# Microscopic origin of the effective spin-spin interaction in a semiconductor
quantum dot ensemble

Frederik Vonhoff  Andreas Fischer  Kira Deltenre  Frithjof B. Anders
Department of Physics, TU Dortmund University, Otto-Hahn-Straße 4, 44227
Dortmund, Germany

###### Abstract

We present a microscopic model for a singly charged quantum dot (QD) ensemble
to reveal the origin of the long-range effective interaction between the
electron spins in the QDs. Wilson’s numerical renormalization group (NRG) is
used to calculate the magnitude and the spatial dependency of the effective
spin-spin interaction mediated by the growth induced wetting layer.
Surprisingly, we found an antiferromagnetic Heisenberg coupling for very short
inter-QD distances that is caused by the significant particle-hole asymmetry
of the wetting layer band at very low filling. Using the NRG results obtained
from realistic parameters as input for a semiclassical simulation for a large
QD ensemble, we demonstrate that the experimentally reported phase shifts in
the coherent spin dynamics between single and two color laser pumping can be
reproduced by our model, solving a longstanding mystery of the microscopic
origin of the inter QD electron spin-spin interaction.

Spins confined in semiconductor QDs have been discussed as candidates for
implementation of quantum bits (qubits) in quantum information technologies
[1, 2] since it allows integration into conventional semiconductor logic
elements. For information processing, qubit initialization and readout [3, 4]
are as important as manipulations of the spins. Optical control experiments in
QD ensembles [5, 6] as well as the measurements of the dephasing time as
function of the laser spectral width [7] in such samples provided strong
evidence for a long-range electron spin-spin interactions between the
different QDs in the ensemble of unknown microscopic origin. Initially, it was
speculated [5] that it might be caused by an optically induced [8] Ruderman-
Kittel-Kasuya-Yosida (RKKY) interaction [9, 10, 11]. Since the laser pulse
duration is of the order of ps such an optically induced interaction, however,
would decay rather rapidly and is not compatible with the observed spin
coherence on a scale of several ns.

In order to make use of the intrinsic long range spin-spin interaction between
the localized spins in QDs by spin manipulation protocols, its origin needs to
be understood. In this letter, we propose a microscopic mechanism based on the
analysis of a multi-impurity Anderson model [12, 13, 14]. We start from a
localized electron bound state in each QD that weakly hybridize with the
conduction band (CB) of the thin wetting layer (WL) [15, 16, 17] that is left
below the QDs in the Stranski-Krastanow growth protocol. The basic setup is
sketched in Fig. 1. By including all virtual charge fluctuations in the
leading order, significant corrections to the conventional textbook expression
starting from an effective local moment picture have been reported [18, 12]
for short distances. We believe that implementing backgates below the WL would
allow to manipulate the properties of the low concentration WL electron gas
and hence to control the effective spin-spin interaction.

Recently a growth procedure for InGaAs QD ensembles was proposed [19] to
eliminate the WL. In such samples, the effective spin-spin interaction between
the QD electron spins should be absent. While those types of QD ensembles are
produced to decrease the energy loss when used as photon emitters, a detailed
investigation of their spin properties in electron doped samples is still
missing to the best of our knowledge.

Figure 1: Sketch of two quantum dots that are linked by the InAs wetting
layer.

Model.— The WL is treated as a free two-dimensional CB

$\displaystyle H_{\rm WL}$ $\displaystyle=$
$\displaystyle\sum_{\vec{k}}\frac{\hbar^{2}\vec{k}^{2}}{2m^{*}}c^{\dagger}_{\vec{k}\sigma}c_{\vec{k}\sigma}$
(1)

in the isotropic effective mass approximation with $m^{*}/m_{0}=0.023$ [20].
The minimal model for the $N_{\rm QD}$ QDs in the ensemble is given by

$\displaystyle H_{\rm QD}$ $\displaystyle=$ $\displaystyle\sum_{i}^{N_{\rm
QD}}\left(\sum_{\sigma}\varepsilon^{d}_{i}d^{\dagger}_{i\sigma}d_{i\sigma}+U_{i}n^{d}_{i\uparrow}n^{d}_{i\downarrow}\right),$
(2)

where $\varepsilon^{d}_{i}$ denotes the single particle energy of the bound
electron state with spin $\sigma$ in the $i$-th QD and $U_{i}$ the
corresponding Coulomb repulsion preventing the bound state to be doubly
occupied. The creation (annihilation) operator
$d^{\dagger}_{i\sigma}(d_{i\sigma}$) adds (removes) an electron to (from) the
$i$-th QD. The electronic wave function is localized but covers the whole
diameter of the QD. The parameters $U_{i}$ and $\varepsilon^{d}_{i}$ depend on
the individual shape of the QD. Using the experimental estimates [21], we
approximate $U_{i}\approx$4\text{\,}\mathrm{meV}$$ for a QD with a diameter
$D=$25\text{\,}\mathrm{nm}$$.

The concentration of the donors in the QD ensemble used in Ref. [5] was
selected to match the QD density in the sample. Each QD can capture at least
one electron if the bound state $\varepsilon^{d}_{i}$ is below the CB of the
WL. It remains empty when the chemical potential $\mu<\varepsilon^{d}_{i}$.
Spin spectroscopy experiments [22], however, indicate that not all QDs are
filled with one electron. This might be due to local imperfections that shift
individual $\varepsilon^{d}_{i}$ to higher energies. Also doubly occupied QD
states are possible when $\varepsilon^{d}_{i}<\mu-U_{i}$. Excluding this
doubly occupied QD ground state configuration puts a lower bound on
$\varepsilon^{d}_{i}$. We assume that a fraction $q_{\rm WL}$ of the total
donor excess electrons is filling up the WL such that the chemical potential
$\mu>0$ lies within the CB of the WL. The upper bound of the chemical
potential $\mu=\varepsilon_{F}$ is reached for $q_{\rm WL}=1$ yielding
$\varepsilon_{F}\approx$1\text{\,}\mathrm{meV}$$ for a dot density of
${10}^{10}$cm-2[5].

The bound state of the $i$-th QD can tunnel with a finite tunneling amplitude
$V^{i}_{m}$ into the Wannier orbital $m$ of the WL. The resulting
hybridization term between the bound state of each QD and the WL takes the
form

$\displaystyle H_{\rm hyb}$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{N_{\rm
QD}}\sum_{\sigma}\sum_{m}\left(V^{i}_{m}d^{\dagger}_{i\sigma}c_{m\sigma}+h.c.\right),$
(3)

where $c_{m\sigma}$ is the annihilation operator of the Wannier orbital at
site $\vec{R}_{m}$ of the 2D WL and whose spatial Fourier transform is
$c_{\vec{k}\sigma}$. The total Hamiltonian of the coupled QD problem is given
by $H=H_{\rm WL}+H_{\rm QD}+H_{\rm hyb}$ which is just a realization of a
multi-impurity problem [13].

The effect of the WL onto the dynamics of the localized QD states is
determined by the hybridization function matrix [12, 13]

$\displaystyle\Delta_{ij}(z)$ $\displaystyle=$
$\displaystyle\sum_{lm}\frac{1}{N}\sum_{\vec{k}}\frac{[V^{i}_{l}]^{*}V^{j}_{m}e^{i\vec{k}(\vec{R}_{m}-\vec{R}_{l})}}{z-\varepsilon_{\vec{k}}}.$
(4)

In the wide band limit, i. e. $V_{0}/D\ll 1$ where $D$ is the band width of
the WL CB, $t_{ij}=\Re\Delta_{ij}(\mu)$ generates an effective hopping between
the QDs $i$ and $j$. The averaged distance between the QDs is of the same
order of magnitude as the Fermi wave length $\lambda_{F}$. The distance
variations between the WL sites, however, are small compared to $\lambda_{F}$.
Consequently, we can replace $\vec{R}_{m}-\vec{R}_{l}$ by the distance between
the two centers of the QDs, i. e. $\vec{R}_{i}-\vec{R}_{j}$, and include the
spatial extension of the QD 111We checked that by an explicit calculation
assuming a Gaussian distribution of $V^{i}_{m}$ centered at $\vec{R}_{j}$ with
a width given by the QD radius of
$r_{\rm{QD}}\approx$15\text{\,}\mathrm{nm}$$. by defining
$\bar{V}_{i}=\sum_{m}V^{i}_{m}$. We introduce the average hybridization matrix
element $V_{0}^{2}=\langle\bar{V}_{i}^{2}\rangle$ and define a reference
energy scale $\Gamma_{0}=\pi V_{0}^{2}\rho_{0}$, $\rho_{0}$ being the constant
density of state of the 2D WL CB. The charge fluctuation scale $\Gamma_{0}$
determines the order of magnitude of $\Delta_{ij}(z)$. The antiferromagnetic
(AF) part of the RKKY interaction can be estimated [12, 13] as
$J_{\rm{AF},ij}^{\rm RKKY}\approx 4t_{ij}^{2}/U\propto 4\Gamma_{0}^{2}/U$
serving as a first estimate for $\Gamma_{0}\approx
10-100$\mathrm{\SIUnitSymbolMicro eV}$$.

In a conventional metal, the AF part of the RKKY interaction is compensated by
the FM contribution for $|\vec{R}|\to 0$ [24, 18]. The relevant parameter
regime of our model, however, does not fit the conventional regime discussed
in the literature where $D,U>\bar{J}_{K}$, $\bar{J}_{K}$ being the averaged
local Heisenberg coupling $J_{K}$ between the QD electron spin and the local
CB spin density [25]. The energy $\varepsilon^{d}_{i}$ is located below the
lower band edge of the CB. Many Wannier orbitals contribute to $\bar{V}_{i}$,
and virtual charge fluctuations in the localized orbitals below the CB
continuum are allowed. We expect deviations between the correct effective
spin-spin interaction within our model and the prediction from the
conventional two stage mapping of the QD model: first onto a Kondo ensemble
model by applying a Schrieffer-Wolff transformation (SWT) [26] that is
$O(V^{2}_{0})$ followed by a perturbative estimation of the $J^{\rm
RKKY}_{ij}$ [25] in $O(V_{0}^{4})$. Significant corrections to $J^{\rm
RKKY}_{ij}$ have been already reported [18, 12] at short distances since
additional terms in $O(V_{0}^{4})$ neglected in the two stage mapping become
important.

Numerical renormalization group approach.— To circumvent the inconsistency
problem of the two stage perturbative approach, we apply the numerical
renormalization group (NRG) [27, 28] to the Hamiltonian with $N_{\rm QD}=2$ to
determine the distance dependent effective Heisenberg interaction between the
localized electron spins. For each distance $R=|\vec{R}_{1}-\vec{R}_{2}|$, the
two QD problem is mapped onto an effective two band model [24, 29, 12], where
we used $\Gamma_{0}$, $\mu=\varepsilon_{F}$ and
$\varepsilon_{i}^{d}=\varepsilon^{d}$ as the adjustable parameters such that
each QD remains singly charged. Note that the NRG mapping of the strong
asymmetric bands onto a Wilson chain [27, 28] leads to modification of the
hopping parameters [30] and of the NRG fixed point (FP) spectrum 222See
Supplemental Material for technical additional details.. This adjustment only
depends on the support of the CB density of states and, therefore, is
independent of the QD distance $R$.

Figure 2: NRG results: (a) $\langle\vec{S}_{1}\vec{S}_{2}\rangle(T,R)$ vs $T$
for two distances $R$ and the fit to an interacting two spin model for
extracting $J$. (b) $J^{\rm NRG}_{\rm
RKKY}(R)/(\Gamma_{0}^{2}\varepsilon_{F})$ vs $Rk_{F}/\pi$ for different
$\Gamma_{0}$ and $\varepsilon_{F}$. We added a fit to Eq. (Microscopic origin
of the effective spin-spin interaction in a semiconductor quantum dot
ensemble) as solid magenta line. NRG parameters $\Lambda=5$, high energy
cutoff $D=$1\text{\,}\mathrm{eV}$$,
$\varepsilon^{d}=$-1.5\text{\,}\mathrm{meV}$,$ and $\bar{\beta}=40$.

Using the NRG, we calculated the temperature dependent spin-spin correlation
function $\langle\vec{S}_{1}\vec{S}_{2}\rangle(T,R)$ as shown for two
distances in Fig. 2(a). The sign of
$\langle\vec{S}_{1}\vec{S}_{2}\rangle(T,R)$ determines the sign of $J_{\rm
RKKY}^{NRG}(R)$, and $|J_{\rm RKKY}^{NRG}(R)|$ is obtained from the fit to the
universal functions of the spin-spin correlation function of a $s=1/2$ toy
model, $S_{12}(T)=\mu_{\rm eff}^{2}(e^{-\beta J}-1)/(1+3e^{-\beta J})$, where
the parameter $\mu_{\rm eff}^{2}$ includes possible Kondo screening effects of
the localized spins 333$\mu_{\rm eff}^{2}=s(s+1)$ for unscreened spins.. We
added the fit functions as solid lines to Fig. 2(a). The extracted $J_{\rm
RKKY}^{NRG}(R)=J$ are shown in Fig. 2(b) for different fixed set of
parameters.

For two localized moments coupled by a Heisenberg interaction $J_{K}$ to the
local spin-density of 2D CB with a quadratic dispersion in 2D, Klein and
Fischer [25] derived the analytic expression

$\displaystyle J^{\rm RKKY}_{ij}$ $\displaystyle=$
$\displaystyle-\rho_{0}\bar{J}^{2}_{K}\frac{\bar{v}k_{F}^{2}}{4\pi}$
$\displaystyle\times\left[J_{0}(k_{F}R)N_{0}(k_{F}R)+J_{1}(k_{F}R)N_{1}(k_{F}R)\right]$

for the effective RKKY interaction, where $J_{l}(x)(N_{l}(x))$ is the Bessel
(Neumann) function of order $l$ and $\bar{v}$ the area of the 2D unit cell. We
added a fit to the expression Eq. (Microscopic origin of the effective spin-
spin interaction in a semiconductor quantum dot ensemble) to our NRG data as
solid magenta line in Fig. 2(b). The NRG results follow excellently the
analytic predictions for larger distances. The magnitude of $J_{\rm
RKKY}^{NRG}(R)$ is proportional to $\Gamma_{0}^{2}\varepsilon_{F}$ as
expected, but the absolut value differs for those predicted by Eq.
(Microscopic origin of the effective spin-spin interaction in a semiconductor
quantum dot ensemble). $J_{\rm RKKY}^{NRG}(R)$ remains invariant under
rescaling $\alpha\Gamma_{0},\alpha U,\alpha\varepsilon^{d}$ for
$Rk_{F}/\pi>0.5$. For short distances, $Rk_{F}/\pi<0.25$, however, we observe
significant deviations: The NRG reveals an AF RKKY interaction in contrary to
the conventional RKKY result. The origin of this surprising effect can be
linked to the large effective single-particle hopping between the QD orbitals
induced by the very strong particle-hole asymmetry of the CB for small WL
fillings. The conventional two stages perturbation theory is inconsistent and
requires additional corrections in $O(V_{0}^{4})$ [18, 12].

Spin polarization in laser pulsed QD ensembles.— After establishing the
distance dependent effective interaction between the electron spins in the
different QDs, we investigate its influence on an ensemble of singly charged
QD in an external magnetic field subject to a two color laser pumping with
circularly polarized light [5]. We used a semiclassical simulation [33, 34]
for the spin dynamics of a QD ensemble model [7]

$\displaystyle H_{\rm array}$ $\displaystyle=$
$\displaystyle\sum_{i}^{N_{\mathrm{QD}}}H_{1}^{(i)}+\sum_{i<j}J_{ij}\vec{S}^{(i)}\vec{S}^{(j)},$
(6)

where the central spin model (CSM) $H_{1}^{(i)}$,

$\displaystyle H_{1}^{(i)}=$ $\displaystyle
g_{e}^{(i)}\mu_{\mathrm{B}}\vec{B}_{\mathrm{ext}}\vec{S}^{(i)}+\sum_{k=1}^{N_{i}}g_{N,k}^{(i)}\mu_{\mathrm{N}}\vec{B}_{\mathrm{ext}}\vec{I}_{k}^{(i)}$
$\displaystyle+\sum_{k=1}^{N_{i}}A_{k}^{(i)}\vec{I}_{k}^{(i)}\vec{S}^{(i)},$
(7)

includes the electron and nuclear Zeeman term in the external magnetic field
$\vec{B}_{\mathrm{ext}}$ as well as the hyperfine coupling between the $N_{i}$
nuclei spins denoted by $\vec{I}_{k}^{(i)}$ and the electron spin
$\vec{S}^{(i)}$ [35, 36, 37, 38, 39]. We generated random 2D ensembles of
$N_{\mathrm{QD}}$ QDs with a dot density of ${10}^{10}$cm-2 [5] and assigned
$J_{ij}=x_{i}x_{j}J^{\rm NRG}_{\rm RKKY}(R,\varepsilon_{F},\Gamma_{0})$ from
the NRG data depicted in Fig. 2(b) where $x_{i}$ accounts for the variations
of $\bar{V}_{i}$. We attributed $N_{\rm QD}/2$ randomly selected QDs to be
resonant to one of the two different laser frequencies and, therefore, also
assigned the corresponding $g_{e}^{(i)}$. We set the average
$\bar{g}_{e}=0.55$, the ratios between the two subsets
$g_{e}^{1}/g^{2}_{e}=1.03$, and
$z=\bar{g}_{e}\mu_{B}/(\bar{g}_{N}\mu_{N})=1/800$.

We run three different types of semiclassical simulations for the spin
dynamics with $I_{k}=3/2$ nuclear spins: (i) a box model with
$A_{k}=A_{0}=\mathrm{const.}$, $N_{\rm QD}=10000$, and $N_{i}=10^{6}$ nuclear
spins, (ii) a simulation for a frozen nuclear spin dynamics (FOA) [36] for
$N_{\rm QD}=10000$ and $N_{i}$ as in (i) and (iii) an $A_{k}$ distribution
with $N_{\mathrm{QD}}=1000$ and $N_{i}=1000$ including the full nuclear spin
dynamics. In all cases, we kept the characteristic energy scale
$T^{*}=(\sum_{k}(A_{k}^{(i)})^{2}\langle
I^{2}_{k}\rangle)^{-1/2}=$1\text{\,}\mathrm{ns}$$ fixed. The technical aspects
of the simulation including the treatment of the laser pulse using a trion
excitation cycle can be found in the literature [34, 7] which is summarized in
the Supplemental Material.

The experiments reported in Ref. [5] were performed in a transversal magnetic
field of $1\text{\,}\mathrm{T}$, and laser pulses of two different colors were
applied pumping two different resonant QD subsets. In order to reproduce one
of the experimental key results, we first determined a reference curve for
$\langle S_{z}(t)\rangle$ of subset 1 subjected to a circular polarized laser
pulse at $t=0$ that shows damped coherent oscillations characterized by the
Larmor frequency. Then we run the same setup but apply a second circular
polarized laser pulse with a color resonating with subset 2 at a moment when
$\langle S_{z}(t)\rangle$ of subset 1 reached a minimum, i. e. at a delay of
$\Delta t\approx$0.111\text{\,}\mathrm{ns}$$.

Figure 3: Phase shift $\phi$ in the coherent electron spin oscillation between
a single color pumping reference curve and a two color laser pumped QD
ensemble. The experimental data are taken from Ref. [5], the simulations are
for $A_{k}=\mathrm{const.}$ (box model), a frozen Overhauser field
approximation (FOA) including some disorder of the $x_{j}$ as well as a
distribution of the $A_{k}$.

We tracked the relative phase shifts of the coherent oscillations between the
reference curve $\langle S_{z}(t)\rangle$ and those in the presence of a laser
pulse onto subset 2. The sign of the phase shift depends only on the relative
sign of the circular polarization. The result is plotted in Fig. 3. As in the
experiment, the phase shift increases roughly linearly in time. The slope is
determined by overall magnitude of $J^{\rm RKKY}_{ij}(R)$. The results that
resembles the experiments the best were obtained for
$\Gamma_{0}=$140\text{\,}\mathrm{\SIUnitSymbolMicro eV}$$ and
$\varepsilon_{F}=$0.5\text{\,}\mathrm{meV}$$ corresponding to $q_{\rm WL}=0.5$
and $U/\Gamma_{0}=29$. Leaving the ratio $U/\Gamma_{0}=29$ fixed, we can also
reproduce the experimental phase shifts with
$\Gamma_{0}=$1.25\text{\,}\mathrm{meV}$,\varepsilon^{d}=$-15\text{\,}\mathrm{meV}$$,
$U=$40\text{\,}\mathrm{meV}$$ indicating the physics is driven by an effective
lokal Kondo coupling [26] plus corrections in $O(V^{4}_{0})$ for short
distances. Note that the individual hybridization strength between the bound
electron QD state and the local Wannier orbital remains small and is of the
order $V_{m}^{i}=10-40\,$\mathrm{\SIUnitSymbolMicro eV}$$. Only by the
summation to $\bar{V}_{i}$, a significant contribution arises.

We run the simulation for fixed $x_{i}=1$ and for the FOA additional with a
Gaussian normal distribution with $\delta x=0.2$ to reveal the influence of
the local derivation of the QD hybridization matrix element from its average
value. The phase shifts are nearly independent of the distribution of local
hybridization matrix elements $\bar{V}_{i}$. The major disorder contributions
stems from the randomly distribution of distances to the neighbouring QDs. As
a result, each QD senses a slightly different effective coupling converting
the quadratic increase of the phase shift in a two QD model to a linear
increase for short times. Tracking the full dynamics of the Overhauser field
is not required on the time scale of $3\text{\,}\mathrm{ns}$: The frozen
Overhauser approximation and the case $A_{k}^{(i)}=A_{0}$ yields the almost
same results as the full nuclear spin dynamics but with a reduced $N_{{\rm
QD}}$. We also run the simulation for a $J^{\rm RKKY}_{ij}(R)$ after rescaling
$\Gamma_{0},U,\varepsilon^{d}$ with a factor of 10 to accommodate for errors
in the rough parameter estimates. The slope of the phase shift $\phi$ came out
about 50% to large requiring a reduction of $\Gamma_{0}$. The origin of the
increase is caused by contribution of the first minimum in $J^{\rm
RKKY}_{ij}(R)$ at around $RK_{F}/\pi\approx 0.2$ where we observe significant
deviations from the analytic prediction. Overall, the simulation data agree
remarkably well with the experimental findings even over a large range of
parameters.

Conclusion.— In this letter, we present a QD WL model to explain the
microscopic origin of the inter QD electron spin-spin interaction conjectured
in Ref. [5]. We used Wilson’s NRG [28] to extract the strength and distance
dependency of this spin-spin interaction mediated by the WL via an RKKY based
mechanism. While the long distance behavior is in agreement with the low-
filling perturbative RKKY result [25], we found significant deviations for
shorter distances. The unconventionally AF RKKY interaction at short distances
can be connected to the large effective inter QD hopping between the QD
orbitals [12, 13] as a consequence of the very low band filling. The
conventional RKKY approach [25] starts already from an effective local moment
picture and does not include all corrections in $O(V_{0}^{4})$.

We used our NRG data as input for a semiclassical simulation of the spin
dynamics in a coupled QD ensemble which is able to reproduce the experimental
phase shifts reported for two-color pumping setup [5] very accurately for a
variety of different realistic parameters. Although the interaction in the
effective model might appear substantial, one has to bare in mind that all WL
Wannier orbital below a QD contribute in Eq. (3) only by a very weak coupling.

Our theory solves the long standing mystery about the microscopic origin of
the inter QD spin-spin interaction. We predict that this interaction can be
eliminated by a QD growth process that excludes an InAs WL [19]. By gating the
2D WL, it might be possible to modulate the 2D electron gas and, therefore,
control the effective Heisenberg interactions $J^{\rm RKKY}_{ij}(R)$ by a
change of gate voltage.

###### Acknowledgements.

The authors would like to thank fruitful discussions with Alex Greilich who
also provided the experimental raw data of Ref. [5] added to Fig 2(b). We
acknowledge financial support by the Deutsche Forschungsgemeinschaft through
the transregio TRR 160 within the Projects No. A4 and No. A7.

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11institutetext: Instituto de Astrofísica de Canarias (IAC), C/ Vía Láctea
s/n, E-38205 La Laguna, Tenerife, Spain 22institutetext: Universidad de La
Laguna, Departamento de Astrofísica, C/ Astrofísico Francisco Sánchez s/n,
E-38206 La Laguna, Tenerife, Spain 33institutetext: Dipartimento di Fisica,
Sezione di Astronomia, Università di Trieste, Via Tiepolo 11, I-34143 Trieste,
Italy 44institutetext: INAF/Osservatorio Astronomico di Trieste, Via Tiepolo
11, I-34143 Trieste, Italy

# Biases in the estimation of velocity dispersions and dynamical masses for
galaxy clusters

A. Ferragamo 1122<EMAIL_ADDRESS>J. A. Rubiño-Martín 1122 J. Betancort-
Rijo 1122 E. Munari 3344 B. Sartoris 3344 R. Barrena 1122

###### Abstract

Using a set of 73 numerically simulated galaxy clusters, we have characterised
the statistical and physical biases for three velocity dispersion and mass
estimators, namely biweight, gapper and standard deviation, in the small
number of galaxies regime ($N_{\rm gal}\leq 75$), both for the determination
of the velocity dispersion and the dynamical mass of the clusters via the
$\sigma$–$M$ relation. These results are used to define a new set of unbiased
estimators, that are able to correct for those statistical biases. By applying
these new estimators to a subset of simulated observations, we show that they
can retrieve bias-corrected values for both the mean velocity dispersion and
the mean mass.

##  1 Introduction

Several authors have used the velocity dispersion mass proxy to study and
characterise scaling relations between SZ and dynamical mass (ruel14, ;
sifon16, ; amodeo17, ). In order to have sample with enough statistical power,
it is necessary to estimate the velocity dispersion for hundreds of galaxy
clusters (GCs). Although this goal can be achieved through spectroscopic
follow-up (e.g. nostro16, ; rafa18, ), these studies are extremely expensive
in terms of observational time and data reduction. For these reasons, it is
extremely difficult to estimate radial velocities for more than few members
(typically $\sim 20$) for each cluster target.

In this article we present our study of statistical and physical biases
introduced in the estimation of velocity dispersion and dynamical mass.

##  2 Statistical biases in Velocity Dispersion estimation

For our analysis we use a sample of 73 simulated massive clusters selected
from the simulations described in munari13 . Our selected sample contains
clusters with masses $M_{200}>2\times 10^{14}M_{\odot}$ and located at five
redshifts between $0.12\leq z\leq 0.82$.

There are several method to estimate mean and scale of a distribution. We
focus our attention on the estimators that, in the last decades, became
standard tools in GC analyses, namely biweight and gapper, compared with the
standard deviation. A detailed description of these estimators can be found in
beers90 .

Figure 1: Left panel: mean velocity dispersion $S_{\rm X}/S_{\rm
std}(<R_{200})$ as a function of the number of galaxies $N_{\rm gal}$. Right
panel: corrected velocity dispersion estimators $S^{\prime}_{\rm X}/S_{\rm
std}(<R_{200})$ as a function of $N_{\rm gal}$. The dispersion $S_{\rm
X}(N_{\rm gal})$ is calculated for the standard deviation (green line),
biweight (blue line), and gapper (red line) estimators.

In order to investigate bias and variance of the three scale estimators
$S_{\rm X}(N_{\rm gal})$, being X$=$ “std”, “bwt” or “gap”, for each one of
the three cases, we have explored the low number of galaxies regime between
$N_{\rm gal}=8$ and $N_{\rm gal}=75$. We have generated 2500 configurations of
randomly selecting galaxies within the projected circle of radius $R_{200}$
and then we averaged the $73\times 2250$ velocity dispersions (750 for each
mayor axis).

Left panel of Fig. 1 shows the average $S_{\rm X}(N_{\rm gal})$ normalised
with respect to $S_{\rm std}(<R_{200})$, which represents the velocity
dispersion of all the galaxies in the simulation within a circle of projected
radius $R_{200}$, and calculated using the standard deviation estimator. In
the low-$N_{\rm gal}$ regime, all estimators are biased showing very different
behaviours. The standard deviation estimator (green line) shows a dependence
with the number of elements used for the estimation. However, this dependence
can be theoretically predicted to be $1-1/(4(N_{\rm gal}-1))$. The biweight
(blue line) shows a stronger drop for $N_{\rm gal}\leq 30$ underestimating the
reference dispersion by up to $4$ % at $N_{\rm gal}=7$. Finally, the gapper
(red line) shows an estimate of the velocity dispersion almost constantly
biased at any $N_{\rm gal}$.

In order to construct an unbiased velocity dispersion estimators,
$S^{\prime}_{\rm X}(N_{\rm gal})$, we use a parametrisation of the curves in
left panel of Fig. 1:

$S^{\prime}_{\rm X}(N_{\rm gal})\equiv S_{\rm X}(N_{\rm
gal})\left(1+\left(\left(\frac{D}{(N_{\rm
gal}-1)}\right)^{\beta}+B\right)\right).$ (1)

Table 1 shows the best-fit values for the parameters $D$, $\beta$ and $B$, for
each one of the three estimators (biweight, gapper and standard deviation).

Table 1: Best-fit parameters to be used in the parametric function given in equation 1, describing the bias of the three estimators. See text for details.  | $BWT$ | $GAP$ | $STD$
---|---|---|---
$D$ | $0.72\pm 0.03$ | $0$ | $0.25$
$B$ | $-0.0225\pm 0.0002$ | $-0.0080\pm 0.0002$ | $-0.0037\pm 0.0003$
$\beta$ | $1.28\pm 0.03$ | $1$ | $1$

Right panel of Fig. 1 shows that the corrected estimators $S^{\prime}_{\rm
X}(N_{\rm gal})$ are actually unbiased by construction.

##  3 Biases by interlopers contamination

Any spectroscopic sample of cluster members is contaminated by galaxies
belonging to the large scale structure that surrounds the cluster itself. This
population of ‘pseudo cluster members’, called interlopers, modifies the
velocity distribution and therefore affects the estimation of velocity
dispersion. According to the definition of interlopers given in Pratt19 , one
must distinguish between two very different types of contaminants: i. galaxies
gravitationally bound to the clusters that are outside the virial sphere
(according to the definition given in mamon10 ), but that, due to projection
effects appear within a smaller radius; ii. background/foreground galaxies
with similar redshifts to the cluster, but belonging to the large scale
structure that surrounds the cluster itself. Detailed study of these
interlopers is beyond the scope of this work. Here, we illustrate the
robustness of the three estimators in exam, by fixing the fraction of
contaminants at any $N_{\rm gal}$. Fig. 2 shows the results obtained for five
different fractions of interlopers: 5% (dashed lines), 10% (dot-dashed lines),
15%(three-dot-dashed lines), 20% (two-dot-long-dashed lines) and 30% (dotted
lines). We note that the three estimators are similarly affected by interloper
contamination. The effect of the interlopers contamination consists of a
velocity dispersion overestimation which is as high as their relative fraction
(up to 30%) representing the most damaging source of biases in the velocity
estimation estimation.

Figure 2: Effect of interlopers on $S^{\prime}_{\rm X}/S_{\rm std}(<R_{200})$,
as a function of the number of galaxies. The velocity dispersion,
$S^{\prime}_{\rm X}(N_{\rm gal})$, is first computed using a pure galaxie
sample (solid lines). We also evaluated the response of biweight (blue lines),
standard deviation (green lines), and gapper (red lines) using samples
contaminated by $0\%$ (solid line), $5\%$ (dashed lines), $10\%$ (dot dashed
lines), $15\%$ (three-dot-dashed lines), $20\%$ (two-dot-long-dashed lines)
and $30\%$ (dotted lines) of interlopers at any $N_{\rm gal}$.

##  4 Physical biases in Velocity Dispersion estimation

Observational strategies and technical limitations generally force us to
observe only massive clusters members sampled in a fraction of $R_{200}$. We
studied how these limitations might produce biased velocity dispersion
estimates.

###  4.1 Effects due to the selected fraction of massive galaxies

In the ideal case we can observe any cluster member regardless of its
brightness. However, the telescope aperture limits the detection magnitude and
prevents us from detecting faint objects, for a fixed integration time.
Therefore, cluster samples contain only a fraction of the brightest galaxy
members, which are also the most massive.

In order to simulate this effect, we mimicked observational conditions by
selecting three percentages of all visible galaxies in the simulation, i.e. 50
%, 33 %, and 25 %, by sorting the cluster members by mass and dividing the
sample in 2, 3, and 4 mass bins, starting from the most massive object. For
each case, we reproduce the procedure explained in Sect. 2.

Figure 3: Left panel: Mean (bias) of $S^{\prime}_{\rm std}(N_{\rm gal})/S_{\rm
std}(<R_{200})$ as a function of the number of galaxies $N_{\rm gal}$,
calculated by choosing galaxies within $100\%$ (black solid line), $1/2$ (blue
solid line), $1/3$ (red solid line), and $1/4$ (green solid line) of the
complete cluster member samples. Right panel: Average velocity dispersion
profile within a given enclosed radius $r$, $<S^{\prime}_{\rm std}(<r)/S_{\rm
std}(<R_{200})>$, normalised to $R_{200}$. The red line represents the mean at
each radius of the individual 73 simulated GC profiles (grey lines). The
dashed blue line represents the sifon16  profile, which is almost coincident
with our derived profile.

Fig. 3 (left panel) shows $S^{\prime}_{\rm std}(N_{\rm gal})/S_{\rm
std}(<R_{200})$ as function of $N_{\rm gal}$ calculated with the corrected
standard deviation estimator, and using galaxies picked up from 100 % (black
line), $1/2$ (blue line), $1/3$ (red line) and $1/4$ (green line) of the
complete cluster member samples. We see that the velocity dispersion is
sensitive to the fraction of massive galaxies used to estimate it, reaching a
bias of almost 2% using only the most massive fraction. However, it is almost
insensitive to $N_{\rm gal}$. For this reason, we can use the curves in the
left panel of Fig. 3 to correct this physical effect.

###  4.2 Effect of aperture sub-sampling

There are evidence in the literature that the velocity dispersion has a radial
dependence (e.g., mamon10, ; sifon16, ). This implies that sampling galaxies
in different fractions of the cluster’s virial radius should lead to a biased
velocity dispersion. In order to investigate and quantify this effect, we
averaged the 73 velocity dispersions calculated using all galaxies inside a
cylinder of variable radius $0.2\leq r/R_{200}\leq 1.2$. Fig. 3 (right panel)
shows that the velocity dispersion is (on average) overestimated in region
smaller than $R_{200}$ and slightly underestimated for $r>R_{200}$, in analogy
with that recover by sifon16 .

##  5 Statistical bias in the estimation of $M_{200}$

Figure 4: Left panel: mean of $M\left(S^{\prime}_{\rm X}(N_{\rm
gal})\right)/M\left(S^{\prime}_{\rm std}(<R_{200})\right)$, which represent
the standard mass estimator, described in munari13 , applied to normal and
unbiased velocity dispersion estimators, standard deviation (green), gapper
(red), and biweight (blue). Right panel: mean of the corrected mass estimator
$M^{\prime}\left(S^{\prime}_{\rm X}(N_{\rm gal})\right)/M\left(S^{\prime}_{\rm
std}(<R_{200})\right)$.

When we use velocity dispersion as a mass proxy we apply a scaling relation
$\sigma_{\rm 1D}-M$, it is a power law that has been previously calibrated
either with simulations evrard08 ; saro13 ; munari13 . The left panel of Fig.
4 shows how applying the scaling relation $\sigma_{\rm 1D}-M$ to the unbiased
velocity dispersion estimator $S^{\prime}_{\rm X}$ the resulting masses
$\left<M(S^{\prime}_{\rm X})/M(S_{\rm std}(<R_{200}))\right>$ are biased.

We can calculate analytically the functional form of the mass estimator as a
function of the number of galaxies $\frac{1-2\alpha}{4\alpha^{2}(N_{\rm
gal}-1)}$, with $A=1177.0$ km s-1 and $\alpha=0.364$ parameters of the
munari13  scaling relation.

As in Sect. 2, we fitted a parametric form description of the bias as a
function of $N_{\rm gal}$ by using three parameters ($E^{\prime}$,$F^{\prime}$
and $\gamma^{\prime}$, listed in Table 2):

$M^{\prime}\left(S^{\prime}_{\rm X}(N_{\rm gal})\right)=M\left(S^{\prime}_{\rm
X}(N_{\rm
gal})\right)\left[\frac{1-E^{\prime}\alpha}{(E^{\prime}\alpha)^{2}(N_{\rm
gal}-1)^{\gamma^{\prime}}}+F^{\prime}\right]^{-1}.$ (2)

Table 2: Best-fit parameters for the function describing the bias in $\left<M(S^{\prime}_{\rm X})/M(S_{\rm std}(<R_{200}))\right>$ for simulated clusters, as described in equation 2. | $BWT$ | $GAP$ | $STD$
---|---|---|---
$E^{\prime}$ | $1.31\pm 0.03$ | $1.50\pm 0.03$ | $1.53\pm 0.03$
$F^{\prime}$ | $1$ | $1$ | $1$
$\gamma^{\prime}$ | $1.24\pm 0.03$ | $1.17\pm 0.04$ | $1.11\pm 0.04$

The right panel of Fig. 4 shows the bias of $M^{\prime}(S^{\prime}_{\rm X})$.
The new mass estimator is actually unbiased by construction.

Finally, in order to test the corrections described in previous sections, we
have applied them to a set of mock cluster catalogues. To do this we have
simulated a realistic observational strategy based on the observations
described in (nostro16, ; rafa18, ). We generated $100$ mock samples out of
the $73$ GCs object simulated in this study. For each of these samples we
calculated the mean ratio between the estimated and the reference velocity
dispersion of each cluster, as well as the velocity dispersion calculated
using $S^{\prime}_{\rm X}$ defined in eq. 1 with the parameters in Table 1.
Averaging over all the mock samples we obtained a biased mean velocity
dispersion

$\begin{split}\left<S_{\rm std}(N_{\rm gal},r)/S_{\rm
bwt}(<R_{200})\right>&=0.96\pm 0.02,\\\ \left<S^{\prime}_{\rm std}(N_{\rm
gal},r)/S_{\rm bwt}(<R_{200})\right>&=1.00\pm 0.02.\end{split}$ (3)

The normal $S_{\rm std}$ estimator shows to be biased, whereas the
$S^{\prime}_{\rm X}$ lead to an unbiased estimation of the velocity
dispersion.

Using the bias corrected velocity dispersions, we calculated cluster masses,
$M(S^{\prime}_{\rm X})$ and $M^{\prime}\left(S^{\prime}_{\rm std}\right)$,
obtaining

$\begin{split}\left<M\left(S^{\prime}_{\rm std}(N_{\rm
gal},r)\right)/M\left(S_{\rm bwt}(<R_{200})\right)\right>&=1.13\pm 0.07,\\\
\left<M^{\prime}\left(S^{\prime}_{\rm std}(N_{\rm
gal},r)\right)/M^{\prime}\left(S_{\rm bwt}(<R_{200})\right)\right>&=1.00\pm
0.06.\end{split}$ (4)

As described above, these normal mass estimator is overestimated, while primed
mass estimator, $M^{\prime}$ is actually unbiased.

##  6 Conclusions

We have used 73 simulated GCs from hydrodynamic simulations including AGN
feedback and star formation, in order to characterise the statistical and
physical biases in three velocity dispersion (and mass) estimators frequently
used in the literature: the biweight, the gapper, and the standard deviation.
We have focused our study in the (common) case of a low number of galaxy
members ($N_{\rm gal}\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower
4.0pt\hbox{$\sim$}}\hss}\hbox{$<$}}}75$).

We showed that each of these estimators (dispersion and mass) presents a
statistical bias. Therefore, we propose bias corrected velocity dispersion
($S^{\prime}_{\rm X}$) and mass ($M^{\prime}(S^{\prime}_{\rm X})$) estimators.

We have tested the robustness of the new estimators against the contamination
by interlopers. We found that the velocity dispersion estimators
$S^{\prime}_{\rm X}$ are similarly affected by the contamination for all the
three cases in this low-$N_{\rm gal}$ limit.

We observed that the most likely sources of physical bias are i) the selection
effect the fraction of massive galaxies used to estimate the velocity
dispersion; and ii) the fraction of the virial radius explored. We showed that
in the first case the bias is estimated to be around 2 % when considering only
$1/4$ of the most massive galaxies. Concerning the effect produced by the
sampling aperture, we find a dispersion radial profile in agreement with
sifon16  results.

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