
antiferromagnetic interaction for considerable dipolar interaction strength
($h_{d}>0.2$) and $A_{r}<6.0$ in this case also. While for weak dipolar
interaction strength, the hysteresis loop area is negligibly small
irrespective of $A_{r}$, which indicates the signature of superparamagnetic
behaviour. Remarkably, the dipolar interaction induces ferromagnetic coupling
for $h_{d}>0.2$ and a very large aspect ratio ($A_{r}>6.0$). Consequently, the
coercive field and remanence have huge values and get enhanced with an
increase in $h_{d}$. One can explain it using the fact that the dipolar
interaction forces the magnetic moment to get aligned along the sample’s
longer axis due to the increased value of shape anisotropy. For extremely
large $A_{r}$ and strongest interacting MNPs ($h_{d}=1.0$), remanence is close
to 1.0, indicating extremely large ferromagnetic coupling. It is evident that
for sizeable dipolar interaction strength, there is a transition point at
$A_{r}\leq 6.0$; below this value, antiferromagnetic coupling dominates. For a
larger value of $A_{r}$, ferromagnetic coupling prevails. These results
suggest that the nature of the interaction can be tuned from antiferromagnetic
to ferromagnetic just by changing the aspect ratio provided the dipolar
interaction strength is very large. It has got profound applications in
spintronics based applications where one needs large tunnel magnetic
resistance, which could be maximized with the antiferromagnetic arrangement of
MNPs Bupathy  _et al._ (2019). The latter can thus be achieved for the sample
having a smaller aspect ratio and sizeable dipolar interaction.

Temperature also plays an essential role in determining the magnetic
hysteresis response in an array of dipolar interacting MNPs. So, now we study
the temperature dependence of hysteresis for various aspect ratio and the
applied magnetic field direction. In Fig. (4), we plot the hysteresis curve
for the square sample ($L_{x}\times L_{y}=20\times 20$) with various values of
$T$ for weakly interacting MNPs ($h_{d}=0.2$). The magnetic field is applied
along the $x$-direction. In the absence of thermal fluctuations ($T=0$ K), the
system behaves like a blocked state. The corresponding hysteresis curve shows
the coercive field value of $0.48H_{K}$ and remanence $\sim 0.5$, which are in
perfect agreement with the Stoner and Wohlfarth model Stoner and Wohlfarth
(1948). It is also in qualitative agreement with the work of V. Russier
Russier (2001). There is a decrease in the hysteresis loop area with an
increase in temperature. It can be explained using the fact the with an
increase in thermal fluctuations; magnetic moments tends to orient randomly,
resulting in the minimal value of coercive field and remanence provided the
dipolar interaction strength is weak. These observations are robust with
respect to the aspect ratio and direction of the applied magnetic field.
Therefore curves are not shown for other values of $A_{r}$ and direction of
the applied magnetic field to avoid duplications.

In the case of sizeable dipolar interaction strength, the hysteresis response
should depend on the direction of the applied magnetic field and aspect ratio
for a given temperature. Therefore, we study the magnetic hysteresis with
$h_{d}=0.4$ as a function of temperature for three representative values of
aspect ratio $A_{r}=1.0$, 1.56 and 100 in Fig. (5). $T$ is varied between 0 to
300 K and the magnetic field along $x$ and $y$-direction. For small $A_{r}$
and $T>20$ K, antiferromagnetic interaction is dominant resulting in
negligible remanence and the coercive field value. The system tends to behave
as a blocked state for the small value of $T$. These observations are
independent of applied magnetic field’s direction. On the other hand, in the
case of huge aspect ratio, the hysteresis curve depends strongly on the
applied magnetic field’s direction. The hysteresis loop area is very small
when the field is applied along the $x$-direction. The dipolar interaction
induces ferromagnetic coupling when the field is applied along the long axis
of the sample ($y$-direction), which results in a tremendous value of coercive
field and remanence. Consequently, the hysteresis loop area is very large
compared to the field applied along the sample’s shorter axis ($x$-direction).
The hysteresis loop decreases with an increase in temperature because of
enhancement in thermal fluctuations.

To see further the effect of temperature on the hysteresis for enormous
dipolar interaction strength, we plot magnetic hysteresis curves for
$h_{d}=0.6$ and 0.8 in Fig. (6) and Fig. (7), respectively. All other
parameters are the same as that of Fig. (5). Irrespective of the applied
magnetic field’s direction, the hysteresis curve shows the dominance of
antiferromagnetic interaction for a small value of $A_{r}(\leq 6.0)$. The
hysteresis curve has a negligibly small value of the coercive field and
remanence when the magnetic field is applied along the sample’s shorter axis
($x$-axis). It is due to the fact that the natural tendency of magnetic
moments is to get aligned along the long axis of the sample; the applied
magnetic field along $x$-direction forces them to orient along the system’s
shorter axis, which results in a minimal hysteresis loop area. In the case of
the magnetic field applied along the long axis of the sample ($y$-axis), the
hysteresis loop area is huge, indicating the ferromagnetic coupling dominance.
As the dipolar interaction strength is very large, the temperature does not
affect the hysteresis behaviour irrespective of the applied magnetic field’s
direction.

In Fig. (8), we study the temperature dependence of hysteresis for the
strongest dipolar interacting MNPs ($h_{d}=1.0$). All other parameters are the
same as that of Fig. (7). The hysteresis curve has a negligible value of
hysteresis loop area irrespective of the applied magnetic field’s direction
for $A_{r}\leq 6.0$. We observe a non-hysteresis straight line for $A_{r}$’s
very large value with the applied field is along the shorter side of the
sample($x$-direction). Remarkably, the hysteresis shows the dominance of
ferromagnetic interaction for an extremely large value of $A_{r}$ when the
field is applied along the long axis of the system. In this case, the
hysteresis loop area is enormous. The extremely large value of remanence
($\sim 1.0$) also strengthens the fact the ferromagnetic coupling is the
maximum. Temperature does not have an impact on the hysteresis as dipolar
interaction strength is the largest in this case.

To quantify the magnetic hysteresis, we now study the variation of coercive
field $\mu_{o}H_{c}$ (scaled by $H_{K}$), remanence $M_{r}$ and hysteresis
loop area $E_{H}$ as a function of aspect ratio $A_{r}$ and dipolar
interaction strength $h_{d}$ in Fig. (9). The magnetic field is applied along
the $y$-direction and temperature is taken as $T=300$ K. The coercive field
and remanence are negligibly small for weakly interacting MNPs and small value
of $A_{r}$ indicating the dominance of superparamagnetic behaviour. The
antiferromagnetic coupling between the magnetic moment is favoured for large
$h_{d}$ and small values of $A_{r}$, which is manifested in the minimal value
of $M_{r}$ and $\mu_{o}H_{c}$. On the other hand, there is an enhancement in
the coercive field and remanence magnetization with an increase in dipolar
interaction strength for large $A_{r}$, indicating the ferromagnetic coupling
dominance. When the dipolar interaction strength and aspect ratio are
enormous, there is weak dependence of coercive field and remanence on $A_{r}$.
Remarkably, the variation of $E_{H}$ with aspect ratio and dipolar interaction
strength is precisely similar to that of coercive field variation. Our results
for $h_{d}=0$ is in perfect aggreement with the works of Carrey et al. Carrey
_et al._ (2011). We could not compare the entire range of aspect ratio and
$h_{d}$ as they have concentrated only on the non-interacting case
($h_{d}=0$). Therefore, our results can be used as a benchmark in this
context. These results could be used for choosing correct values of dipolar
interaction strength, aspect ratio for optimizing heat dissipation and tunnel
magnetoresistance for magnetic hyperthermia and spintronics based
applications.

Finally, we study the variation of coercive field and remanence as a function
of dipolar interaction strength and temperature for various values of aspect
ratio. In Fig. (10), we plot the variation of $\mu_{o}H_{c}$ (scaled by
$H_{K}$) and $M_{r}$ as a function of $T$ and $h_{d}$ for three representative
values of aspect ratio $A_{r}=1.0$, 1.56 and 100. The coercive field and
remanence magnetization are negligibly small irrespective of $h_{d}$ and $T$
for square sample. In the case of weak dipolar interaction and a large value
of temperature, the coercive field and remanence have minimal values, which
indicates the dominance of superparamagnetic character. The results with
square sample is in qualitative agreement with the work of Figueiredo et al.
Figueiredo and Schwarzacher (2007). We could not compare for other aspect
ratio as they have concentrated on square geometry. It is also in qualitaive
agreement with the work of Ewerlin et al. Ewerlin  _et al._ (2013). While for
large dipolar interaction strength, there is a negligible value of remanence
and coercive field as the antiferromagnetic coupling is dominant in this case.
Similar observations can be made for $A_{r}=1.56$ except at smaller dipolar
interaction strength and temperature values where remanence and coercive field
is higher. Remarkably, the coercive field and remanence increase with an
increase in $h_{d}$ for a given temperature, provided the aspect ratio is
huge. It is bacause of the enhanced ferromagnetic coupling. In the case of
extremely large value of dipolar interaction strength, coercive field and
remanence magnetization are very large and they depend very weakly on the
temperature. The variation of $E_{H}$ with $T$ and $h_{d}$ is also found to be
the same as that of coercive field variation. Therefore we have not shown the
corresponding curves of $E_{H}$ variation.

##  IV Summary and conclusion

Now we summarize and discuss the main results of the present work. We have
used kinetic Monte Carlo simulation technique to study the magnetic hysteresis
in a two-dimensional arrangement of MNPs. We have systematically probed the
hysteresis response as a function of dipolar interaction strength $h_{d}$,
system size ($L_{x}\times L_{y}$) or the aspect ratio $A_{r}$, temperature $T$
and direction of the applied magnetic field. In the absence of thermal
fluctuations and negligible dipolar interaction, the hysteresis curve obeys
the Stoner and Wohlfarth model irrespective of aspect ratio and direction of
the applied magnetic field. The superparamagnetic behaviour is dominant for
weak dipolar interaction strength and large value of temperature, manifested
in the minimal hysteresis loop area. Interestingly, the dipolar interaction is
found to induce antiferromagnetic coupling provided the aspect ratio $A_{r}$
is relatively small ($\leq 6.0$) and $h_{d}\geq 0.2$, irrespective of the
direction of the applied magnetic field. Therefore, the corresponding
hysteresis curve shows a negligible value of remanence and coercive field
Ewerlin  _et al._ (2013). On the other hand, for large dipolar interaction
strength and $A_{r}$, the ferromagnetic coupling is dominant with the magnetic
field applied the sample’s longer axis ($y$-direction). Temperature is also
found to affect the hysteresis properties of the underlying system. In the
case of weak dipolar interaction, the system moves from the blocked state to a
superparamagnetic state as the temperature is increased. On the other hand,
the temperature does not affect the hysteresis mechanism for large dipolar
interaction strength.

The coercive field $\mu_{o}H_{c}$, remanence $M_{r}$ and hysteresis loop area
$E_{H}$ are also useful quantifier to analyse the hysteresis properties of
dipolar interacting MNPs. In the case of negligible and weak dipolar
interaction, $\mu_{o}H_{c}$ and $M_{r}$ are extremely small with a large value
of temperature, irrespective of $A_{r}$ and direction of the applied magnetic
field. We can interprete it due to the dominance of superparamagnetic
behaviour. Remarkably, the dipolar interaction is found to induce
antiferromagnetic coupling between the magnetic moments for $h_{d}>0.2$ and
small values of $A_{r}$ ($\leq 6.0$). As a consequence, the corresponding
hysteresis curves have negligibly small values of $M_{r}$ and $\mu_{o}H_{c}$.
When the magnetic field is applied along the long axis of the sample, there is
an increase in coercive field and remanence with an increase in dipolar
interaction strength for large $A_{r}$, indicating the ferromagnetic coupling
dominance. In the case of weak dipolar interaction, $\mu_{o}H_{c}$ and $M_{r}$
decrease with an increase in temperature because of enhancement in thermal
fluctuations. While for large dipolar interaction strength, they depend weakly
on temperature. Remarkably, There is an intimate relationship between $E_{H}$
and $\mu_{o}H_{c}$ as the variation of $E_{H}$ with $h_{d}$, $T$ and $A_{r}$
is found to be the same as that coercive field variation.

In conclusion, we analysed the hysteresis properties in a two-dimensional
assembly of dipolar interacting MNPs using kinetic Monte Carlo simulations.
Our results suggest that the dipolar interaction affects the hysteresis
mechanism drastically in such an array of MNPs. Interestingly, the nature of
interaction can be tuned by varying the aspect ratio and direction of the
applied magnetic field. We believe that these results are extremely useful for
spintronics based applications. The observation made in the present article
should help the experimentalists to manipulate the hysteresis response and
other useful properties of interacting MNPs in a more controlled manner. These
results should also be taken into account in interpreting of the experiments
and efficient usage of magnetic hyperthermia. Therefore, we hope that the
present work could pave the way for joint efforts in experimental,
theoretical, and computational studies for these extremely useful and
versatile systems.

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Figure 1: (a) Schematic of the two-dimensional assembly of MNPs used in the
present work, $D$ is the particle diameter and $a$ the lattice spacing. (b)
Magnetic hysteresis curves for MNPs arranged in a square lattice as a function
of dipolar interaction strength at $T=300$ K. The external alternating
magnetic field is applied along $x$-direction [(b)] and $y$-direction [(c)] at
$T=300$ K. For non-interacting and weakly interacting MNPs ($h_{d}\leq 0.2$),
the hysteresis loop area is minimal, indicating the dominance of
superparamagnetic behaviour. Interestingly, the hysteresis curve has all the
antiferromagnetic interaction characteristics for considerable dipolar
interaction strength ($h_{d}>0.2$). The hysteresis response is independent of
the applied magnetic field’s direction (in-plane) in a square lattice.

Figure 2: Magnetic hysteresis curves as a function of dipolar interaction for
six values of aspect ratio $A_{r}=1.56$ [(a)], 4.0 [(b)], 6.25 [(c)], 25.0
[(d)], 100 [(e)], and 400 [(f)] at $T=300$ K. The dipolar interaction strength
$h_{d}$ is varied from 0 to 1.0 and the alternating magnetic field is applied
along the $x$-direction. There is a negligible hysteresis loop area with small
$h_{d}$, which indicates the dominance of superparamagnetic behaviour. For
large dipolar interaction strength and $A_{r}\leq 6.25$, the hysteresis curve
has a negligible value of remanence and coercive field, which indicates the
dominance of antiferromagnetic coupling. Non-hysteresis is observed for
extremely large aspect ratio ($A_{r}>6.25$) and dipolar interaction strength
$h_{d}>0.8$.

Figure 3: Magnetic hysteresis curves as a function of dipolar interaction
strength $h_{d}$ with the magnetic field is applied along the $y$-direction,
and $T=300$ K. We have considered six values of aspect ratio $A_{r}=1.56$
[(a)], 4.0 [(b)], 6.25 [(c)], 25.0 [(d)], 100 [(e)], and 400 [(f)]. For
$A_{r}\leq 6.25$ and large dipolar interaction strength $h_{d}$,
antiferromagnetic interaction dominates the magnetic hysteresis, i.e., the
negligible remanence value. Interestingly for large aspect ratio and
appreciable dipolar interaction strength, the hysteresis curve has a very
large value of coercive field, and remanence. They get enhanced with an
increase in $h_{d}$. These observations indicate the dominance of
ferromagnetic interaction.

Figure 4: Magnetic hysteresis curve as a function of temperature $T$ in a
square arrangement of MNPs ($A_{r}=1.0$) for weakly dipolar interacting
($h_{d}=0.2$). The alternating magnetic field is applied along the
$x$-direction. In the absence of thermal fluctuations ($T=0$ K), the
hysteresis curve has the coercive field $\mu_{o}H_{c}\sim 0.48H_{K}$ and
remanence $M_{r}\sim 0.5$ as expected, which is in perfect agreement with the
Stoner and Wohlfarth model. The hysteresis loop area decreases with an
increase in temperature because of enhancement in thermal fluctuations. These
observations are independent of applied magnetic field’s direction (in-plane)
and the aspect ratio $A_{r}$ (curves not shown).

Figure 5: Magnetic hysteresis curve as a function of temperature $T$ for
moderate dipolar interaction ($h_{d}=0.4$) and three representative values of
aspect ratio $A_{r}=1.0$, 1.56 and 100. The magnetic field is applied along
$x$-direction [(a), (b) and (c)] and $y$-direction [(d), (e) and (f)]. In the
case of small $A_{r}$ and non-zero temperature, antiferromagnetic interaction
is dominant resulting in negligible remanence and coercive field, which is
independent of direction of the applied magnetic field. While for large
$A_{r}$, the hysteresis curve has small hysteresis loop area with the field
applied along the shorter axis of the sample ($x$-direction). Interestingly,
there is a large coercive field and remanence with the magnetic field applied
along $y$-direction and $A_{r}$ is very large, which indicates the dominance
of ferromagnetic coupling. There is always a decrease in hysteresis loop area
with temperature.

Figure 6: Temperature dependence of magnetic hysteresis curve with large
dipolar interaction $h_{d}=0.6$. We have considered three typical values of
aspect ratio $A_{r}=1.0$, 1.56 and 100. The magnetic field is applied is
applied along $x$-direction [(a), (b) and (c)] and $y$-direction [(d), (e) and
(f)]. Irrespective of the applied magnetic field’s direction, the hysteresis
curve has all the characteristics of antiferromagnetic interaction for small
$A_{r}$. There is a large hysteresis loop area with the magnetic field applied
along the sample’s long axis ($y$-direction) and extremely large $A_{r}$.
There is a weak dependence of hysteresis on temperature.

Figure 7: Magnetic hysteresis as a function of temperature for very large
dipolar interaction strength $h_{d}=0.8$. We have considered there
representative values of $A_{r}=1.0$, 1.56 and 100; the magnetic field is
applied along $x$-direction [(a), (b) and (c)] and $y$-direction [(d), (e) and
(f)]. In the case of small $A_{r}$ and irrespective of applied magnetic
field’s direction, the hysteresis curve has negligible remanence and the
coercive field, indicating the dominance of antiferromagnetic coupling. The
hysteresis loop area is minimal with the magnetic field applied along the
sample’s shorter axis ($x$-direction), provided the aspect ratio is huge. The
hysteresis loop area is very large when the direction of the applied magnetic
field is along the longer axis of the system ($y$-direction) and $A_{r}$ is
enormous.

Figure 8: Magnetic hysteresis curves as a function of temperature for the
strongest dipolar interacting MNPs ($h_{d}=1.0$) with three values of
$A_{r}=1.0$, 1.56 and 100. The magnetic field is applied along $x$-direction
[(a), (b) and (c)] and $y$-direction [(d), (e) and (f)]. In the case of the
magnetic field applied along the $x$-direction, non-hysteresis is observed
irrespective of aspect ratio. The hysteresis loop area is huge, indicating the
dominance of ferromagnetic interaction when the field is applied to the longer
axis of the sample, provided the aspect ratio is enormous. Temperature does
not affect the hysteresis because of substantial large dipolar interaction.

Figure 9: Variation of the coercive field $\mu_{o}H_{c}$ (scaled by single-
particle anisotropy field $H_{K}$), remanence $M_{r}$ and hysteresis loop area
$E_{H}$ as a function of dipolar interaction strength $h_{d}$ and aspect ratio
$A_{r}$. The magnetic field is applied along the $y$-direction, and
temperature $T$ is taken as 300 K. For small aspect ratio and appreciable
$h_{d}$, $\mu_{o}H_{c}$ and $M_{r}$ have negligible values. There is an
increase in value of $\mu_{o}H_{c}$ and $M_{r}$ with an increase in $h_{d}$,
indicating ferromagnetic behaviour. Interestingly, the variation of $E_{H}$
with $h_{d}$ and $A_{r}$ is the same as that of coercive field variation.

Figure 10: Variation of the coercive field $\mu_{o}H_{c}$ (scaled by $H_{K}$)
and remanence $M_{r}$ as a function of temperature $T$ and dipolar interaction
strength $h_{d}$ for three representative values of aspect ratio $A_{r}=1.0$,
1.56 and 100. Irrespective of $T$ and $h_{d}$, $\mu_{o}H_{c}$ and $M_{r}$ is
negligibly tiny for square sample ($A_{r}=1.0$). Even in the case of
$A_{r}=1.56$, there is a weak dependence of $\mu_{o}H_{c}$ and $M_{r}$ on $T$
and $h_{d}$ except at small temperature and dipolar interaction strength.
Interestingly, there is an increase in $\mu_{o}H_{c}$ and $M_{r}$ with $h_{d}$
for a fixed temperature. There is a rapid fall in these values with
temperature for weak dipolar interaction strength. While for sizeable dipolar
interaction, there is a weak dependence of coercive field and remanence on
temperature.


# Higgs searches at the Tevatron

Fermilab
E-mail on behalf of the CDF and D0 collaborations

###### Abstract:

We present combined CDF and D0 searches for the Standard Model (SM) and
Minimal Supersymmetric Standard Model (MSSM) Higgs boson using up to 6.7 fb-1
of integrated luminosity from 1.96 TeV proton-antiproton collisions at the
Tevatron. Specialized searches for Higgs bosons produced via gluon fusion,
associated boson production, and vector boson fusion, decaying to $b\bar{b}$,
$W^{+}W^{-}$, $\tau^{+}\tau^{-}$, and $\gamma\gamma$ are combined to produce
95% CL upper limits on SM Higgs production as a function of mass. Current
Tevatron limits are shown, including a new exclusion for SM Higgs masses
between 158 and 175 GeV$/c^{2}$. We also present prospects for future
sensitivity.

##  1 Introduction

The discovery of the Higgs boson will resolve the longstanding question of how
the electroweak symmetry is broken, and how fermions and bosons acquire mass.
The standard model of particle physics hypothesizes that a non-zero scalar
field with four degrees of freedom permeates the universe, such that the
$W^{+}$, $W^{-}$, and $Z$ boson gain mass through three of these degrees of
freedom, while the fourth degree of freedom becomes a new scalar boson. The
BEHHGK (pronounced ”beck”) mechanism and boson, named after the authors
(Breit, Englert, Higgs, Hagen, Guralnik, Kibble), was proposed in 1964 [1],
and is often referred to as the Higgs mechanism and boson. Finding the Higgs
boson confirms that the Higgs field exists, and is the subject of an intense
search at the Tevatron collider.

Relationships between measurable electroweak parameters within the standard
model, in conjunction with direct searches [3], constrain the Higgs boson mass
to be between 114 GeV111c is set to 1 in this proceeding and 185 GeV at the
95% CL [2]. With enough data, this entire mass range is accessible to the 1.96
TeV center of mass Tevatron proton-antiproton collider, for either observation
or exclusion of the standard model Higgs boson. The two multipurpose detectors
at the Tevatron, CDF and D0, are able to reconstruct all of the final state
particles and topologies resulting from SM Higgs boson production and decay.
The Tevatron has delivered 9 fb-1 of luminosity each to CDF and D0. Data
collection efficiencies were 85 - 90 % for this data, and an integrated
luminosity of up to 6.7 fb-1 have been analyzed for the Higgs boson searches
covered in these proceedings. The 350 nb-1 delivered to the 7 TeV proton-
proton LHC is not sufficient for Higgs searches, meaning the search for the
Higgs boson is currently unique to the Tevatron.

The CDF and D0 experiments at the Tevatron have been producing combined Higgs
searches since 2006. Each iteration, efforts are made to improve signal
acceptance, such as by loosening lepton and $b$ hadron identification
requirements, adding backup triggers which select events online using
different aspects of the signal signature, and relaxing kinematic selection.
As the signal acceptance is improved, the backgrounds increase and become more
difficult to model, and are separated into categories with similar S
$/\sqrt{B}$, such that high S $/\sqrt{B}$ categories provide the most signal
sensitivity, whereas low S $/\sqrt{B}$ categories provide the best background
constraints. The primary Higgs channels of $H\to W^{+}W^{-}$, $WH\to l\nu
b\bar{b}$, $ZH\to\nu\nu b\bar{b}$, and $ZH\to l^{+}l^{-}b\bar{b}$ from CDF and
D0 correspond to about 500 Higgs bosons produced at each mass, 114 $<m_{H}<$
185 GeV.

##  2 Low mass SM Higgs searches

Searches for a ”low mass” (m${}_{H}<$ 135 GeV) standard model Higgs boson are
presented elsewhere in these conference proceedings for $H\to b\bar{b}$ [4],
$H\to\tau^{+}\tau^{-}$ [5], and $H\to\gamma\gamma$ [6]. Here we report on the
features of the most sensitive low mass Higgs search channels. A low mass
Higgs boson preferentially decays $H\to b\bar{b}$. It is most easily
identified in events produced via associated Higgs production, $WH$ and $ZH$,
when the $W$ and $Z$ decay leptonically, into final states of $WH\to l\nu
b\bar{b}$, $ZH\to\nu\nu b\bar{b}$, and $ZH\to l^{+}l^{-}b\bar{b}$ where
$\ell=e$ or $\mu$. In events with a reconstructed $W$ or $Z$ boson and two or
more additional jets, the di-jet invariant mass is used to search for a
resonance originating from $H\to b\bar{b}$. To reduce the background from $W$
and $Z$ production in association with jets, $b$-jets are ”tagged” by
identifying a secondary vertex or tracks with high impact parameter from $B$
hadron decay. The dijet mass is shown before and after one and two $b$-tags
for $WH\to\ell\nu b\bar{b}$ candidates in Figure 1. Expected sensitivity is
improved by categorizing events according to the number of $b$-tags and the
purity of the applied $b$-tagging algorithms.

Figure 1:  The dijet mass for $WH\to\ell\nu b\bar{b}$ candidates with no
$b$-tags, one $b$-tag, and two $b$-tags (left to right), demonstrating how the
$W$+jets background is reduced while $WH$ signal, $W+b\bar{b}/c\bar{c}$, and
$t\bar{t}$ heavy flavor backgrounds become more prominent.

##  3 High mass SM Higgs searches

Searches for a ”high mass” Higgs boson ($m_{H}>$ 135 GeV) are reported in
detail elsewhere in these proceedings for CDF and D0 [7]. The most sensitive
mode at the Tevatron is $gg\to H\to WW\to\ell\nu\ell\nu$ due to the high cross
section and well identified final state. The high mass analysis benefits by
separating events into categories according to the number of jets and the
number of leptons because of the difference in the kinematics of the signal
production mechanisms and primary background processes. The signal cross
section and uncertainties for the dominant signal process $gg\to H$ use state
of the art NNLL and NNLO calculations as explained in Ref. [12]. Since there
are neutrinos in the final state, the invariant mass of the Higgs boson cannot
be reconstructed. The best variable for distinguishing $H\to WW$ from the
background is the separation between the charged leptons, dR
$=\sqrt{d\eta^{2}+d\phi^{2}}$, which is different for spin-zero $H\to WW$
decay than for spin-one $Z\to WW$ decay as demonstrated in Figure 2. Also
shown is the equivalent plot for same sign leptons, one of the control regions
used to test the background modeling.

Figure 2:  The $\Delta$R for opposite sign (OS) leptons with no accompanying
jets (left) and for the same-sign-lepton control region (right).

##  4 Validation of multivariate analysis techniques

Multivariate techniques are used in all the Tevatron SM Higgs analyses. The
main algorithms used are Neural Networks, Matrix Element probabilities, and
Boosted Decision Trees. By taking advantage of the different correlations for
signal and background among the multiple input variables, these techniques
typically improve analyses by about 20% with respect to simply fitting the
leading two kinematically distinct variables. To help validate these
multivariate techniques we use them to measure higher statistics standard
model processes. Shown in Figure 3 are two measurements using multivariate
techniques. We measure $\sigma_{WW+WZ}=$ 16.6${}^{+3.5}_{-3.0}$ pb in a
$W$+2-jet final state compared to the SM expected cross section of 15.1 $\pm$
0.8 pb [8], and also verify the $W$+jets background process rate and shape by
analyzing a $WH\to\ell\nu b\bar{b}$ sample of events before the $b$-tagging
requirements.

Figure 3:  A matrix element event probability discriminant from a diboson
search for $WW/WZ\to\ell\nu jj$ (left) and a random forest (RF) of boosted
decision trees tested in the pre b-tagged WH control region (right).

##  5 Combined SM Higgs search

The CDF combined search and D0 combined search for the SM Higgs boson was
presented elsewhere in these proceedings [9, 10] and the final results are
shown in Figure 4. The 95% CL upper limit on the cross section divided by the
SM value of the cross section is shown on the y-axis and is equal to one when
standard model sensitivity is achieved. The dotted line is the expected upper
limit on cross section in pseudo-experiments with no signal events, allowing
systematic uncertainties to be fit within each pseudo-experiment, and fitting
for the maximum signal that can be accommodated at the 95% CL. The green and
yellow bands represent the one and two $\sigma$ variations of the expected
limits. The solid line is the upper limit observed in actual data. CDF
achieves expected sensitivity to the Higgs boson for m${}_{H}=$ 165 GeV, while
D0 almost achieves observed sensitivity for m${}_{H}=$ 165 GeV. CDF is able to
exclude 100 ¡ mH ¡ 102 GeV.

CDF and D0 perform a joint search for the Higgs boson where shared systematic
uncertainties are kept correlated (bottom of Fig. 4). The observed exclusion
is 158 $<m_{H}<$ 175 GeV which is consistent with the expected exclusion of
156 $<m_{H}<$ 175 GeV. The combined search is also able to exclude Higgs
masses below 109 GeV. At 115 GeV, just above the LEP limit, the expected
(observed) exclusion is 1.45*SM (1.56*SM). For $m_{H}=$ 165 GeV, where the
high mass exclusion is strongest, $H\to W^{+}W^{-}$ dominates, but for
$m_{H}=$ 115 GeV, the sensitivity comes from a combination of analyses (see
Table 1).

We can collectively view the dozens of discriminant outputs from the multiple
search channels by gathering all histogram bins from all sub-channels, summing
up those bins with the same $S/B$ into the same bins, and then sorting the
bins from lowest to highest $S/B$. The distribution is shown for m${}_{H}=$
165 GeV and m${}_{H}=$ 115 GeV in Figure 5. For 165 GeV, the data clearly
prefer the background-only model, while for m${}_{H}=$ 115 GeV an excess of
events in bins with very high $S/B$ and a deficit of events with slightly
lower $S/B$ result in the observed limit being approximately equal to the
expected limit.

Figure 4: SM Higgs combinations for CDF (left), D0 (right), and the Tevatron (bottom). Table 1: Tevatron analyses ordered by the expected upper limit at 95% CL for a Higgs boson search mass of 115 GeV.  Analysis | Experiment | Expected | Integrated
---|---|---|---
channel |  | limit $\sigma$/$\sigma_{SM}$ @ 115 GeV | Luminosity (fb-1)
$WH\to\ell\nu b\bar{b}$ | CDF | 3.4 | 5.7
$ZH/WH\to\not\\!\\!E_{T}b\bar{b}$ | CDF | 4.0 | 5.7
$ZH/WH\to\not\\!\\!E_{T}b\bar{b}$ | D0 | 4.2 | 6.4
$WH\to\ell\nu b\bar{b}$ | D0 | 4.8 | 5.3
$ZH\to\ell\ell b\bar{b}$ | CDF | 5.5 | 5.7
$ZH\to\ell\ell b\bar{b}$ | D0 | 5.7 | 6.2
$H\to WW$ | CDF | 10.6 | 5.9
$H\to WW$ | D0 | 12 | 6.7
$H\to\tau\tau$ | D0 | 16 | 4.9
$ZH/WH\to qqb\bar{b}$ | CDF | 18 | 4
$H\to\gamma\gamma$ | D0 | 18.5 | 4.9
$H\to\gamma\gamma$ | CDF | 21 | 5.4
$H\to\tau\tau$ | CDF | 25 | 2.3
$ttH$ | D0 | 45 | 2.1
Total | CDF + D0 | 1.45 | 5.8

Figure 5:  On left, for m${}_{H}=$ 165 GeV, the data from all search channels
sorted into bins of varying $S/B$. On right, same for m${}_{H}=$ 115 GeV .

##  6 Beyond SM Higgs searches

Beyond standard model Higgs bosons searches are presented in detail elsewhere
in these conference proceedings [11]. The MSSM predicts enhanced Higgs
production cross sections and decays to down-type fermions for high values of
tan $\beta$, the ratio of the vacuum expectation value for up-type and down-
type fermions, thereby enhanacing rates for $H\to b\bar{b}$ and $H\to\tau\tau$
final states. The Tevatron has searched for such signatures primarily using
the invariant mass of the $b$-jets or the invariant mass of the visible $\tau$
decay products as discriminants and set limits close to the theoretically
interesting value of tan $\beta=$ 30, which is approximately the ratio of mtop
to mb. A small excess of 2 $\sigma$ (including a trials factor) is seen in the
CDF $bb$ search (Fig. 6).

Figure 6:  Search for the MSSM in $H+b\to bb+b$ search channel at CDF shown
with best fit to signal plus background hypothesis.

##  7 Outlook

Figure 7 shows the expected a priori signal sensitivity as a function of mass
and analyzed integrated luminosity per experiment. This includes a factor of
1.5 improvement in the expected limit based on a range of improvements from
lepton identification efficiency, b-tagging efficiency, triggering, and jet
energy resolution. The full dataset for Run II of the Tevatron will be
collected by September 2011, and is expected to be about 10 fb-1 of analyzed
luminosity per experiment. The projected sensitivity with this dataset is 3
$\sigma$ for m${}_{H}=$ 115 GeV. The sensitivity would be expected to be at
least 2.4 $\sigma$ across the full Higgs boson mass range of 100 to 185 GeV.
The possibility of running the Tevatron for another 3 years would allow for
about 17 fb-1 of data to be analyzed per experiment. This would increase the
expected sensitivity to 3 $\sigma$ across the full mass range, 4 $\sigma$ at
115 GeV.

Figure 7:  Projections of expected Higgs sensitivity, assuming a factor of 1.5
improvement in the expected limits at each mass point, and assuming D0 and CDF
have equal sensitivity and equal analyzed integrated luminosity. Note that
much of this factor of 1.5 has already been achieved at high mass.

##  8 Summary

The CDF and D0 experiments at the Tevatron have a comprehensive and aggressive
search program for the Higgs boson which can be divided into three main
categories : high mass, low mass, and beyond standard model searches. The
newest high mass combination from the Tevatron excludes 158 $<m_{H}<$ 175 GeV
using a dataset of up to 6.7 fb-1. The low mass primary channels achieved
1.45*SM (1.56*SM) expected (observed) exclusion for $m_{H}=$ 115 GeV and a 95%
CL exclusion for masses between 100 GeV and 109 GeV. Tevatron MSSM Higgs
searches are setting limits at theoretically motivated values of tan $\beta$,
and a $H\to b\bar{b}+b$ search with 2.2 fb${-1}$ has a 2 $\sigma$ excess at
m${}_{H}=$ 140 GeV. The Tevatron is scheduled to deliver up to 12 fb-1 by its
conclusion in September 2010, providing about 10 fb${-1}$ of analyzable data
for Higgs analysis. With this dataset, a full combination of Higgs searches
will have at least 2.4 $\sigma$ level sensitivity for 100 $<m_{H}<$ 185 GeV.
This would increase to 3 $\sigma$ were the Tevatron to run an additional three
additional years as is the Tevatron Run III proposal.

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