# Chemical and magnetic impurity effects on electronic properties of
semiconductor quantum wires

Alireza Saffarzadeh  E-mail<EMAIL_ADDRESS>1Department of
Physics, Payame Noor University, Nejatollahi St., 159995-7613 Tehran, Iran
2Computational Physical Sciences Laboratory, Department of Nano-Science,
Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531,
Tehran, Iran

###### Abstract

We present a theoretical study of electronic states in magnetic and
nonmagnetic semiconductor quantum wires. The effects of chemical and magnetic
disorder at paramagnetic temperatures are investigated in single-site coherent
potential approximation. It is shown that the nonmagnetic impurity shifts the
band of carriers and suppresses the van Hove singularities of the local
density of states (LDOS) depending on the value of impurity concentration. The
magnetic impurity, however, broadens the band which depends on the strength of
exchange coupling, and in the high impurity concentration, the van Hove
singularities in the LDOS can completely disappear and the curves become
smooth.

##  I Introduction

The nature of the dimensionality and of the confinement associated with a
particular nanostructure such as a quantum well, quantum wire, or quantum dot
have a pronounced effect on its physical properties. Quantum wires and quantum
dots are under active experimental investigation because devices based on them
offer important opportunities as the building blocks for the next generation
of electronic and opto-electronic devices ranging from ultrafast optical
switching to ultradense memories Uskov . Quantum wire structures have density
of states features which are very useful for laser applications with
possibility of smaller current threshold density than in lasers produced from
higher dimensional structures. However, it is clear that disorder affects
these attractive features of quantum wires Taylor ; Nik0 ; Harris . Over the
past decade, using the coherent potential approximation (CPA), the effects of
boundary roughness and the presence of islands on the electronic properties of
nonmagnetic semiconductor (NMS) quantum wires have been studied Nik1 ; Nik2 ;
Hong . Furthermore, it has been demonstrated by Ohno YOhno  that it is indeed
possible to inject spin from a diluted magnetic semiconductor (DMS) to a NMS,
which is necessary in order to carry out qubit (quantum bit) operations
required for quantum computing DiVince .

DMS’s Chapman  are semiconductors of the general type
$\mathrm{A}_{1-x}\mathrm{M}_{x}\mathrm{B}$, where AB is either a II-VI or a
III-V semiconductor and M a magnetic element, most commonly Mn . Substitution
of a small fraction $x$ of the element A by Mn impurities (and in the case of
II-VI semiconductors an additional charge dopant, such as P on the B site)
leads to the appearance of a semiconductor with ferromagnetic properties
Furdyna ; Ohno . The magnetic state in these materials has been attributed to
the exchange interaction of the localized Mn moments with the spin of the
charge carriers introduced by the Mn dopants, or in the case of II-VI
semiconductors, by the additional dopant.

In recent years, considerable works have been devoted to the understanding of
physical properties of DMS quantum wires, both theoretically Kim ; Kyry ;
Chang ; Xu ; Moradian  and experimentally Chen1 ; Chen2 ; Jeon . One of the
most important physical quantities in these quantum structures, is the density
of states of charge carriers which depends on the dimensionality of the
structure. The behavior of this quantity is very important in determining the
electrical, thermal, and other properties of the system. Furthermore, DMS’s
belong to the class of disordered systems, hence, any first principle
consideration should take into account the randomness of impurities. Among
different kinds of randomness, two kinds of them can strongly affect the
transport properties of charge carriers in DMS quantum wires, i.e., the random
substitution of the magnetic atoms and the random direction of the impurity
spins.

The aim of this work is to investigate the effects of chemical (spin-
independent) potential and magnetic disorder on the electronic properties of
semiconductor quantum wires. Based on the single-site CPA for the magnetic and
nonmagnetic impurities at paramagnetic temperatures Taka99 , we study the
local density of states (LDOS) for charge carriers, in terms of the impurity
concentrations and spin scattering strengths.

The paper is organized as follows. The model, Hamiltonian and formalism are
given in section II. In section III, we present the results of the numerical
calculations for the NMS and DMS quantum wires. A brief conclusion is given in
section IV.

##  II Model and method

We consider a semiconductor quantum wire described by the tight-binding model
on a square lattice in which one of the dimensions (the $x$ direction) is much
larger than the other (the $y$ direction), i.e., a long-strip lattice in two
dimension. The sites of the lattice are denoted by ($m$,$n$) where $m$ is an
integer number and $N_{y}$ is number of atoms in the $y$ direction, hence
$1\leq n\leq N_{y}$. In fact, we have divided the wire into $N_{y}$ atomic
lines along the $y$ axis and each line lies along the $x$ axis. We set the
site energies to be infinite along the lines $n=0$ and $n=N_{y}+1$; thus, the
carriers are confined along the $y$ direction. The one-electron Hamiltonian
for this system is given by

$H=\sum_{i}u_{i}(\mathrm{M,A})+\sum_{i,j,\sigma}t_{ij}\mid
i,\sigma\rangle\langle j,\sigma\mid\ ,$ (1)

where $u_{i}$ depends on whether $i\equiv(m,n)$ is a magnetic (M) or
nonmagnetic (A) site. For the A-site

$u_{i}^{\mathrm{A}}=\varepsilon_{\mathrm{A}}\sum_{\sigma}\mid
i,\sigma\rangle\langle i,\sigma\mid\ ,$ (2)

and for the M-site

$u_{i}^{\mathrm{M}}=\sum_{\sigma,\sigma^{\prime}}\mid
i,\sigma\rangle[\varepsilon_{\mathrm{M}}\delta_{\sigma\sigma^{\prime}}-I{\bf
S}_{i}{\bf\cdot\tau}_{\sigma\sigma^{\prime}}]\langle i,\sigma^{\prime}\mid\ ,$
(3)

In the above equations, $\mid i,\sigma\rangle$ is an atomic orbital with spin
$\sigma$ (=$\uparrow$ or $\downarrow$) at site ($m$,$n$),
$\varepsilon_{\mathrm{A}}$ and $\varepsilon_{\mathrm{M}}$ are the on-site
energies for A- and M-sites, and the hopping energy $t_{ij}=t$ if $i$ and $j$
are nearest neighbors and zero otherwise. The second term on the right hand
side of Eq. 3 is the $\mathbf{k}$-independent exchange interaction in which
${\bf S}_{i}$ is the local spin operator of the $\mathrm{M}$-atom and
${\bf\tau}$ is the usual Pauli matrix for carrier’s spin. We regard the spin
of magnetic ion as a classical spin, while the value of exchange interaction
strength $IS$ (=$I\times S$) is finite.

The single-electron Hamiltonian can be written as

$H=\mathcal{H}_{eff}+V\ ,$ (4)

where the effective Hamiltonian $\mathcal{H}_{eff}$, which describes the
effective medium, is expressed as

$\mathcal{H}_{eff}=\sum_{i,j,\sigma}[t_{ij}+\delta_{i,j}\Sigma_{i}(\omega)]\mid
i,\sigma\rangle\langle j,\sigma\mid\ .$ (5)

Here, $\Sigma_{i}(\omega)$ is the site-dependent self-energy, and the
perturbation term is given as

$\displaystyle V$ $\displaystyle=$ $\displaystyle H-\mathcal{H}_{eff}$ (6)
$\displaystyle=$ $\displaystyle\sum_{i}v_{i}\ ,$

where $v_{i}=v_{i}^{\mathrm{A}}$ for the A-site and $v_{i}=v_{i}^{\mathrm{M}}$
for the M-site are given by

$v_{i}^{\mathrm{A}}=\sum_{\sigma}\mid
i,\sigma\rangle[\varepsilon_{\mathrm{A}}-\Sigma_{i}]\langle i,\sigma\mid\ ,$
(7) $v_{i}^{\mathrm{M}}=\sum_{\sigma\sigma^{\prime}}\mid
i,\sigma\rangle[(\varepsilon_{\mathrm{M}}-\Sigma_{i})\delta_{\sigma,\sigma^{\prime}}-I{\bf
S}_{i}{\bf\cdot\tau}_{\sigma\sigma^{\prime}}]\langle i,\sigma^{\prime}\mid\ ,$
(8)

It should be emphasized that, we have assumed a spin-independent effective
medium, since the system is at the paramagnetic temperatures ($T\gg T_{c}$).
Thus, $\Sigma_{i}$ does not depend on the spin of carriers and hence, the
electronic states will be independent of the temperature within the
paramagnetic regime. Here, $T_{c}$ is defined as the ferromagnetic critical
temperature where the spontaneous magnetization of magnetic impurities
vanishes, and in $\mathrm{Ga}_{1-x}\mathrm{Mn}_{x}\mathrm{As}$ quantum wires,
for example, $T_{c}$ as high as 350 K has been reported Jeon .

The matrix elements of effective Green’s function, $\bar{G}$, can be
determined from the Dyson equation:

$\bar{G}_{i,i^{\prime}}(\omega)=G^{0}_{i,i^{\prime}}(\omega)+\sum_{j}G^{0}_{i,j}(\omega)\Sigma_{j}(\omega)\bar{G}_{j,i^{\prime}}(\omega)\
,$ (9)

where ${G}^{0}_{i,j}$ is the clean system Green’s function matrix element and
is given by

$\displaystyle G^{0}_{i,i^{\prime}}$ $\displaystyle=$ $\displaystyle\langle
m,n|G^{0}(\omega)|m^{\prime},n^{\prime}\rangle$ (10) $\displaystyle=$
$\displaystyle\frac{1}{N_{x}}\sum_{k_{x}}\sum_{l=1}^{N_{y}}\frac{f_{n,n^{\prime}}(l)}{\omega+i\eta-\epsilon_{l,k_{x}}}e^{ik_{x}(m-m^{\prime})a}\
,$

where

$f_{n,n^{\prime}}(l)=\frac{2}{N_{y}+1}\sin(\frac{l\pi}{N_{y}+1}n)\sin(\frac{l\pi}{N_{y}+1}n^{\prime})\
,$ (11)

and

$\epsilon_{l,k_{x}}=2t\cos(\frac{l\pi}{N_{y}+1})+2t\cos(k_{x}a)\ ,$ (12)

is the clean system band structure.

Here, $N_{x}$ and $k_{x}$ are the number of lattice sites and the wave vector
in the $x$ direction, $a$ is the lattice constant, $l$ is the mode of the
subband, and $\eta$ is a positive infinitesimal. Since the translational
symmetry is absent in the $y$ direction, the self-energy depends on the atomic
line number ($n$), however, it is independent of the number ($m$) of the
atomic site on each line, i.e. $\Sigma_{i}(\omega)\equiv\Sigma_{n}(\omega)$.
In this case, the Dyson equation (9) can be rewritten as

$\displaystyle\bar{G}_{n_{1},n_{2}}(m_{1},m_{2};\omega)=G^{0}_{n_{1},n_{2}}(m_{1},m_{2};\omega)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle+\sum_{n=1}^{N_{y}}\sum_{m}G^{0}_{n_{1},n}(m_{1},m;\omega)\Sigma_{n}(\omega)\bar{G}_{n,n_{2}}(m,m_{2};\omega)\
.$ (13)

The CPA replaces the real system with an effective periodic medium Soven ;
Gonis . For this purpose, the potential of all sites is replaced by an energy-
dependent coherent potential, except one site which is denoted by _impurity_.
The effective medium is determined self-consistently in such a way that the
Green’s function of the effective medium is equal to the configurationally
averaged Green’s function of the real medium. Therefore, the effective
scattering of a carrier at the impurity site is zero, on average.

In order to derive the CPA equation for the coherent potential, we write the
Green’s function of the real system, $G\equiv(\omega-H)^{-1}$, in terms of the
effective Green’s function, $\bar{G}$, and the total scattering matrix, $T$,
as

$G=\bar{G}+\bar{G}T\bar{G}\ ,$ (14)

where, $T=V(1-\bar{G}V)^{-1}$. Using the multiple scattering theory Gonis ,
one can express $T$ as the multiple scattering series

$T=\sum_{i}t_{i}+\sum_{i}\sum_{j\neq i}t_{i}\bar{G}t_{j}+\sum_{i}\sum_{j\neq
i}\sum_{k\neq j}t_{i}\bar{G}t_{j}\bar{G}t_{k}+\cdots\ .$ (15)

Here, $t_{i}(\equiv t_{m,n})$ is the single-site $t$ matrix which represents
the multiple scattering of carriers due to the isolated potential
$v_{i}(\equiv v_{m,n})$ in the effective medium, and is expressed as

$t_{i}=v_{i}(1-\bar{G}v_{i})^{-1}\ .$ (16)

The averaging of Eq.(14) and the use of single-site CPA condition $\langle
t_{i}\rangle_{\mathrm{av}}=0$ for any site $i$ in the wire, leads to $\langle
T\rangle_{\mathrm{av}}=0$ and thus, we obtain $\langle
G\rangle_{\mathrm{av}}=\bar{G}$, as mentioned above.

The present system includes both substitutional disorder and spin scattering.
Therefore, the CPA equation for the coherent potential is given by

$\langle t_{m,n}\rangle_{\mathrm{av}}=(1-x)t_{m,n}^{\mathrm{A}}+x\langle
t_{m,n}^{\mathrm{M}}\rangle_{\mathrm{spin}}=0\ ,$ (17)

where, $t^{\mathrm{A}}_{m,n}(t^{\mathrm{M}}_{m,n})$ represents the complete
scattering associated with the isolated potential
$v^{\mathrm{A}}_{m,n}(v^{\mathrm{M}}_{m,n})$ in the effective medium, and
$\langle\cdots\rangle_{\mathrm{spin}}$ denotes average over the spin
scattering at the M-site. In the classical spin treatment, the potential for
which a carrier is subjected at the M-site is regarded as
$\varepsilon_{\mathrm{M}}-IS$ or $\varepsilon_{\mathrm{M}}+IS$, depending on
whether the localized spin on the M-site and the carrier spin are parallel or
antiparallel with each other. At the paramagnetic temperature at which the
orientation of localized spin is completely random, the probability of each
state is 1/2. Therefore, the associated $t$-matrices for an arbitrary site of
each atomic line, such as $(0,n)$, are given by

$t_{0,n}^{\mathrm{A}}=\frac{\varepsilon_{\mathrm{A}}-\Sigma_{n}}{1-(\varepsilon_{\mathrm{A}}-\Sigma_{n})F_{n}}\
,$ (18)

and

$\displaystyle\langle t_{0,n}^{\mathrm{M}}\rangle_{\mathrm{spin}}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left[\frac{\varepsilon_{\mathrm{M}}-IS-\Sigma_{n}}{1-(\varepsilon_{\mathrm{M}}-IS-\Sigma_{n})F_{n}}\right]$
(19) $\displaystyle+$
$\displaystyle\frac{1}{2}\left[\frac{\varepsilon_{\mathrm{M}}+IS-\Sigma_{n}}{1-(\varepsilon_{\mathrm{M}}+IS-\Sigma_{n})F_{n}}\right]\
,$

where

$\displaystyle F_{n}(\omega)$ $\displaystyle=$
$\displaystyle\bar{G}_{n,n}(m,m;\omega)=\bar{G}_{n,n}(0,0;\omega)$
$\displaystyle=$
$\displaystyle\frac{a}{2\pi}\int_{-\pi/a}^{\pi/a}\bar{G}_{n,n}(k_{x};\omega)\,dk_{x}\mathrm{~{}~{}for~{}~{}}n=1,\cdots,N_{y}$

In Eq. (II), we should emphasize that the effective Green’s function,
$\bar{G}$, depends on the one-dimensional wave vector $k_{x}$ via $G^{0}$ (see
Eq.(10)). Hence, to obtain any specific matrix element
$\bar{G}_{n_{1},n_{2}}$, we must integrate over the first Brillouin zone of
the one-dimensional lattice Gonis .

Equation (17) leads to a cubic equation for the self-energy of $n$th atomic
line, $\Sigma_{n}(\omega)$, as

$\mathcal{A}\Sigma_{n}^{3}+\mathcal{B}\Sigma_{n}^{2}+\mathcal{C}\Sigma_{n}+\mathcal{D}=0\mathrm{~{}~{}for~{}~{}}n=1,\cdots,N_{y}\
,$ (21)

with

$\displaystyle\mathcal{A}$ $\displaystyle=$ $\displaystyle-F_{n}^{2}$ (22)
$\displaystyle\mathcal{B}$ $\displaystyle=$
$\displaystyle(\varepsilon_{\mathrm{A}}+2\varepsilon_{\mathrm{M}})F_{n}^{2}-2F_{n}$
(23) $\displaystyle\mathcal{C}$ $\displaystyle=$
$\displaystyle[(IS)^{2}-2\varepsilon_{\mathrm{A}}\varepsilon_{\mathrm{M}}-\varepsilon_{\mathrm{M}}^{2}]F_{n}^{2}$
(24)
$\displaystyle+[x(\varepsilon_{\mathrm{M}}-\varepsilon_{\mathrm{A}})+2(\varepsilon_{\mathrm{M}}+\varepsilon_{\mathrm{A}})]F_{n}-1$
$\displaystyle\mathcal{D}$ $\displaystyle=$
$\displaystyle[\varepsilon_{\mathrm{M}}^{2}\varepsilon_{\mathrm{A}}-(IS)^{2}\varepsilon_{\mathrm{A}}]F_{n}^{2}$
(25)
$\displaystyle+[x(IS)^{2}+(x-2)\varepsilon_{\mathrm{M}}\varepsilon_{\mathrm{A}}-x\varepsilon_{\mathrm{M}}^{2}]F_{n}$
$\displaystyle+x(\varepsilon_{\mathrm{M}}-\varepsilon_{\mathrm{A}})+\varepsilon_{\mathrm{A}}\
.$

It should be noted that, Eq. (II) is a system of linear equations which should
be solved self-consistently, using Eqs. (II) and (21), to obtain the coherent
potentials $\Sigma_{n}(\omega)$. Then, the LDOS per site in the $n$th atomic
line of the quantum wire, $g_{n}(\omega)$, is calculated by

$g_{n}(\omega)=-\frac{1}{\pi}\,\mathrm{Im}\,F_{n}(\omega)\ .$ (26)

We should remind the reader that, the LDOS is a function of the energy and the
space coordinate, which illustrates the spatial distribution of the states at
the particular location (here, $n$), and it is well known that many important
physical properties and characteristics of a mesoscopic system are determined
by the LDOS, which is experimentally measurable.

When Eq. (21) is solved for a real energy $\omega$, we obtain three roots. We
only choose the correct root corresponding to $\omega+i\eta$, i.e., the
imaginary part of $F_{n}$ must be negative in order to give a positive LDOS.
The existence and uniqueness of such a solution depend on the initial guess
for the self-energies $\Sigma_{n}$. We believe that, the best guess for
$\Sigma_{n}$, is zero. One should note, however, that the real and imaginary
parts of the roots and the speed of convergence depend on the parameters of
the system. Except for a few values of the energy, we have rapid convergence
when the final value of $\Sigma_{n}$ for a given energy is taken as the
initial $\Sigma_{n}$ for the next energy. Furthermore, in the case of weak
exchange interaction, the convergence is faster than in the strong one.

It is important to note that, if we set $IS=0$, the present formalism will be
applicable for a NMS quantum wire of general type
$\mathrm{A}_{1-x}\mathrm{D}_{x}\mathrm{B}$ which has been doped with donor or
acceptor nonmagnetic impurities. Here, D indicates the nonmagnetic impurity
atom, and in the above equations, we should set
$\varepsilon_{\mathrm{M}}=\varepsilon_{\mathrm{D}}$. In next section we
present the numerical results of LDOS for both the NMS and DMS quantum wires.

##  III Results and discussion

In our numerical calculations, we measure the energies in units of $t$ and we
set $\varepsilon_{\mathrm{A}}=0$, since we can shift the chemical potential
without loss of physics. We present the numerical results for both the NMS and
DMS quantum wires with $N_{y}=5$. In both cases, we have assumed the carrier
density is very low; hence, we have ignored the interaction between carriers.
In practice, we have done the numerical calculations for a case in which only
single carrier moves in the conduction (or valance) band of the quantum wire.

In quantum wires, quantum effects influence the electronic properties of the
system. Due to the confinement of free carriers in the transverse direction of
the wire, their transverse energy is quantized into a series of discrete
values. In practice, when the size or dimension of a material diminishes to
the nanoregion, the charge carriers begin to experience the effects of
confinement, meaning that their motion becomes limited by the physical size of
the region or domain in which they move.

Figure 1: The energy dispersion and the LDOS curves of a clean semiconductor
quantum wire. (a) The dispersion curve as a function of normalized wave
vector, and (b)-(d) the LDOS as a function of energy, for the atomic lines
$n=1,5$, $n=2,4$ and $n=3$ respectively. Energy is measured in units of $t$.

In Fig. 1, the energy dispersion and the LDOS curves of a clean quantum wire
are shown for comparison. In such system, there are states with zero group
velocity which are responsible for the singularities of spectral density.
Therefore, the van Hove singularities appear as sharp features in the LDOS and
due to the presence of five atomic lines, five subbands are observed Hugle .
Due to the symmetry of the system, the LDOS for $n=1$ is the same as for $n=5$
(which are at the edges of wire), and $n=2$ with $n=4$. Also, the number of
sharp peaks depends on the atomic line number.

Figure 2: The LDOS of a NMS quantum wire with $\varepsilon_{\mathrm{D}}=-0.75$
as functions of energy and impurity concentration, for the atomic lines (a)
$n=1,5$, (b) $n=2,4$ and (c) $n=3$ respectively. Energy is measured in units
of $t$.

To illustrate the impurity effects in NMS quantum wires, the LDOS as a
function of energy is plotted in Fig. 2 for $\varepsilon_{\mathrm{D}}=-0.75$
and in Fig. 3 for $\varepsilon_{\mathrm{D}}=+0.75$. An impurity with negative
(positive) site energy is analogous to an acceptor (a donor) center at the
host crystal. At zero concentration of impurity ($x=0$), the band for all
atomic lines, is equivalent to the one in the clean system (see Fig.
1(b)-1(d)). For the case of $\varepsilon_{\mathrm{D}}<0$
($\varepsilon_{\mathrm{D}}>0$), with increasing $x$ the band shifts towards
lower (higher) energies. Also, with increasing $x$, the relative sharpness of
peaks reduces and at $x=0.5$ which represents the state of maximum
substitutional disorder, the LDOS is completely symmetric with respect to the
center of band and the sharpness disappears. With further increasing $x$, the
peaks again become sharp, and for $x=1$, the LDOS with a shift equal to
$\varepsilon_{\mathrm{D}}$, is completely equivalent to the band of clean
system. The reason of such behavior is that, for $x>0.5$ the majority of atoms
have $\varepsilon_{\mathrm{D}}\neq 0$ ($\varepsilon_{\mathrm{D}}=-0.75$ or
$\varepsilon_{\mathrm{D}}=+0.75$ depending on the type of impurity). Thus, one
can imagine the situation in which the atoms with $\varepsilon_{\mathrm{D}}=0$
act as impurity atoms in a host crystal with $\varepsilon_{\mathrm{D}}\neq 0$.
These features can be seen in all atomic lines. Therefore, the nonmagnetic
impurity can change the behavior of LDOS and there is not band broadening in
the NMS quantum wires.

Figure 3: The same as Fig. 2 but for $\varepsilon_{\mathrm{D}}=+0.75$.

We now investigate the influence of magnetic impurities on the electronic
properties of semiconductor quantum wires. Doping of magnetic atoms such as Mn
into GaAs or InAs quantum wires, introduces not only magnetic moments but also
free carriers. Therefore, in such quantum wires, we should consider both the
effects of chemical potential and magnetic disorder on the LDOS at the
paramagnetic temperatures. Figures 4 and 5 show how the carrier band changes
with $x$ in the case of weak and strong exchange interaction, respectively. We
have shown the results for $\varepsilon_{\mathrm{M}}=-0.75$, $IS=-0.4$ (weak
exchange interaction) and $IS=-1.2$ (strong exchange interaction) as sampling
cases of III-V-based DMS’s. For this kind of impurity, with increasing $x$ the
sharpness of peaks decreases continuously and at the concentration $x=1.0$ at
which the LDOS is completely symmetric with respect to the center of band, the
van Hove singularities completely disappear (particularly for $IS=-1.2$), and
the curves of all atomic lines become smooth. Such features have not been
observed in the bulk case Taka99 . It should be noted that the results are
same for different signs of $IS$, because the system is in paramagnetic phase
and the spin of magnetic atoms is treated classically.

Figure 4: The LDOS of a DMS quantum wire with $\varepsilon_{\mathrm{M}}=-0.75$
and $IS=-0.4$ as functions of energy and impurity concentration, for the
atomic lines (a) $n=1,5$, (b) $n=2,4$ and (c) $n=3$ respectively. Energy is
measured in units of $t$.

On the other hand, the results show that, the LDOS of atomic lines is
different with each other. This feature indicates that the electronic
transport depends on the atomic line number. The difference in the density of
states of the atomic lines predicts a nonuniform charge distribution in such
quantum wires. Therefore the atomic lines will have different contributions in
carrier transport and cause the quantum interferences, which can be important
in the process of charge transport through nano-scale devices. Similar feature
in the coherent electron conductance of a quantum point contact in the
presence of a scanning probe microscope tip, has been reported both
experimentally Topinka  and theoretically He .

Figure 5: The same as Fig. 4 but for $IS=-1.2$.

Now, we discuss the band-edge energy shift and the band broadening due to the
random distribution of $\mathrm{M}$ ions and the fluctuation of localized
spins. The energy shift and the broadening of the band can be calculated using
the band-edge energies, $\omega_{b}$, at which the density of states goes to
zero and the imaginary part of the self-energy vanishes. The approximate
solution for the lower (upper) band edge is given by Taka99

$\omega_{b}=\Sigma_{n}^{l(u)}(\omega_{b})\mp W_{0}\ ,$ (27)

where $\Sigma_{n}^{l(u)}(\omega_{b})\equiv\Sigma_{b}^{l(u)}$ is the energy
shift of the lower (upper) band edge, and $W_{0}\simeq 3.74\,t$ is the half-
bandwidth of the clean system. Using Eqs. (II) and (27), we numerically obtain
$F_{n}(\omega_{b})t\simeq\mp 0.81$. Substituting these values into Eq. (21),
we obtain real $\Sigma_{b}^{l(u)}$ or the energy shifts of the band edges for
the system. In Fig. 6, the band-edge energy shift and the band broadening are
depicted as a function of the impurity concentration for various values of
$IS$, with $\varepsilon_{\mathrm{M}}=0$ for simplicity. In such a case, the
LDOS is symmetric around $\omega=0$, hence the magnitudes of the band-edge
energy shifts are equal and the band broadening is given by
$2|\Sigma_{b}^{l}|$ or $2|\Sigma_{b}^{u}|$. The results clearly show that, in
the presence of magnetic impurities, the bands are broadened with the increase
of $x$ due to the impurity spin fluctuation. This band broadening depends on
the strength of magnetic disorder and is much larger when the interaction is
stronger (compare Fig. 4 and 5).

Figure 6: The energy shift of the lowest band edge and the band broadening for
various values of $IS$, with $\varepsilon_{\mathrm{M}}=0$ as a function of
impurity concentration. Energy is measured in units of $t$.

We applied our theory for both the NMS and DMS quantum wires, by employing
parameters for simulating the semiconducting alloys. The parameters of the
tight-binding model used in this work can be chosen from realistic orbitals of
atomic species which are suitable for an experimental realization of the
modeled quantum wires. Such parameters have been used to study bulk DMS’s
Taka99 ; Taka02 . However, we should note here that, our investigation is
based on a single-band model and a single-site approximation, neglecting many
features such as multiband effects, off-diagonal disorder, Coulomb
interactions between free carriers, and correlated defects. If such features
affect the physical properties of a real system, the above mentioned model
should be improved. Otherwise, one cannot expect to obtain accurate results
for that system, using the experimental parameters.

One could consider the present system as an infinite stack of atomic slices
along the $x$ direction. This means that we consider the disorder in the $y$
direction. Thus, in such a case, we should define a position-dependent self-
energy for each slice. In other words, the self-energy, in the cross section
of the wire, is different from one site to the other; however, the slices have
equal self-energy values. The single-site CPA condition should be used for
each slice to obtain the self-energies, and, in order to derive the Green’s
function of such a system, one can use a method similar to that for semi-
infinite leads Nik1 ; Nik2 ; MacK .

The calculations presented here can be extended for more than one atomic strip
and study the transport of free carriers parallel to the strips (layers). In
such a case, the transport properties of the system depend on the strip
number, which can be attractive for planar devices with submicron dimensions.

##  IV Conclusions

We have studied the effects of magnetic and nonmagnetic impurities on the
electronic states of semiconductor quantum wires. Using the CPA for random
distribution of impurity atoms, we investigated the influence of impurity
concentration and the strength of exchange interaction on the LDOS. In NMS
quantum wires, the acceptor (donor) impurities only shift the bands towards
lower (higher) energies, and the van Hove singularities in the LDOS depend on
the value of impurity concentration. For DMS quantum wires, the magnetic
impurities broaden the bands and reduce continuously the peaks sharpness, even
in the case of weak exchange coupling. The results presented here predict that
the DMS quantum wires can be used in laser operation and the physics of
spintronic nanodevices.

## Acknowledgments

The author would like to thank Professor R. Moradian for helpful discussions.
This work was supported by the Payame Noor University.

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compat=1.1.0

11institutetext: Departament de Física Quàntica i Astrofísica and Institut de
Ciències del Cosmos (ICCUB), Facultat de Física, Universitat de Barcelona,
Martí i Franquès 1, 08028 Barcelona, Spain

# Thermal modification of open heavy-flavor mesons from an effective hadronic
theory

Glòria Montaña 11<EMAIL_ADDRESS>
###### Abstract

We have developed a self-consistent theoretical approach to study the
modification of the properties of heavy mesons in hot mesonic matter which
takes into account chiral and heavy-quark spin-flavor symmetries. The heavy-
light meson-meson unitarized scattering amplitudes in coupled channels
incorporate thermal corrections by using the imaginary-time formalism, as well
as the dressing of the heavy mesons with the self-energies. We report our
results for the ground-state thermal spectral functions and the implications
for the excited mesonic states generated dynamically in the heavy-light
molecular model. We have applied these to the calculation of meson Euclidean
correlators and transport coefficients for $D$ mesons and summarize here our
findings.

##  1 Introduction

Heavy-ion collision experiments in the Relativistic Heavy Ion Collider (RHIC),
at BNL, and in the Large Hadron Collider (LHC), at CERN, recreate in the
laboratory the conditions for which the quark-gluon plasma (QGP) was created
in the early Universe, short after the Big Bang. Fundamental quantities that
permit to characterize this medium are thermal spectral and transport
properties.

Heavy quarks, namely the charm and bottom quarks, are produced through the
hard-scattering processes at the initial stage of the collisions and interact
with the constituents of the QGP. When the temperature of the system goes
below the confining transition temperature heavy quarks hadronize into heavy
hadrons, mostly into open heavy-flavor ($D$ and $\bar{B}$) mesons for the high
energies achieved at RHIC and LHC, where low baryonic densities are reached.
The spectral properties of the open heavy-flavor mesons are modified in the
hot hadronic medium from their interaction with the light mesons forming the
medium.

Finite temperature corrections to the scattering of $D^{(*)}$ and
$D_{s}^{(*)}$ mesons with the Goldstone bosons ($\pi$, $K$, $\bar{K}$ and
$\eta$ mesons) have been implemented within a unitarized thermal approach in
Montana:2020lfi ; Montana:2020vjg . This framework exploits both chiral
symmetry and heavy-quark spin-flavor symmetry (HQSFS) to construct the
effective Lagrangian that describes the interactions between the heavy and
light degrees in vaccuum. HQSFS implies that the interactions are independent
of the heavy-quark spin and flavor. The unitarization technique in coupled
channels allows us to extend the energy region where the effective theory can
be applied, as well as to generate resonances and bound states from the two-
body dynamics. In this paper we have studied the thermal modification of these
states in the hadronic phase, below the critical temperature $T_{c}$. The
finite temperature is incorporated using the imaginary-time formalism (ITF)
and the in-medium scattering amplitudes and ground-state self-energies are
computed self-consistently.

In Section 2 of this contribution to the proceedings of the Virtual Tribute to
Quark Confinement and the Hadron Spectrum 2021 conference we present our
latest results on the thermal modification of the properties (masses and
widths) of the $D$ and $\bar{B}$ mesons, as well as of the resonances and
bound states that are dynamically generated from their scattering with the
light mesons. The effective theory results for the thermal scattering
amplitudes and the spectral functions were used to compute open-charm meson
Euclidean correlators in Ref. Montana:2020var  and off-shell transport
coefficients in Ref. Torres-Rincon:2021yga . The main results are summarized
in Sections 3 and 4 of this contribution, respectively. Further details can be
found in Refs. Montana:2020lfi ; Montana:2020vjg ; Montana:2020var ; Torres-
Rincon:2021yga .

##  2 Thermal effective field theory for open heavy-flavor mesons

The interaction in vacuum between the open heavy-flavor mesons
($D^{(*)}=\\{D^{(*)},D_{s}^{(*)}\\}$) and the pseudoscalar Goldstone bosons
($\Phi=\\{\pi,K,\bar{K},\eta\\}$) is described in Refs. Montana:2020lfi ;
Montana:2020vjg  by means of an effective Lagrangian that is based on chiral
symmetry and HQSFS Guo:2009ct ; Liu:2012zya ; Tolos:2013kva ; Guo:2018tjx .
The corresponding tree-level amplitudes of the $D\Phi$ (or $D^{*}\Phi$)
scattering for an incoming channel $i$ and an outgoing channel $j$ read

$\displaystyle V^{ij}(s,t,u)=$ $\displaystyle\
\frac{1}{f_{\pi}^{2}}\Big{[}\frac{C_{\rm
LO}^{ij}}{4}(s-u)-4C_{0}^{ij}h_{0}+2C_{1}^{ij}h_{1}$ $\displaystyle\
-2C_{24}^{ij}\Big{(}2h_{2}(p_{2}\cdot p_{4})+h_{4}\big{(}(p_{1}\cdot
p_{2})(p_{3}\cdot p_{4})+(p_{1}\cdot p_{4})(p_{2}\cdot p_{3})\big{)}\Big{)}$
(1) $\displaystyle\ +2C_{35}^{ij}\Big{(}h_{3}(p_{2}\cdot
p_{4})+h_{5}\big{(}(p_{1}\cdot p_{2})(p_{3}\cdot p_{4})+(p_{1}\cdot
p_{4})(p_{2}\cdot p_{3})\big{)}\Big{)}\Big{]},$

where $s=(p_{1}+p_{2})^{2}$, $t=(p_{1}-p_{3})^{2}$ and $u=(p_{1}-p_{4})^{2}$
are the usual Mandelstam variables. The values of the low-energy constants
(LECs) of the terms at NLO in the chiral expansion, $h_{0,...,5}$, were fitted
in Guo:2018tjx  to lattice results of scattering lengths and finite-volum
spectra. Keeping lowest order in the heavy-quark mass expansion, there are no
tree-level diagrams connecting $D\Phi$ and $D^{*}\Phi$ channels, and the LECs
take the same values for these two sectors. The isospin coefficients $C_{ij}$
for the different strangeness and isospin sectors and the values of the LECs
are given in Tables III and IV in Ref. Montana:2020vjg . Relying on heavy-
quark flavor symmetry (HQFS) the interaction potential of Eq. (2) can be
extended to the open-bottom sector upon the replacement $D\rightarrow\bar{B}$
and the rescaling of the LECs with the heavy meson masses,

$\frac{h_{i}^{D}}{\hat{m}_{D}}=\frac{h_{i}^{B}}{\hat{m}_{B}}\
,\quad\textrm{for}\;i=\\{0,1,2,3\\}\ ,\quad\textrm{and}\quad
h_{4,5}^{D}\hat{m}_{D}=h_{4,5}^{B}\hat{m}_{B}\
,\quad\textrm{for}\;i=\\{4,5\\}\ .$ (2)

The $s$-wave projection of the amplitude in Eq. (2) is unitarized by solving
the Bethe-Salpeter equation in coupled channels (Fig. 1(a)),

$T_{ij}(s)=V_{ij}(s)+V_{ik}(s)G_{k}T_{kj}(s)\ ,$ (3)

where $T$ is the unitarized amplitude, $V$ is the $s$-wave interaction kernel
and $G$ is the loop function constructed from the two meson propagtors,

$G^{l}(s)=i\int\frac{d^{4}q}{(2\pi)^{4}}\frac{1}{q^{2}-m_{D}^{2}+i\epsilon}\frac{1}{(p-q)^{2}-m_{\Phi}^{2}+i\epsilon}\
,$ (4)

which is regularized with a hard momentum cutoff $\Lambda$. Equation (3) has
an algebraic solution in the on-shell approximation,
$T(s)=V(s)[1-V(s)G(s)]^{-1}$, which is a matrix equation for the case of two
or more coupled channels.

$D_{i}$$D_{j}$$\Phi_{j}$$\Phi_{i}$

$=$ $D_{i}$$D_{j}$$\Phi_{j}$$\Phi_{i}$ $+$
$D_{i}$$D_{j}$$\Phi_{j}$$\Phi_{i}$$D_{k}$$\Phi_{k}$

(a)

D

$=$ $D$ $+$ $\pi$$D$D

(b)

Figure 1: (a) Bethe-Salpeter equation and (b) Dyson-equation for the heavy-
meson propagator at finite temperature. Big red circles represent the
$T$-matrix, small blue circles correspond to the interaction kernel and thick
orange lines depict dressed propagators. Figure taken from Ref.
Montana:2020lfi .

At finite temperature the loop function $G$ is calculated through the ITF, in
which the time dimension is Wick rotated and compactified in the range
$[0,\beta=1/T]$, with $T$ the temperature of the system, and the energy
integral transforms into a summation over discrete Matsubara frequencies,
$q^{0}\rightarrow i2\pi nT$ for bosons Das:1997gg ; Weldon:1983jn . We also
introduce the Lehmann representation for the meson propagators

$\mathcal{D}_{M}(i\omega_{n},\textbf{q})=\int
d\omega^{\prime}\frac{S_{M}(\omega^{\prime},\textbf{q})}{i\omega_{n}-\omega^{\prime}}\
,$ (5)

where $S_{M}$ is the spectral function and $M$ denotes the meson species ($D$
or $\Phi$). After the Matsubara summation and the analytical continuation to
real energies, $i\omega_{m}\rightarrow E+i\varepsilon$, the thermal loop
function reads

$G_{D\Phi}(E,\textbf{p};T)=\int\frac{d^{3}q}{(2\pi)^{3}}\int d\omega\int
d\omega^{\prime}\frac{S_{D}(\omega,\textbf{q};T)S_{\Phi}(\omega^{\prime},\textbf{p}-\textbf{q};T)}{E-\omega-\omega^{\prime}+i\varepsilon}[1+f(\omega,T)+f({\omega^{\prime}},T)]\
,$ (6)

where $f(\omega,T)=[\exp(\omega/T)-1]^{-1}$ is the equilibrium Bose-Einstein
distribution function.

The spectral function of the heavy meson is given by

$S_{D}(\omega,\textbf{q};T)=-\frac{1}{\pi}{\rm
Im\,}\mathcal{D}_{D}(\omega,\textbf{q};T)=-\frac{1}{\pi}{\rm
Im\,}\Bigg{(}\frac{1}{\omega^{2}-\textbf{q}^{2}-m_{D}^{2}-\Pi_{D}(\omega,\textbf{q};T)}\Bigg{)}\
,$ (7)

where the self-energy follows from closing the light-meson line in the
corresponding $T$-matrix element (Fig. 1(b)),

$\Pi_{D}(E,\textbf{p};T)=\frac{1}{\pi}\int\frac{d^{3}q}{(2\pi)^{3}}\int
d\Omega\frac{E}{\omega_{\Phi}}\frac{f(\Omega,T)-f(\omega_{\Phi},T)}{E\,^{2}-(\omega_{\Phi}-\Omega)^{2}+i\varepsilon}{\rm
Im\,}T_{D\Phi}(\Omega,\textbf{p}+\textbf{q};T)\ .$ (8)

In Refs. Montana:2020lfi ; Montana:2020vjg  we have approximated the spectral
function of the light meson to a $\delta$-type spectral function because the
in-medium modification of the pions is small. At $T\lesssim 150\rm\,MeV$ the
largest contribution to the self-energy comes from the interaction of the
heavy mesons with the pions, as the abundance of heavier light mesons, i.e.
kaons, antikaons and eta mesons, is thermally suppressed. This has been
discussed in detail in Ref. Montana:2020vjg . In the results below only the
pion contribution is considered in Eq. (8) to calculate the heavy-meson self-
energy.

Figure 2: Real and imaginary parts of the two-meson loop function at finite
temperature. Top plots: $D\Phi$ case. Bottom plots: $D_{s}\Phi$ case. Data
taken from Montana:2020vjg .

Equations (6) to (8) are coupled to each other and require a self-consistent
solution. The results obtained for the loop function for $D\Phi$ and
$D_{s}\Phi$ channels are shown in the top and bottom panels of Fig. 2,
respectively, for different temperatures. At $T=0$ (black lines) the imaginary
part becomes non-zero in the energy region above threshold
($m_{D/D_{s}}+m_{\Phi}$), where the unitary cut develops. At $T>0$ an
additional cut opens up for energies below $m_{D/D_{s}}-m_{\Phi}$. This is the
so-called Landau cut and it is related to thermal production and absorption
processes Das:1997gg ; Weldon:1983jn . From the loop functions in Fig. 2 it is
clear that the Landau cut is much larger for channels involving pions (for $K$
and $\eta$ mesons it cannot be distinguished with the scale used for the
plots), and its importance grows with increasing temperatures because of the
larger abundance of thermal pions. In addition, at finite temperature, the
boundaries of the cuts are smoothened due to the widening of the heavy-meson
spectral function, as discussed in Montana:2020lfi ; Montana:2020vjg .

Figure 3 shows the self-consistent results for the unitarized scattering
amplitudes at finite temperature in the sectors in which dynamically generated
states with $J^{P}=0^{+}$ appear. In the charm sector with strangeness and
isospin $(S,I)=(0,1/2)$ (first panel) the two resonances are identified with
the experimental $D_{0}^{*}(2300)$ state, for which a two-pole structure at
$T=0$ was described in vaccuum in Albaladejo:2016lbb . The lower resonance is
seen in the $D\pi$ scattering amplitude, while the higher one is visible in
the $D_{s}\bar{K}$ channel. The thermal effects on the generation of these
states are moderate and the structures are still clearly seen at
$T=150\leavevmode\nobreak\ \textrm{MeV}$. In the sector with $(S,I)=(1,0)$
(second panel) the bound state associated to the $D_{s0}^{*}(2317)$ in vacuum
gains a substantial thermal width as the temperature is increased. The bottom
counterparts of these states have not been reported experimentally yet. We
predict here their thermal modification to be of the same order as in the
charm sector, as expected from HQFS (third and fourth panels). The thermal
scattering amplitudes for $D^{*}\Phi$ and $D_{s}^{*}\Phi$ and the
corresponding resonances with $J^{P}=1^{+}$ are reported in Montana:2020vjg .

Figure 3: $T$-matrix at finite temperature for the diagonal transitions in the
sectors with charm $C=1$ (bottom $B=1$), spin $J=0$ and strangeness and
isospin $(S,I)=\\{(0,1/2),(1,0)\\}$. Figure extracted from Montana:2021vks
with data partially taken from Montana:2020lfi . Figure 4: Thermal evolution
of the mass (left panel) and the width (right panel) of the open heavy-flavor
ground states. Data partially taken from Montana:2020vjg .

The thermal modification of the ground states $D$, $D_{s}$, $\bar{B}$ and
$\bar{B}_{s}$ is extracted from their respective spectral functions. Figure 4
displays the evolution of the masses and the widths as a function of
temperature in the left and right panels, respectively. The heavy ground
states, which are stable degrees of freedom of the model in vacuum, acquire a
thermal width due to their interaction with the thermal pions of the bath.
Furthermore the masses decrease of the order of $\sim 20-50$ MeV at the
highest temperature considered ($T=150\leavevmode\nobreak\ \textrm{MeV}$). The
thermal modification of the open-charm vector counterparts is given in
Montana:2020vjg .

##  3 Open-charm Euclidean correlators

In this section we present the results for the Euclidean meson correlators of
open-charm mesons that were obtained in our previous work Montana:2020var .
Temporal Euclidean meson correlators in a hot medium can be computed on the
lattice for a discrete number of points in Euclidean time $\tau$. The
temperature of the system is related to the temporal extent of the lattice,
$T=1/(aN_{\tau})$. The spectral function is related to the Euclidean
correlator through the integral relation:

$G_{E}(\tau;T)=\int_{0}^{\infty}d\omega\,K(\tau,\omega;T)\,\rho(\omega;T)\ ,$
(9)

where the spectral function $\rho(\omega;T)$ is convoluted with a temperature
dependent kernel,

$K(\tau,\omega;T)=\cosh\Bigg{[}\omega\Bigg{(}\tau-\frac{1}{2T}\Bigg{)}\Bigg{]}\sinh^{-1}\Bigg{(}\frac{\omega}{2T}\Bigg{)}\
.$ (10)

The extraction of a continuous spectral function from the lattice QCD
correlators requires the inversion of Eq. (9). However such an inversion of a
discrete number of data points with statistical errors is an ill-posed problem
and the methods that are usually employed rely on specific assumptions for the
spectral function. In Ref. Montana:2020var  we took the opposite approach. In
order to compare the effective field theory results with the lattice QCD
calculations for open-charm mesons at finite temperature of Ref. Kelly:2018hsi
we insert the thermal spectral function in Eq. (9) and obtain the
corresponding Euclidean correlator.

The full meson spectral function $\rho(\omega;T)$ contains information of the
ground state and the additional contribution of excited bound states and the
continuum of scattering states. The ground-state spectral function is obtained
using Eq. (7) and the continuum is taken into account through a
parametrization of the free meson spectral function in the non-interacting
limit, while the contribution of the possible excited states is neglected in
Montana:2020var . Thus, the full spectral function reads

$\rho(\omega;T)=M_{D}^{4}S_{D}(\omega;T)+a\rho_{\rm cont}(\omega;T)\ ,$ (11)

where the contribution of the continuum is weighted with respect to the
ground-state contribution through the factor $a$.

Figure 5: Left panel: Full spectral functions of the $D$-meson for unphysical
meson masses. Right panel: Ratio of the Euclidean correlator and the
reconstructed correlator at $T_{r}=44\leavevmode\nobreak\ \textrm{MeV}$. The
results are shown for temperatures below $T_{c}=185\leavevmode\nobreak\
\textrm{MeV}$ Kelly:2018hsi  and for various values of the weight parameter
$a$. Figure extracted from Montana:2021vks  with data taken from
Montana:2020var .

The full spectral functions obtained for the $D$ mesons in a thermal medium of
pions of mass $m_{\pi}=384\leavevmode\nobreak\ \textrm{MeV}$, i.e. the
unphysical pion mass used in the lattice calculations of Ref. Kelly:2018hsi ,
are shown in the left panel of Fig. 5. The results are given for various
temperatures below the pseudocritical temperature on the lattice
($T_{c}=185\leavevmode\nobreak\ \textrm{MeV}$) and for the weight parameter
$a=\\{0,1,10\\}$ in Eq. (11).

In order to discern the modification of the correlator due to the temperature
dependence of the spectral function alone and cancel that of the integration
kernel, the ratio with a recostructed correlator is usually computed. This is
shown in the right panel of Fig. 5, together with the results of the lattice
simulations of Ref. Kelly:2018hsi . The inclusion of the continuum improves
the behavior of the ratio at small $\tau$ and the results are in good
agreement with the lattice QCD data within the error bars for the lowest
temperatures considered.

##  4 $D$-meson transport coefficients

In this section we summarize our latest results on open-charm transport
coefficients of Ref. Torres-Rincon:2021yga , where we implemented the thermal
scattering amplitudes and the $D$-meson spectral functions in order to take
into account thermal and off-shell effects. After some justified
approximations, the off-shell kinetic equation derived in Torres-
Rincon:2021yga  following the Kadanoff-Baym approach kadanoff1962quantum
reduces to an off-shell Fokker-Planck equation for the Green’s function,

$\frac{\partial}{\partial t}G_{D}^{<}(t,k)=\frac{\partial}{\partial
k^{i}}\left\\{\hat{A}(k;T)k^{i}G_{D}^{<}(t,k)+\frac{\partial}{\partial
k^{j}}\left[\hat{B}_{0}(k;T)\Delta^{ij}+\hat{B}_{1}(k;T)\frac{k^{i}k^{j}}{|{\textbf{k}|}^{2}}\right]G_{D}^{<}(t,k)\right\\}\
.$ (12)

In this equation the drag force $\hat{A}$ and the diffusion coefficients
$\hat{B}_{0}$ and $\hat{B}_{1}$ are defined off shell, as they are calculated
using spectral functions, as well as thermal scattering amplitudes. They are
defined in Torres-Rincon:2021yga  as

$\displaystyle\hat{A}(k^{0},\textbf{k};T)$
$\displaystyle\equiv\bigg{\langle}1-\frac{\textbf{k}\cdot\textbf{k}_{1}}{\textbf{k}^{2}}\bigg{\rangle}\
,$ (13) $\displaystyle\hat{B}_{0}(k^{0},\textbf{k};T)$
$\displaystyle\equiv\frac{1}{4}\bigg{\langle}\textbf{k}_{1}^{2}-\frac{(\textbf{k}\cdot\textbf{k}_{1})^{2}}{\textbf{k}^{2}}\bigg{\rangle}\
,$ (14) $\displaystyle\hat{B}_{1}(k^{0},\textbf{k};T)$
$\displaystyle\equiv\frac{1}{2}\bigg{\langle}\frac{[\textbf{k}\cdot(\textbf{k}-\textbf{k}_{1})]^{2}}{\textbf{k}^{2}}\bigg{\rangle}\
,$ (15)

where $\langle\mathcal{F}\rangle$ stands for the average of the quantity
$\mathcal{F}$ weighted with the heavy-meson spectral function, the $T$-matrix
and the appropriate Bose-Einstein factors. Full details of the calculation and
the results for these transport coefficients are given in Torres-
Rincon:2021yga .

Figure 6: Spatial diffusion coefficient below and above $T_{c}$ resulting from
different approaches (see Torres-Rincon:2021yga  for details). The empty boxes
correspond to a range of values of $2\pi TD_{s}$ for the central value of $T$.
Figure extracted from Montana:2021vks  with data taken from Torres-
Rincon:2021yga .

The spatial diffusion coefficient $D_{s}$ is related to $\hat{B}_{0}$ in the
static limit,

$2\pi TD_{s}(T)=\lim_{{\textbf{k}}\rightarrow 0}\frac{2\pi
T^{\,3}}{\hat{B}_{0}({E_{k},\textbf{k}};T)}\ ,$ (16)

where $E_{k}$ is the quasiparticle energy. The result for this transport
coefficient is shown in Fig. 6 below $T_{c}$ for the case of the off-shell
thermal calculation (black solid line) and it is compared with the on-shell
approximation (green dashed line), in which the spectral function is
approximated to a $\delta$ and the thermal $T$-matrix is replaced by the
corresponding one in vacuum. The main difference is due to the contribution of
the Landau cut when thermal scattering amplitudes are employed. We stress
that, when thermal corrections are included, the $D$-meson spatial diffusion
coefficient matches smoothly the recent lattice QCD calculations and Bayesian
analyses performed above $T_{c}$.

##  5 Conclusions

In this contribution we have presented our recent results for heavy mesons at
finite temperature, including the calculation of Euclidean meson correlators
and transport coefficients. We have described the interactions of heavy mesons
in a bath of light mesons using a thermal effective field theory based on
chiral and heavy quark symmetries. The thermal corrections to the heavy-meson
self-energies and the unitarized amplitudes have been obtained in a self-
consistent manner. We have shown recent results of the impact that the thermal
medium has on the properties of the ground-states (masses and widths) as well
as the on the states generated dynamically from the two-meson interaction.

Our results for the Euclidean correlators are compatible with the existing
lattice QCD data below $T_{c}$. It would be interesting to compare both
approaches at smaller temperatures and meson masses closer to the physical
values, as well as to apply techniques for the reduction of excited-state
contamination from the correlators in the analysis of the lattice QCD data.
From the effective theory side, one might need to include the contribution of
the thermal kaons to the $D$-meson self-energy, specially taking into account
the reduction of the mass gap between kaons and pions on the lattice.

Finally, we have shown that the use of thermal scattering amplitudes to
compute charm transport coefficients entails the appearance of a new term
related to the Landau cut of the $T$-matrix at finite temperature. This
contribution is absent if vacuum scattering amplitudes are employed, as
commonly done in the past. In particular we have seen that its consideration
gives rise to a smooth matching of the transport coefficents with lattice QCD
calculations and Bayesian analyses around $T_{c}$.

This work has been supported by the projects CEX2019-000918-M (Unidad de
Excelencia “María de Maeztu"), PID2019-110165GB-I00 and PID2020-118758GB-I00
financed by MCIN/AEI/10.13039/501100011033. The author also acknowledges
support from the FPU17/04910 Doctoral Grant from the Spanish Ministerio de
Ciencia InnovaciÃn y Universidades.

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