Isabelle Bouchoule

Léa Dubois

Léo-Paul Barbier

Losses in interacting quantum gases: ultra-violet divergence and its regularization

We investigate the effect of losses on an interacting quantum gas. We show that, for gases in dimension higher than one, assuming together a vanishing correlation time of the reservoir where dissipation occurs, and contact interactions leads to a divergence of the energy increase rate. This divergence is a combined effect of the contact interactions, which impart arbitrary large momenta to the atoms, and the infinite energy width of the reservoir associated to its vanishing correlation time. We show how the divergence is regularized when taking into account the finite energy width of the reservoir, and, for large energy width, we give an expression for the energy increase rate that involves the contact parameter. We then consider the specific case of a weakly interacting Bose Einstein condensate, that we describe using the Bogoliubov theory. Assuming slow losses so that the gas is at any time described by a thermal equilibrium, we compute the time evolution of the temperature of the gas. Using a Bogoliubov analysis, we also consider the case where the regularization of the divergence is due to the finite range of the interaction between atoms.

The effect of the coupling of a many-body quantum system to an environment attracted a lot of attention in the last years, in the context of cold atoms experiments. Engineered coupling was proposed to realize particular many-body states [1,2], including strongly correlated phases or highly entangled states [3]. It can also be used as a resource for quantum computation [4]. A particular coupling to an environment, that has received a lot of attention recently, is realized when the gas suffers from losses. Losses can produce highly correlated phases [5][6][7][8], induce Zenon effect [9][10][11], drive phase transition [12], lead to non-thermal states [13][14][15][16], and produce cooling [17][18][19]. In all the works mentioned above, the coupling to the environment is described assuming that the correlation time of the environment is much smaller than any characteristic evolution time of the system. Then the time evolution of the system obeys a universal Lindblad equation (see the review [20]) describing the coupling to an environment of vanishing correlation time. In this paper we show that this approximation is not always correct.

For a homogeneous single atom loss process, the universal Lindblad equation reads, for a gas in the continuous space,

dρ dt = -(i/ )[H 0 , ρ] + Γ ˆdd r - 1 2 {ψ + r ψ r , ρ} + ψ r ρψ + r
(1) where H 0 is the Hamiltonian of the quantum gas, ρ is its density matrix, d is the dimension of system, ψ r annihilates an atom at position r, and Γ is the loss rate. For simplicity we consider here a single specy gas. Eq. ( 1) is universal in the sense that the loss process is characterized by a single parameter Γ, details of the reservoir being irrelevant.

The evolution under the above Lindblad equation is simple if one assumes the state of the gas is uncorrelated, for instance within a mean-field approximation : the population of each single particle state decreases exponentially [9]. However, interactions between atoms introduce correlations, which highly complicates the calculation of the effect of losses. In cold atoms experiments, the range of the interaction potential between atoms is typically much smaller than all length scales in the problem. Then the effect of interactions is well modeled by a contact interaction term. This description of interactions is also a universal model: details of the interaction potential is irrelevant and interactions are described by a single parameter, the scattering length. In this paper, we show that the combination of the two above universal models leads to unphysical predictions in dimensions higher than one: for a gas with contact interaction evolving under Eq. ( 1), the increase rate of the energy diverges.

The divergence of the energy increase rate originates from the following process. The contact interaction in the gas is responsible for singularities of the many body wavefunction when two atoms meet [21], leading, in dimension higher than one, to a diverging kinetic energy, this divergence being counterbalanced by the interaction energy such that the total energy is finite. The Lindblad dynamics of Eq. (1) assumes that loss events are instantaneous with respect with the gas dynamics: within the quantum trajectory description equivalent to the Lindblad dynamics [20] a loss event corresponds to the instantaneous action of the jump operator ψ r . Thus, just after a loss event occurred, the many-body wavefunction of the remaining atoms is equal to its value just before the loss event. This wave function presents a singularity when the position of an atom approaches the position of the lost atom. The divergence of the kinetic energy associated with this singularity, is no longer counterbalanced by interaction energy: it amounts to an infinite value of the energy in the system. Note that the infinitely large increase of the energy is made possible by the infinite energy available in the reservoir involved in the loss process: the vanishing correlation time is associated to an infinite energy width.

Several mechanisms could lead to a regularization of the above divergence. First, the finite range of the interaction between the atoms will introduce a cutoff that prevents the divergence of the kinetic energy. Second, the reservoir has in practice a finite energy width which limits the maximum energy a loss event can deposit in the system. In this paper, we consider both regularizations, with an emphasis on the effect of the finite reservoir energy-width.

We first propose a model for the loss mechanism, with a finite energy width E res . Using an analysis of the 2-atoms case, we then derive the expected value of the energy density increase rate for a gas with contact interactions, valid for large E res . We find a general expression, that involves the contact parameter. Although our derivation concentrates on the Bosonic case for simplicity of notations, our results are general. To compute the evolution of the system beyond this limit of large E res , one needs a many-body model of the system that includes correlations between atoms introduced by interactions. We will concentrate on the case of a weakly interacting Bose-Einstein condensate and we use the Bogoliubov description. Within this framework, we compute the evolution of the energy. Assuming a loss rate much smaller than the relaxation rate of the gas, the system can be described locally at any time by a thermal equilibrium state. We compute the expected evolution of the temperature under the effect of losses. Within the Bogoliubov treatment, we also consider the case where the regularization comes from the finite range of the interactions.

Model for the loss process. We consider a gas made of particles of mass m in dimension d = 1 (1D), d = 2 (2D) or d = 3 (3D), and we use periodic boundary conditions in a box of size L d . A homogeneous one-body loss process occurs if, at each point, the atoms are coupled to a continuum. In a gas confined in 1D or 2D, the frozen dimension(s) could serve as the continuum, if atoms are coupled to an untrapped state. In 3D, the loss mechanism could be the de-excitation of the atoms, if the latter are in a metastable state, in which case the momentum of the emitted photon provides the continuum for the loss mechanism. Here instead we will consider a simpler, yet equivalent model [22], where the loss mechanism is induced by a noisy coupling to an untrapped internal state, the different Fourier components playing the role of the continuum. This would correspond to the effect of a noisy magnetic field for magnetically trapped atoms [19]. More precisely, we consider a coupling to the reservoir which writes

V = ˆdd rΩ(t)ψ r b + r + h.c. = p Ω(t)Ψ p B + p + h.c., (2)
where ψ r , resp. b r , annihilates an atom of the system, resp. of the reservoir, at position r, Ψ p , resp. B p , annihilates an atom of the system, resp. of the reservoir, of momentum p, h.c. is the abbreviation of "hermitian conjugate" and Ω(t) is a noisy function. p takes discrete values whose coordinates are multiple of 2π /L, and we note p = |p|. We define the energy-dependent rate Γ(E) from the spectral density of Ω(t) according to

Γ(E) = 1 2 ˆdτ e -iEτ / Ω * (τ )Ω(0) (3)
and we note Γ 0 = Γ(0). We assume a Gaussian correlation function such that Γ(E) = Γ 0 e -E 2 /(2E 2 res ) , where E res is the energy width of the loss process, corresponding to a correlation time /E res . The energy of the state of momentum p in the reservoir is p 2 /(2m) where m is the mass of the atoms, up to a constant term that could be compensated by a shift in E of Γ(E) and that we take equal to zero. If there would be a single atom in the system, its loss rate, obtained within a Born-Markov approximation [22], would be Γ 0 . The Lindblad equation ( 1) is obtained by making E res → ∞ at a fixed value of Γ 0 : E res then no longer play a role and the parameter Γ 0 entirely characterizes the loss process. However, as shown below, in presence of contact interactions between atoms, such an approximation leads to a divergence of the energy increase rate in dimension d > 1. In this paper, we consider a finite value for E res .

Two atoms case. Let us first investigate the behavior expected for a system comprising initially 2 atoms. In addition to the kinetic energy term, the Hamiltonian contains a contact interaction term. We go in the center-of-mass frame so that the total momentum is vanishing and we consider a state of energy E 0 . The two-atoms wave function writes ϕ(r

1 , r 2 ) = ϕ(r 1 -r 2 ), with ´´d d r 1 d d r 2 |ϕ(r 1 -r 2 )| 2 = 1.
For simplicity of notation, we will consider identical Bosonic atoms and use second quantization representation, such that this state reads |ϕ = (1/2) p ϕ(p)Ψ + p Ψ + -p |0 , where, for p = 0, ϕ(p) = √ 2 ´dd re ip.r/ ϕ(r). The contact interaction imposes the short distance behavior ϕ(r) u 0 (|r| -a 1D ) in 1D, ϕ(r) u 0 ln(|r|)/a 2D ) in 2D and ϕ(r) u 0 (1/|r| -1/a 3D ) in 3D, where the parameter u 0 depends on d, E 0 and L. In momentum space, this asymptotic form leads to the large p behavior [21,23]

|ϕ(p)| 2 |p|→∞ α d 4 |u 0 | 2 p 4 , (4)
where

α d = 8 in 1D, α d = 8π
γ(p) = |ϕ(p)| 2 Γ(p 2 /m -E 0 ). ( 5
)
We can then compute the initial rate of change of the energy for the trapped atoms:

dE/dt = -ΓE 0 + p p 2 /(2m)|ϕ(p)| 2 Γ(p 2 /m -E 0 )
. Here Γ = p γ(p) is the total loss rate. We will assume that E res is large enough so that there exists a momentum p 0 such that E 0 p 2 0 /m E res . The first inequality ensures that the finite energy width of the reservoir does not affect the loss events towards states of momentum smaller than p 0 . The second inequality ensures that, for final states of momenta larger than p 0 , |ϕ(p)| 2 takes its large p asymptotic behavior given Eq. ( 4). Then, the contribution to dE/dt of decay processes towards momentum states of the remaining atom of momentum larger than p 0 is

dE/dt) |p|>p0 = Γ 0 α d 4 L d |u 0 | 2 (2π ) d m B, (6)
where 7) where ν = 6.769 . . . . In 1D, the result does no longer depend on E res : the energy change rate has a well defined finite value when E res → ∞. In the following we consider only gases in dimension d > 1. Then B presents a UV divergence when E res → ∞, which leads to the diverging energy change rate announced in the introduction. The finite value of E res regularizes this divergence. For large enough E res however, B is large enough such that Eq. 6 gives the main contribution to dE/dt and in the next paragraph we assume dE/dt is simply given by Eq. 6. Note finally that this two-atoms result could have been derived for two different atoms, such as two different fermions, providing losses affect both atomic specy in the same way.

B=      ´∞ p0 dpΓ(p 2 /m)/(p 2 Γ 0 ) 1/p 0 in 1D π ´∞ p0 dp p Γ(p 2 /m)/Γ 0 π 4 ln( √ mE res /p 0 ) in 2D 2π ´∞ p0 dpΓ(p 2 /m)/Γ 0 ν √ mE res in 3D. (
Many-body case: role of the contact. The results above can be generalized to many-body systems containing N atoms since, for large enough E res , the physics will be dominated by the 2-body physics presented above. More precisely, one expects the above results to hold provided one does a sum over the pairs of atoms. The relevant quantity will be the contact C, which quantifies the number of pairs in the gas [21,23,24]. The contact is defined by the amplitude of the 1/p 4 tails of the momentum distribution. More precisely, C = lim p→∞ W (p)p 4 , where the momentum distribution is normalized to ´dd pW (p) = N . In the two atoms case discussed above,

W (p) = |ϕ(p)| 2 L d /(2π ) d such that C = 4-d α d |u 0 | 2 L d /(2π) d
. Thus, Eq. ( 6) generalizes to a many-body system as

dE/dt = Γ 0 C m B (8)
We emphasize the broad applicability of this expression: it is valid both in 2D and 3D, and for Fermions or Bosons. Its validity domain is however restricted to very large E res . To go beyond this approximation, and to estimate its applicability range, one should know the details of the many-body physics. In the following we do the calculation in the case of a weakly interacting Bose gas described by the Bogoliubov theory.

Exact treatment for a gas described by Bogoliubov In this section we suppose the gas is a Bose condensed gas of density n. Beyond-mean field physics is captured, to first approximation, by the Bogoliubov theory. In this theory, the Hamiltonian reduces to

H BG = e 0 L d + p =0 p a + p a p , (9)
where a + p creates a Bogoliuov excitation of momentum p whose energy is p = p 2 /(2m)(p 2 /(2m) + 2gn), and e 0 is the ground state energy density. Bogoliubov operators are bosonic operators which fulfill [a p a + p ] = 1, and they are related to the atomic operators by the Bogoliubov transform

Ψ p = u p a p + v p a + -p Ψ + -p = v p a p + u p a + -p (10)
where u 2 p -v 2 p = 1 and v 2 p = (f p + f -1 p -2)/4, and f p = p 2 /(2m p ). We set u 0 = 1 and v 0 = 0 such that the above equation also holds for p = 0. Note that we use the symmetry breaking Bogoliubov approach that does not converse atom number [25].

Using the Bogoliubov transformation, the coupling to the reservoir, given Eq. ( 2), reads

V = p a p (u p Ω(t)B + p + v p Ω * (t)B -p ) + h.c. (11)
We compute the master equation describing the time evolution of the density matrix of the system, ρ: using second order perturbation theory, omitting fast oscillating terms, whose effect averages out, and making the Born-Markov approximation, we obtain [22]

dρ dt = -(i/ )[H 0 , ρ] -p Γ( p 2 2m -gn -p )u 2 p 1 2 {a + p a p , ρ} -a p ρa + p +Γ( p 2 2m -gn + p )v 2 p 1 2 {a p a + p , ρ} -a + p ρa p ( 12
)
where the function Γ(E) is given in Eq. ( 3). For non interacting atoms, v p = 0, u p = 1, gn = 0 and p = p 2 /(2m), such that the above equation reduces to Eq. ( 1), as expected. Correlations between atoms introduced by the interactions are responsible for the anomalous terms in v 2 p . The first effect of losses is to decrease the density n. The difference between dn/dt and -Γ 0 n is of the order of the density of atoms in the modes of wave vector p > √ mE res . We assume that E res is large enough so that we can make the approximation dn/dt -Γ 0 n. Let us now investigate the evolution of the energy E = H 0 . We use the Bogoliuobv approximation H 0 H BG , where H BG is given in Eq. ( 9), such that Temperature is rescaled to the time-dependent chemical potential µ gn gn0e -Γ 0 t , where n is the atomic density, equal to n0 at t = 0, g the interaction strength and Γ0 is the single atom loss rate. The initial value is kBT (0)/(gn0) = 0.1. In (a) interactions are contact interactions but the reservoir has a finite energy width Eres, parameterized by the dimensionless parameter ξ1 = Eres/(gn0). In (b), we assume a reservoir of infinite energy width, or equivalently of vanishing correlation time, but interactions have a finite range σ (see text) and ξ2 = /(σ √ mgn0).

dE dt = -Γ 0 nA + p p d a + p a p dt BG (13)
where A = L d de 0 /dn + p a + p a p d p /dn and (d a + p a p /dt) BG is the evolution of a + p a p within the Bogoliubov approximation. Inverting the Bogoliubov transform Eq. ( 10), we find that a + p and a p depend explicitly on time, via the dependence of u p and v p on n. However, we assume that losses are slow enough so that one has adiabatic following: d(a + p a p )/dt 0 [22]. Then (d a + p a p /dt) BG reduces to (d a + p a p /dt) BG = T r(a + p a p dρ/dt) and injecting Eq. ( 12) we obtain

d a + p ap dt BG = -Γ( p 2 2m -gn -p )u 2 p a + p a p +Γ( p 2 2m -gn + p )v 2 p 1 + a + p a p . (14)
In the case of a reservoir of infinite energy width, for which Γ(E) = Γ 0 for any E, the above equation reduces to (d a + p a p /dt) BG = Γ 0 (-a + p a p +v 2 p ). We recover here the results derived for 1D Bose gases in the quasicondensate regime [26], although those results were derived very differently. In particular, since v 2 p (mgn) 2 /p 4 at large p, we find that a + p a p develops 1/p 4 tails. In dimension 1, such tails are responsible for a failure of Tan's relation [16]. In dimension 2 and 3, such tails lead to the unphysical result that dE/dt diverges. Proper physical results are obtained in higher dimensions only taking into account the finite energy width of the reservoir. For very large E res , dE/dt is dominated by the second term of the r.h.s of Eq. ( 13), itself dominated by the large p terms for which v 2 p (mgn) 2 /p 4 , p p 2 /(2m), and a + p a p 0. Evaluating the sum, and using the fact that the contact within Bogoliubov theory is C = L d (mgn) 2 /(2π ) d , we recover Eq. (8).

The system is ergodic in dimension d > 1: beyond Bogoliubov terms in H 0 include couplings between Bogoliubov modes which, in absence of losses and as long as local observables are concerned, ensure relaxation towards a thermal state. Here we assume that Γ 0 is much smaller than the relaxation rate so that, ρ relaxes at any time to the density matrix of a thermal state. The latter is characterized by the atomic density n and the energy density, or equivalently by n and the temperature T . The energy of the gas fulfills E = E th (n, T ), where E th (n, T ) is evaluated injecting the occupation factors a + p a p = (e p /(k B T ) -1) -1 into Eq. ( 9). The time evolution of the gas is entirely characterized by the functions n(t) = n 0 e -Γ0t and T (t). In order to compute T (t), we evaluate dT /dt in the following way. Since E is conserved by the thermalization process, the calculation of dE/dt with Eq. ( 13) and ( 14) is valid, providing one injects a + p a p = (e p/(kB T ) -1) -1 in the r.h.s of Eq. ( 14). Once dE/dt has been computed one can compute dT /dt using dE/dt = -Γ 0 n(∂E th /∂n) T + dT /dt(∂E th /∂T ) n . Calculations are detailed in the appendix.

In Fig. 1 we present time evolution of the temperature of the system, for different values of E res . We find that the ratio k B T /(gn) is a growing function of time. This contrasts with the prediction obtained for phonons, which are the Bogoliubov modes of momentum p √ mgn : in absence of rethermalization between Bogoliubov modes, and for E res gn, one expects that, for phonons, k B T /(gn) takes the asymptotic value k B T /(gn) = 1 [13,27]. The growing of k B T /gn is due to the contribution of high-p Bogoliubov modes. The growing rate increases with E res , as expected: we expect that dT /dt diverges as E res goes to infinity.

Regularization by a finite interaction range. The Bogoliubov analysis can serve also to describe the regularization of the UV divergence due to finite interaction range. We consider here a two-body interaction potential V (r) = ge -r 2 /(2σ 2 ) /((2π) 3/2 σ 3 ), where r is the distance between the two atoms and σ is the interaction range. The Bogoliubov transform given in Eq. ( 10) is still valid, using the Bogoliubov spectrum [START_REF] Bogolubov | Quantum Statistical Mechanics: Selected Works of N N Bogolubov[END_REF] p = p 2 /(2m)(p 2 /(2m) + 2gne -p 2 σ 2 /(2 2 ) ). One can then compute the effect of losses as above. In the limit of infinite E res , the divergence of dE/dt is regularized by the finite interaction range σ. For very small σ, dE/dt is dominated by the large p term of the sum in Eq. ( 13), for which v 2 p (mgn) 2 (e -p 2 σ 2 / 2 )/p 4 . Evaluating the sum, we recover Eq. ( 8) with B = π 3/2 /σ. As above, from the calculation of dE/dt due to losses, we compute the time evolution of the temperature in the system. Fig. 1 shows the time evolution of the temperature, for different values of σ. The evolution of k B T /(gn) is qualitatively similar to what is observed for contact interactions but finite energy width of the reservoir, 2 /(mσ 2 ) playing the role of E res .

Conclusion.

Remarkably, although losses are ubiquitous in experiments, the description and the understanding of their effect is still at its infancy. Before this work, effect of losses has been studied using the universal Lindblad equation Eq. (1). However, studies where made either in 1D, in which case the divergence of the energy increase rate does not exists, or for a gas confined in the lowest band of a lattice, in which case the lattice period provides a cut-off that prevents the divergence, or using a mean-field approximation that neglect correlations between atoms. This paper provides the first prediction for the effect of losses on an interacting quantum gas in higher dimension and in the continuum. Predictions of this paper could be tested experimentally using an engineered noisy coupling to an untrapped state [19] whose energy width E res can be varied. Alternatively, the measurement of the temperature evolution could be used to infer the energy width of the reservoir. This work raises many questions. How can we extend the results obtained with Bogoliubov to quasi-condensate describing 2D gases at thermodynamic limit ? How can the results presented in this paper be extended to 2-body or 3-body losses ? How can we extend the calculations done in this paper to other models of quantum gases, such as two-components fermionic gases ? Our work put in question the fundamental relation giving the time evolution of the density under the effect of losses. In this paper, we assumed a reservoir energy width large enough so that dn/dt remains close to -Γ 0 n. However, one expects a small correction involving 2-body processes, whose calculation deserves investigation.

5 FIG. 1 .

 51 FIG.1. Evolution of the temperature of a 3D weakly interacting Bose-Einstein condensate under the effect of losses. Temperature is rescaled to the time-dependent chemical potential µ gn gn0e -Γ 0 t , where n is the atomic density, equal to n0 at t = 0, g the interaction strength and Γ0 is the single atom loss rate. The initial value is kBT (0)/(gn0) = 0.1. In (a) interactions are contact interactions but the reservoir has a finite energy width Eres, parameterized by the dimensionless parameter ξ1 = Eres/(gn0). In (b), we assume a reservoir of infinite energy width, or equivalently of vanishing correlation time, but interactions have a finite range σ (see text) and ξ2 = /(σ √ mgn0).

  2 in 2D and 32π 2 in 3D. The 2-atoms state |ϕ is coupled by V to the states |p = Ψ + p B + -p |0 whose energy, equal to the sum of the kinetic energies of the lost atom and the remaining atom, is p 2 /m. For weak enough Γ 0 , one can use Born-Markov approximation to compute the loss rate towards the state |p [22]. Using p|V |ϕ = Ω(t)ϕ(p), one finds a loss rate

Acknowledgment This work was supported by Palm grant 20P555 and ANR grant ANR-20-CE30-0017-01 The authors thanks D. Petrov and J. Dubail for useful discussions.


Toshiyuki Tabata
email: toshiyuki.tabata@screen-lasse.com

Pierre - Edouard Raynal
email: pierre-edouard.raynal@screen-lasse.com

Fabien Rozé
email: fabien.roze@screen-lasse.com

Sébastien Halty
email: sebastien.halty@screen-lasse.com

Louis Thuries
email: louis.thuries@screen-lasse.com

Fuccio Cristiano

Emmanuel Scheid
email: scheid@laas.fr

Fulvio Mazzamuto
email: fulvio.mazzamuto@screen-lasse.com

Copper Large-scale Grain Growth by UV Nanosecond Pulsed Laser Annealing

Keywords: laser anneal, BEOL, grain growth, copper

UV nanosecond pulsed laser annealing (UV NLA) enables both surface-localized heating and short timescale high temperature processing, which can be advantageous to reduce metal line resistance by enlarging metal grains in lines or in thin films, while maintaining the integrity and performance of surrounding structures. In this work UV NLA is applied on a typical Cu thin film, demonstrating a mean grain size of over 1 μm and 400 nm in a melt and sub-melt regime, respectively. Along with such grain enlargement, film resistivity is also reduced.

I. INTRODUCTION

In advanced BEOL interconnects, reducing the trench geometry limits metal grain growth with the consequence of increasing electron scattering at grain boundaries. It results in an exponentially increasing line resistivity while scaling down the lines, and degradation of RC delay. In such scaling era, alternative metals such as Ru, Co, and Mo are introduced because of their potential benefits in line resistivity, which come from a complex combination of bulk resistivity, line width, mean-free-path of electrons, electro-migration reliability (i.e., melting point), integration compatibility, and especially the use of a specific set of barrier and liner [START_REF] Tokei | Sub-5nm Interconnect Trends and Opportunities[END_REF][START_REF] Nogami | Overview of interconnect technology for 7nm node and beyond -New materials and technologies to extend Cu and to enable alternative conductors[END_REF][START_REF] Hu | Electromigration and resistivity in on-chip Cu, Co and Ru damascene nanowires[END_REF][START_REF] Bhosale | Co Reflow and Barrier Scaling Studies for Beyond 7 nm Node BEOL Applications[END_REF]. However, even if the narrowest interconnects are formed with alternative metals, copper will not be completely replaced, and it will remain the reference to beat and the preferred candidate for larger interconnects. Thus, it is important to explore new paths to boost his performances and extend his utilization. Extending Cu technology is still possible, especially by engineering the barrier/liner part [START_REF] Lo | Replacing TaN/Ta Bilayer with 2D Layered TaS2 Converted from Ta for Interconnects at Sub-5 nm Technology Nodes[END_REF][START_REF] Motoyama | EM enhancement of Cu interconnects with Ru liner for 7 nm node and beyond[END_REF][START_REF] Nogami | Advanced BEOL Interconnects[END_REF]. On the other hand, nanosecond laser annealing (NLA) demonstrated a benefit on BEOL interconnects by enlarging the mean size of grains in both Cu [START_REF] Lee | Nanosecond Laser Anneal for BEOL Performance Boost in Advanced FinFETs[END_REF][START_REF] Gluschenkov | Laser Annealing in CMOS Manufacturing[END_REF] and Ru [START_REF] Jourdan | UV nanosecond laser annealing for Ru interconnects[END_REF] lines. In fact, NLA allows to reach a much higher surface temperature than that of conventional BEOL limit (i.e., 400 ℃ for minutes), while conserving the functionality of surrounding devices thanks to its short timescale and shallow irradiation absorption. Such opportunity to reduce the interconnect resistance became particularly critical now, when the number of interconnect layers is continuously increasing [START_REF] Kobrinsky | On-die interconnect challenges and opportunities for future technology nodes[END_REF].

In this paper we study high temperature processing realized by UV NLA to enable large-scale grain growth in thin films and lines (e.g., roughly 50-nm-thick). Specifically, we present the formation of large grains, which is, to our knowledge, a record for such a thin film (typically around 100 nm after annealing at 350 to 600 ℃ for minutes [START_REF] Sun | Surface and grain-boundary scattering in nanometric Cu films[END_REF][START_REF] Chawla | Electron scattering at surfaces and grain boundaries in Cu thin films and wires[END_REF]). This opens a potential path to boost performances of future Cu interconnects.

II. EXPERIMENTAL

A 50-nm-thick sputtered-Cu was deposited on 8 nm-thick Ta/5 nm-thick TaN/100 nm-thick SiO2/Si without any capping layer on top. A UV NLA was performed at room temperature in air. Both laser fluence (LF) and process time (t) were varied to control the heat generated in the Cu film. The evolution of the material as a function of annealing condition was captured by in-plane XRD. Then, some selected conditions were analyzed by TEM and Electron Diffraction Mapping (EDM). Finally, the correlation between the film resistance and grain size was deduced.

III. RESULTS AND DISCUSSION

Figure 1 shows the XRD patterns taken at LF1 for different t (t1 < t2 < t3 < t4 < t5), where the reference (i.e., non-annealed) data is also compared. In the as-deposited Cu film, the peaks of Cu (111), (200), and (220) planes are clearly observed, and a slight surface oxidation is also implied by the peak of Cu2O (111). For t1 and t2, the peaks of Cu (111) and (200) disappear, while those of Cu (220) grows. For t3, t4, and t5, the disappeared peaks emerge again, while the peaks of Cu (220) show a significant drop of intensity. This suggests that the Cu film is melting (i.e., homogeneous nucleation) at the latter conditions (t3, t4, t5), but not at the former ones (t1, t2). A similar XRD pattern evolution was observed at a smaller laser fluence (LF2) as well (data not shown).

To get a more in-depth comparison of sub-melt and melt regimes, two UV NLA conditions (i.e., t2 and t4 at LF1) were selected to pursue the analysis of microstructure in the Cu thin film. Figure 2 shows the plane-view TEM images of the nonannealed and annealed samples. In the as-deposited film, only small grains are observed everywhere. In the annealed films, a significant grain growth is observed. On these TEM specimens, electron diffraction patterns were obtained and fitted with those theoretically predicted for a face-centered-cubic (FCC) crystal lattice of Cu. Also, a mean grain size (Av.) was calculated as a weighted average of area ratio within each observed area, considering Σ3CSL facets as grain boundaries. In the asdeposited film (Fig. 3(a)(b)(c)), the small grains (Av. 53.9 nm) are randomly distributed. After annealing at LF1 for t2 (Fig. 3(d)(e)(f)), a significant grain growth (Av. 414 nm) is observed and interestingly the Cu (111) plane is preferably oriented on the top surface (i.e., ND). This is consistent with the fact that the (111) plane has the lowest surface energy in Cu FCC system [START_REF] Mclean | Surface energy anisotropy by an improved thermal grooving technique[END_REF]. In the in-plane directions (i.e., TD and RD), all the observed planes are on a typical tensile-strain-induced slip line (i.e., between [101] and [011] on an FCC-type stereogram) of Cu single crystal [START_REF] Kim | In-situ X-ray diffraction study of singleslipconditioned copper single crystals during uniaxial deformations[END_REF]. After annealing at LF1 for t4 (Fig. 3(g)(h)(i)), very large grains (Av. 1000 nm) are randomly distributed. Then, the grain morphology seems drastically changed, and certain grains grow up to several μm scale in a preferred direction.

At the end, film resistivity was characterized for the same three samples. According to the literature [START_REF] Sun | Surface and grain-boundary scattering in nanometric Cu films[END_REF], the Mayadas-Shatzkes (MS) model, which involves the scattering at grain boundaries, may give a better fit with our experimental data obtained in the 50-nm-thick Cu film, than the Fuchs-Sondheimer (FS) model. To start, the MS model was fitted with the film resistivity of the non-annealed sample by changing only the reflection coefficient, R. Then, R = 0.45 was obtained, which is close to the reported value (R = 0.47 [START_REF] Sun | Surface and grain-boundary scattering in nanometric Cu films[END_REF]). The same R was applied to the other samples. The effective thickness of the Cu and Cu2O layers before and after UV NLA were confirmed by cross-sectional TEM images. Figure 4 shows the fitting result as a function of the mean grain size (Av.). For the annealed samples, the film resistivity shows a deviation from the theoretical values. A possible origin of this deviation is associated to oxygen (O) contamination in our Cu thin films (before and) during the annealing. According to SIMS, the averaged O concentration level became higher when the mean grown grain size did larger (data not shown). In addition, in the melting regime, the Cu surface roughness may be significantly increased during UV NLA, resulting in more pronounced surface scattering of electrons compared to the non-annealed case. These two aspects will be improved in future works by covering the Cu surface with a capping layer and by better controlling the UV NLA conditions.

IV. CONCLUSION

We investigated the effect of UV NLA on a thin Cu film in order to control the grain growth and to find a path to mitigate the exponential growth of the metal resistance in advanced metal interconnects. In the sub-melt condition, the mean grain size (Av.) was increased up to 414 nm, almost 8-times-larger than that of the as-deposited film (Av. 53.9 nm), with a controlled distribution of grain orientations. In the melting condition, the grain growth was extended further (Av. 1000 nm), but the control of the grain orientation was not maintained. The observed grain growth led to a consistent reduction of the film resistivity. Although these results are promising, it is only the first fundamental study on thin films that motivates a further investigation. Particularly, the grain growth control needs to be confirmed in real interconnect structures. Also, it must be assured that the applied thermal budget does not degrade surrounding materials and structures.

Fig. 1 .

 1 Fig. 1. XRD patterns obtained for the non-annealed and annealed Cu thin films. UV NLA was at LF1 for different t (t1 < t2 < t3 < t4 < t5).

Fig. 2 .Fig. 3 .

 23 Fig. 2. Plain-view TEM images taken for the nonannealed and annealed Cu thin films. UV NLA was at LF1 for t2 and t4. Fig. 3. EDM images taken for the non-annealed (i.e., (a), (b), and (c)) and the annealed Cu thin films. UV NLA was at LF1 for t2 ((d), (e), and (f)) or t4 ((g), (h), and (i)). As depicted in (j), ND, TD, and RD stand for Normal Direction, Transverse Direction, and Reference Direction, respectively. Also, a standard triangle of grain orientations is also shown in (k).

Fig. 4 .

 4 Fig. 4. Experimental and theoretical (based on the Mayadas-Shatzkes (MS) model) film resistivity of the non-annealed and annealed samples.

ACKNOWLEDGMENT

This project has received funding from the ECSEL Joint Undertaking (JU) under grant agreement No 875999. The JU receives support from the European Union's Horizon 2020 research and innovation programme and Netherlands, Belgium, Germany, France, Austria, Hungary, United Kingdom, Romania, Israel.


Christina Hazard

James I Prosser

Graeme W Nicol
email: graeme.nicol@ec-lyon.fr

Use and abuse of potential rates in soil microbiology

Keywords: Potential nitrification rates, potential nitrification activity, ammonia oxidation, soil microbiology

Potential rate assays are used in soil microbial ecology to determine the rates of a functional process in environmental samples under a defined set of conditions. While they can be used appropriately to provide mechanistic insights, potential rates are also often used to estimate the abundance of specific taxonomic groups and their in situ activity. These estimates incorrectly assume that all contributing organisms in a community are active at a maximum rate under one set of 'optimal' incubation conditions and that potential rates reflect activity in the soil. While investigators now recognise that populations within communities are physiologically diverse, they often ignore the consequent suboptimal activity, or even inactivity, of the majority of community members performing that function. In this short perspective article, we discuss when potential assays can be informative and highlight the underlying conceptual problems under circumstances where potential assays are misused, using potential nitrification rate (PNR) as an example. PNR was originally developed to estimate the size of active ammonia oxidising communities in environmental samples. It is routinely determined in short-term shaken slurry incubations by measuring assumed maximum rates of nitrate or nitrite production under optimal, non-substrate-limiting conditions. As with other functional processes, it is now recognised that a broad diversity of organisms contribute to aerobic ammonia oxidation in terrestrial and other habitats, and this diversity represents a substantial range of physiologies, including variation in substrate affinity, ammonia tolerance, cell specific activity and substrate preference. Despite this, PNR, and other potential rate assays, are often inappropriately used in an attempt to determine an ecologically relevant measurement of activity in soil. As with any potential assay, PNR has inherent biases towards particular functional groups and its use in investigating the ecology of ammonia oxidisers in natural systems should be carefully considered.

Main text

Measurement of the potential rate of microbially mediated soil processes has a long tradition.

Commonly used examples are measurement of potential nitrification and denitrification rates and substrate-induced respiration. All determine a process rate, usually after addition of a relevant substrate, under what are sometimes considered to be optimal conditions that enable most organisms capable of performing the process to operate with maximum activity and/or specific growth rate. Potential rate obviously differs from 'actual' process rate, which will typically operate under non-optimal environmental conditions. Potential process rates have

value, and have traditionally been used, in the absence of alternatives, to estimate the biomass (or, less justifiably, abundance) of a functional microbial group. However, the assumptions underlying measurement of potential rate are often not considered, obeyed or tested and have been increasingly challenged as our understanding of the composition and diversity of microbial communities in natural environments has increased. The value, limitations and relevance of potential rate measurements in microbial ecology are illustrated below for soil nitrification. However, the factors defining the usefulness and limitations of potential assays will also apply to other processes and functional groups, some of which encompass far greater taxonomic or functional diversity. For example, potential denitrification assays are typically performed using one set of incubation condition only and attempt to target organisms that represent a far greater level of taxonomic and physiological diversity than nitrifiers. In such circumstances, predictive power will likely be even further reduced.

Potential nitrification rate

Nitrification is the sequential oxidation of ammonia (NH3) to nitrite (NO2 -) and nitrate (NO3 -) and is an essential component of reactive nitrogen transformation in soil. While nitrite can accumulate in soil [START_REF] Giguere | Uncoupling of ammonia oxidation from nitrite oxidation: Impact upon nitrous oxide production in noncropped Oregon soils[END_REF][START_REF] Ma | Nitrite behavior accounts for the nitrous oxide peaks following fertilization in a fluvo-aquic soil[END_REF][START_REF] Taylor | Temperature affects the kinetics of nitrite oxidation and nitrification coupling in four agricultural soils[END_REF][START_REF] Venterea | Mechanisms and kinetics of nitric and nitrous oxide production during nitrification in agricultural soil[END_REF], particularly after N deposition [START_REF] Venterea | Nitrite accumulation and nitrogen gas production increase with decreasing temperature in urea-amended soils: Experiments and modeling[END_REF], ammonia oxidation is often assumed to be the rate-limiting step. As a consequence, the activity, diversity and community structure of ammonia oxidisers (AO) receives greatest attention with respect to understanding environmental factors that influence microbially-mediated nitrification.

Potential nitrification rate (PNR) assays (also referred to as potential nitrification activity (PNA)) have been used for many decades as a rapid method for estimating net ammonia oxidation (≈nitrification) rates in soils.

While a range of PNR assays have been used, the most common approach is the shaken soil-slurry method [START_REF] Belser | Specific inhibition of nitrite oxidation by chlorate and its use in assessing nitrification in soils and sediments[END_REF][START_REF] Hart | Nitrogen mineralization, immobilisation and nitrification[END_REF]. This typically involves incubation of a suspension of sieved soil in a solution containing NH4 + , at non-limiting concentration, in a shaken flask, ensuring aeration and mixing, at an incubation temperature that is considered to be optimal. PNR is calculated as the gradient of the linear increase in nitrate concentration in samples taken regularly during a relatively short incubation period (often ≤24 h) [START_REF] Hart | Nitrogen mineralization, immobilisation and nitrification[END_REF]. A potassium phosphate buffer, typically between pH 7.0 and 7.5, can be included in the assay solution, with the intention of optimizing pH for AO. Chlorate, an inhibitor of nitrite oxidation, is also commonly included and NO2 -rather than NO3 -is measured in each slurry sample [START_REF] Belser | Specific inhibition of nitrite oxidation by chlorate and its use in assessing nitrification in soils and sediments[END_REF]. Growth is assumed to be negligible; hence the assumption of a linear, rather an exponential increase in nitrite or nitrate concentration. PNR therefore measures the activity of the community and not specific growth rate. Linear kinetics are often not tested and it is not unusual for calculations to be based on nitrate concentrations measured only at the beginning and end of incubation or even using just the final value. Any growth, or other divergence from linear kinetics, invalidates calculation of a 'constant' PNR.

Employment of 'ideal' or 'optimal' conditions attempts to ensure that all AO within the sample are active, which is further encouraged by addition of substrate (NH4 + ). The assumption of universal activity was implicit in the traditional use of PNR to estimate AO abundance in soil. This assumption was always seen as a major limitation of this approach (Woldendrop and Laanbroek, 1989;[START_REF] Hart | Nitrogen mineralization, immobilisation and nitrification[END_REF], particularly given the low growth rates of ammonia oxidisers. (The only alternative was use of the most probable number method, which also relies on incubation of dilutions of soil suspensions under 'ideal' conditions and similar assumptions.)

Physiological diversity of ammonia oxidisers challenges underlying assumptions of PNR

For more than a century, ammonia oxidation was thought to be performed mainly by ammonia oxidising bacteria (AOB) [START_REF] Prosser | Archaeal and bacterial ammonia-oxidisers in soil: the quest for niche specialisation and differentiation[END_REF]. The past 25 years have, however, demonstrated considerable phylogenetic diversity within AOB and also the capacity for ammonia oxidation by ammonia oxidising archaea (AOA) [START_REF] Könneke | Isolation of an autotrophic ammonia-oxidizing marine archaeon[END_REF] and complete ammonia oxidizers (comammox) of the genus Nitrospira [START_REF] Daims | Complete nitrification by Nitrospira bacteria[END_REF][START_REF] Van Kessel | Complete nitrification by a single microorganism[END_REF]. Evidence from both cultivation-based and ecological studies also provides evidence for substantial physiological diversity in AO. Physiological diversity of cultivated AOB is well established and was used for traditional classification of these organisms [START_REF] Prosser | Autotrophic nitrification in bacteria[END_REF][START_REF] Koops | The Lithoautotrophic Ammonia-Oxidizing Bacteria[END_REF]. Since the first isolation of an ammonia oxidising archaeon (AOA), Nitrosopumilus maritimus (Könneke et al., 2015), several enrichments and isolates of soil AOA have also been obtained. These include, for example, the neutrophile Nitrososphaera viennensis EN76 [START_REF] Tourna | Nitrososphaera viennensis, an ammonia oxidizing archaeon from soil[END_REF]) and the obligate acidophile Candidatus Nitrosotalea devanaterra ND1 [START_REF] Lehtovirta-Morley | Cultivation of an obligate acidophilic ammonia oxidizer from a nitrifying acid soil[END_REF]. In addition, comammox Nitrospira are abundant and contribute to ammonia oxidation in soil [START_REF] Pjevac | AmoA-targeted polymerase chain reaction primers for the specific detection and quantification of comammox Nitrospira in the environment[END_REF]Wang et al., 2019a), and a gammaproteobacterial AOB (Candidatus Nitrosoglobus terrae) was isolated recently [START_REF] Hayatsu | An acid-tolerant ammonia-oxidizing γ-proteobacterium from soil[END_REF]. This taxonomic diversity is reflected in ecophysiological diversity and the maximum specific growth rate, cell specific activity, inhibitory ammonia and nitrite concentrations and affinity for ammonia of AO all vary by over an order of magnitude among different soil isolates, both within and between the major ammonia oxidiser groups (summarised in [START_REF] Lehtovirta-Morley | Isolation of "Candidatus Nitrosocosmicus franklandus", a novel ureolytic soil archaeal ammonia oxidiser with tolerance to high ammonia concentration[END_REF]. In addition, many AO are inhibited by NO2 -, which accumulates when chlorate is used to inhibit nitrite oxidisers, thereby reducing PNR depending on community composition. Thus, even if the incubation conditions (temperature, pH, oxygen concentration) are appropriate for all AO, the PNR will represent only an aggregate of a wide range of cell activities and will vary with community composition and not just abundance.

Physiological diversity does, however, also mean that incubation conditions required for activity of all AO will not be identical. Cultivation-based studies have demonstrated significant differences in effects of temperature [START_REF] Jiang | Comparison of Nitrosospira strains isolated from terrestrial environments[END_REF][START_REF] Tourna | Nitrososphaera viennensis, an ammonia oxidizing archaeon from soil[END_REF][START_REF] Lehtovirta-Morley | Characterisation of terrestrial acidophilic archaeal ammonia oxidisers and their inhibition and stimulation by organic compounds[END_REF], pH [START_REF] Allison | Urease activity in neutrophilic autotrophic ammoniaoxidizing bacteria isolated from acid soils[END_REF][START_REF] Lehtovirta-Morley | Cultivation of an obligate acidophilic ammonia oxidizer from a nitrifying acid soil[END_REF] and salinity [START_REF] Mosier | Ecophysiology of an ammonia-oxidizing archaeon adapted to low-salinity habitats[END_REF][START_REF] Qin | Marine ammonia-oxidizing archaeal isolates display obligate mixotrophy and wide ecotypic variation[END_REF][START_REF] Bello | Differential sensitivity of ammonia oxidising archaea and bacteria to matric and osmotic potential[END_REF]. Similar evidence of physiological diversity is provided by ecological studies [START_REF] Nicol | The influence of soil pH on the diversity, abundance and transcriptional activity of ammonia oxidizing archaea and bacteria[END_REF][START_REF] Ollivier | Abundance and diversity of ammonia-oxidizing prokaryotes in the root-rhizosphere complex of Miscanthus x giganteus grown in heavy metal-contaminated soils[END_REF][START_REF] Gubry-Rangin | Temperature responses of soil ammonia-oxidising archaea depend on pH[END_REF][START_REF] Nacke | Links between seawater flooding, soil ammonia oxidiser communities and their response to changes in salinity[END_REF][START_REF] Taylor | Modeling of soil nitrification responses to temperature reveals thermodynamic differences between ammonia-oxidizing activity of archaea and bacteria[END_REF][START_REF] Lu | Differential responses of soil ammonia-oxidizing archaea and bacteria to temperature and depth under two different land uses[END_REF]) and the ubiquity of AO in the full range of soils, in itself, is evidence that a single set of incubation conditions will be insufficient. For example, ammonia supply is an important factor influencing niche differentiation between AOA and AOB populations in soil. AOB often dominate growth at higher concentrations of added inorganic ammonium (e.g. [START_REF] Jia | Bacteria rather than Archaea dominate microbial ammonia oxidation in an agricultural soil[END_REF][START_REF] Xia | Autotrophic growth of nitrifying community in an agricultural soil[END_REF][START_REF] Hink | Archaea produce lower yields of N2O than bacteria during aerobic ammonia oxidation in soil[END_REF], while AOA appear to prefer ammonium that is derived from a mineralised organic N source, i.e. that is part of the native organic component of the soil or added organic N [START_REF] Levičnik-Höfferle | Stimulation of thaumarchaeal ammonia oxidation by ammonia derived from organic nitrogen but not added inorganic nitrogen[END_REF][START_REF] Hink | The consequences of niche and physiological differentiation of archaeal and bacterial ammonia oxidisers for nitrous oxide emissions[END_REF]. In low pH soils, AOA make a greater relative contribution to ammonia oxidation under specific incubation conditions [START_REF] Gubry-Rangin | Archaea rather than bacteria control nitrification in two agricultural acidic soils[END_REF][START_REF] Zhang | Ammonia-oxidizing archaea have more important role than ammonia-oxidizing bacteria in ammonia oxidation of strongly acidic soils[END_REF] and, as such, a decreasing proportion of the ammonia oxidising community may preferentially oxidise added inorganic N with decreasing soil pH.

It is therefore impossible to identify incubation conditions that will be 'ideal' for all organisms and the majority of AO will have sub-optimal or no activity and will not contribute to measured PNR. To this must be added the assumption that ammonia available and oxidised in PNR assays will also be available for the same populations in soil.

Potential nitrification rate does not reflect ammonia oxidiser activity in situ

The consequences of these limitations can be seen when correlations are determined between published PNR for soils and sediments and AOA and AOB abundances (using amoA gene abundance as a proxy for cell abundance). Results are highly variable. Of 107 studies published from 2007 to 2020 (Supplementary Table 1), only 36% and 75% reported a significant positive correlation between PNR and AOA or AOB abundances, respectively (Figure 1). Ten percent and 46% report a positive correlation with only AOA or only AOB, respectively, 11% correlated with neither and 4% and 5% reported a negative correlation with AOA and AOB, respectively. No correlation has yet been observed between PNR and comammox bacteria (Wang et al. 2019b;Luchibia et al. 2020;[START_REF] Zhang | Ammonia oxidizers in river sediments of the Qinghai-Tibet Plateau and their adaptations to high-elevation conditions[END_REF]. While these studies suggest that some populations of AOA are using added ammonium, the majority of studies indicate that only ammonia oxidising bacteria (AOB) contribute to PNR, often leading to the conclusion that AOA are not contributing to ammonia oxidation in a soil in situ. If we consider the growth characteristics of AO in both culture and soil, it is evident that correlating results from a single defined assay with the abundance of communities containing substantial physiological diversity is unlikely to be meaningful and likely to be misleading.

The observed dominance of AOB over AOA in contributing to PNR activity in many soils may reflect differences in their ability to utilise added inorganic nitrogen. To investigate this, soil was sampled across a pH 4.5 to 7.5 gradient that has been fully characterised with respect to AOA and AOB community structure and abundance [START_REF] Aigle | The application of high-throughput sequencing technology to analysis of amoA phylogeny and environmental niche specialisation of terrestrial bacterial ammonia-oxidisers[END_REF][START_REF] Gubry-Rangin | Niche specialization of terrestrial archaeal ammonia oxidizers[END_REF], where amoA transcript and gene abundance of AOA and AOB decrease and increase directly with soil pH, respectively [START_REF] Prosser | Archaeal and bacterial ammonia-oxidisers in soil: the quest for niche specialisation and differentiation[END_REF]. PNR assays were established with both standard 1 mM phosphate buffered and unbuffered 1.5 mM NH4 + solution, and parallel incubations were performed in the presence of alkynes to differentiate contributions of all AO, AOA and AOB. Acetylene (C2H2) has been extensively used as an inhibitor of all ammonia oxidisers that possess ammonia monooxygenase (AMO) and 1-octyne (C8H2) inhibits growth of AOB, but not AOA, in both culture and soil [START_REF] Taylor | Use of aliphatic n-alkynes to discriminate soil nitrification activities of ammonia-oxidizing thaumarchaea and bacteria[END_REF].

Nitrate accumulation during incubation for 24 h was used to calculate PNR, which increased directly with soil pH across the gradient (Figure 2). However, as this activity was completely inhibited in the presence of both C8H2, and C2H2, it was attributed solely to AOB. This provides strong evidence that AOA do not contribute to PNR in some soils, PNR severely underestimates their abundance and this prevents any meaningful interpretation of their abundance or actual activity in soil at the time of sampling.

As indicated above, the source of ammonium is also an important factor in defining the ecological niches of AO populations within a community. Using concentrations typical of fertilisation application rates, we compared nitrification activity in both pH 4.5 and 7.5 soil microcosms after the addition of ammonium nitrate or urea (which is fully mineralised to ammonium within 0.5 days). After near stoichiometric conversion of ammonium to nitrate over 12 days, inorganic and urea-derived ammonium were oxidised at comparable rates for both 100 and 200 µg N g -1 dry soil applications in the pH 7.5 soil (Figure 3). The greatest change in pH in soil microcosms was associated with those with the highest nitrification rates, indicating that pH change was not an inhibiting factor (Supplementary Figure 1). Nevertheless, these results highlight an additional concern, as changing conditions during incubation will influence activity and kinetics will no longer be linear. In contrast, urea-derived ammonium was oxidised at a faster rate in the pH 4.5 soil, and ammonia oxidation was inhibited by application of 200 µg inorganic N g -1 dry soil (Figure 3). While these data do not differentiate the contribution of different AO groups, they reveal that sources of N other than inorganic NH4 + , as used in PNR, are preferred by some AO populations, again preventing meaningful interpretation of PNR data.

These two experiments, performed using only one experimental agricultural soil system, are clearly not meant to be considered representative of all other managed or natural soil systems. They do, however, demonstrate that PNR does not effectively characterise the contribution of all AO populations to nitrification, but selects for those that are most competitive under one set of incubation conditions. Despite the major limitations discussed above, potential process rates are still frequently used inappropriately to address ecological questions. Attempts to understand the relative contribution of AOA and AOB to nitrification often involves correlation of their abundances with PNR, often after a treatment or perturbation (e.g. fertiliser application or temperature change) (Supplementary Table 1). This approach is also used to study niche specialisation, which implicitly assumes that changes in relative abundance of different phylotypes will result from differences in activity under different conditions. Incubation conditions used to measure PNR will favour different members of the AOA and AOB communities and will differentially influence activities of AOA and AOB. It will therefore not be possible to determine whether relationships between abundance and PNR are due to differences in preferences of the different communities for the specific incubation conditions used or due to links between abundance and actual activity. A more fundamental issue is the assumption, implicit in niche specialisation, that changes in community composition result from differences in actual activity, determined by different in situ, environmental conditions and do not depend on potential activity under a single set of laboratory conditions.

When can potential rate measurements be useful?

Potential functional assays aim to employ incubation conditions that enable maximum or optimal activity for all organisms performing a particular microbial function in an environmental sample. They therefore provide a means of estimating the biomass of functional groups and standard conditions have been developed for such assays. This is exemplified by traditional use of potential nitrification rate, potential denitrification rate and substrate induced respiration as measures of biomass of nitrifiers, denitrifiers and 'total' microbial biomass in soil. This article has described the major disadvantages of this approach and such assays have very limited value as measures of biomass. Alternative methods are available and have largely been replaced for many functional groups by estimates of functional gene abundance. While these have their own limitations, they do not rely on or describe activity.

These traditional potential assays must, however, be distinguished from the use of laboratory incubations, in liquid culture, soil slurries or soil microcosms to study soil microbiology. Many such studies measure activity following substrate amendment and under controlled incubation conditions. Indeed, such studies are often used to investigate the effects of substrate type and concentration and environmental conditions on functional groups, and subgroups, to distinguish their activities and to investigate the influence of soil conditions.

While such studies effectively measure 'potential' activity, this term is rarely used and is never used to imply that all organisms will be active or that the activity of each is optimal and identical. Laboratory incubations are ecologically meaningful if the limitations of the approach are critically evaluated with respect to their ability to assess activities in the environment [START_REF] Taylor | Temperature affects the kinetics of nitrite oxidation and nitrification coupling in four agricultural soils[END_REF]. For example, with respect to nitrification, they have demonstrated differences in the optimum temperature of AOA and AOB activity [START_REF] Taylor | Modeling of soil nitrification responses to temperature reveals thermodynamic differences between ammonia-oxidizing activity of archaea and bacteria[END_REF], ammonia source preference [START_REF] Levičnik-Höfferle | Stimulation of thaumarchaeal ammonia oxidation by ammonia derived from organic nitrogen but not added inorganic nitrogen[END_REF] and nitrification-associated N2O yields [START_REF] Hink | Archaea produce lower yields of N2O than bacteria during aerobic ammonia oxidation in soil[END_REF]. For other functional groups they have, for example, revealed the presence of both high-and low-affinity methane oxidisers in soil [START_REF] Bender | Kinetics of CH4 oxidation in oxic soils exposed to ambient air or high CH4 mixing ratios[END_REF] and greater influence on denitrification by edaphic factors than N-reducing community composition (Attard et al., 2011). In these instances, it is the relative changes in activity that provide valuable insights into the ecophysiology of different functional groups and not inference of a direct measurement of activity in the natural environment.

The use of ex situ potential activity assays for inferring in situ activity is not justified and better measures are available. For example, soil nitrification activity can be measured as gross nitrification rates using 15 N pool dilution methods (Hart et al., 1996). These techniques may be more time-consuming and require specialist equipment, unlike PNR assays, but they provide environmentally relevant results. Where PNR has been used to evaluate the effect of an environmental factor, observations of relative differences in PNR will reflect effects on a particular set of nitrifying populations, and the data must be recognised as only reflecting the changing activities of these populations and not an overall community response. This is, of course, the same situation for any functional group utilising a specific substrate, but curiously this is often ignored. PNR does not characterise the contribution of all AO to nitrification in soil and knowledge of the diversity of AO physiology and factors controlling their growth clearly challenges a long-established assumption that a single incubation assay measures the activity of all AO. Critically, potential rates can have little relevance to actual rates in situ and therefore have limited meaning in many nitrification studies which should discourage the use of PNR in many circumstances.

Our expanding knowledge of the vast diversity and composition of microbial communities in natural environments emphasises that communities contributing to a functional process comprise a diverse array of ecophysiologies with substantial niche differentiation between populations. This challenges the underlying assumptions in using traditional potential rate measurements to estimate microbial biomass or in situ activity. However, controlled and manipulated laboratory studies that consider the underlying assumptions implicit in their use have been, and will continue to be, an important tool for mechanistic studies of soil microbiology.

Figures

Figure 1. Meta-analysis of soil and sediment studies finding significant positive correlations of AOA and/or AOB abundances with PNR. All studies used a shaken slurry method and measured NO3 -or NO2 -with the addition of chlorate to inhibit NO2 -oxidation. A total of 107 studies were included (Supplementary Table S1). with sterile 144 ml serum vial glass bottles using 10 g soil (dry weight equivalent) which was sampled at 0-10 cm depth and sieved (2 mm mesh size). Microcosms were incubated at 30°C for 12 days with 30% soil water content (w/w). Microcosms were aerated every 3 days and destructively sampled before extracting NO3 -in a 1 M KCl solution. Nitrate concentration was determined colorimetrically following [START_REF] Hink | The consequences of niche and physiological differentiation of archaeal and bacterial ammonia oxidisers for nitrous oxide emissions[END_REF] immediately after sampling (NO2 -was not detected). Error bars represent standard error of the mean. The greatest change in pH in soil microcosms was associated with those with the highest nitrification rates, indicating that pH change was not an inhibiting factor (Supplementary Figure 1). Nevertheless, these results highlight an additional concern, as changing conditions during incubation will influence activity and kinetics will no longer be linear.

Figure 2 .

 2 Figure 2. Demonstration of only AOB contributing to PNR. Soils were sampled across a pH gradient and PNR determined using the shaken slurry method. Flasks were established in triplicate using 144 ml serum vial bottles with 6 g soil (dry weight equivalent) and 50 ml of 1.5 mM NH4Cl, with or without 1 mM potassium phosphate (pH 7.2) buffer. (a) Samples were taken after incubation for 0, 3, 6, 9, 12 and 24 h to determine nitrate concentrations (nitrite was not detected). Means (±SE) with different letters within a buffer treatment are significantly different based on Tukey's post hoc test following one-way ANOVA (p<1.1 x 10 -7 ). Means annotated with a line are significantly different between buffer treatments based on two sample t-tests (p>0.01<0.05). (b) Example of complete inhibition of nitrification activity in flasks incubated in the presence of 0.01% C2H2 or 0.03% C8H2 with pH 7.5 soil (unbuffered).Nitrification activity was not detected in any soil in the presence of either alkyne.

Figure 3 .

 3 Figure 3. Nitrate (NO3 -) production in soil microcosms amended with varying concentrations of ammonium sulphate or urea (0 -200 ug N g -1 dry soil) in pH 7.5 and 4.5 soil ((a) and (b), respectively). Soil microcosms were prepared (in triplicate for each treatment and timepoint)

  Figure 1.

Acknowledgements

The authors would like to thank the reviewers for their insightful and constructive comments on the manuscript. This work was funded by the AXA Research Fund.

Supplementary Figure 1. Soil pH in soil microcosms amended with varying concentrations of ammonium sulphate or urea (0 -200 ug N g-1 dry soil) in pH 7.5 and 4.5 soil ((a) and b), respectively). The standard errors of the mean pH of triplicate microcosms were ≤0.06 and are not visible behind markers. 1. Analysis of 107 studies correlating PNR with the abundance of AOA, AOB or comammox bacteria as (inferred from qPCR analysis of amoA gene abundance).

Supplementary Table

Published studies were found with a SCOPUS search (TITLE-ABS-KEY ammonia AND oxidi* AND archaea AND bacteria AND potential AND nitrification) combined with manual refinement for only soil and sediment studies that used a shaken soil-slurry method.

Publications are grouped by soil or sediment and NO2 -or NO3 -measured as the end product.

The majority of correlations were positive (X) although nine negative correlations (-X) were found. There were no correlations with comammox bacteria.