# Bistable carbon-vacancy defects in $h$-BN

Song Li  Wigner Research Centre for Physics, P.O. Box 49, H-1525 Budapest,
Hungary  Adam Gali  Wigner Research Centre for Physics, P.O. Box 49, H-1525
Budapest, Hungary  Department of Atomic Physics, Institute of Physics,
Budapest University of Technology and Economics, Műegyetem rakpart 3., H-1111
Budapest, Hungary

###### Abstract

Single photon emitters in hexagonal boron nitride have been extensively
studied recently. Although unambiguous identification of the emitters is still
under intense research, carbon related defects are believed to play a vital
role for the emitter producing zero-phonon-lines in the range of $1.6$ to
$2.2$ eV. In this study, we systematically investigate two configurations of
carbon-vacancy defects, VNCB and CNVB, by means of density functional theory
calculations. We calculated the reaction barrier energies from one defect to
the other to determine relative stability. We find the barrier energies are
charge dependent and CNVB could easily transform to VNCB in neutral and
positive charge states while it is stable when negatively charged. Formation
energy calculations show that the VNCB is the dominant defect over CNVB.
However, neither VNCB nor CNVB has suitable fluorescence spectra that could
reproduce the observed ones. Our results indicate that the origin of the
$1.6$-to-$2.2$-eV emitters should be other carbon-related configurations.

##  I Introduction

Point defects as emerging single photon emitters (SPEs) in two-dimensional
(2D) hexagonal boron nitride (hBN) have been intensively studied for possible
applications in quantum sensing, computing and nanophotonics (Tran  _et al._ ,
2016a; Gottscholl  _et al._ , 2020; Chejanovsky  _et al._ , 2021; Mendelson
_et al._ , 2021; Hayee  _et al._ , 2020; Bourrellier  _et al._ , 2016; Bommer
and Becher, 2019; Tran  _et al._ , 2016b). The wide band gap ($\sim$6 eV) and
small spin-orbital coupling of hBN manifests itself as ideal host to
accommodate color centers in a wide region of emission wavelength. The spatial
confinement and dielectric screening of 2D defects enable them showing
desirable properties, for example, bright luminescence, ease of manipulation,
tunable emission through strain (Grosso  _et al._ , 2017; Hayee  _et al._ ,
2020; Mendelson  _et al._ , 2020) and electric fields (Noh  _et al._ , 2018)
motivating researchers to investigate the underlying physical and chemical
nature of the structures in detail. However, direct mapping or
characterization of the defects in experiment is still a challenge. This might
be partially related to various defect types and unintentional impurities
during hBN sample fabrication and post-processing.

Photoluminescence data reveals that there are strong emission bands in
ultraviolet region with zero-phonon-line (ZPL) energy at 4.1 eV (Bourrellier
_et al._ , 2016; Vuong  _et al._ , 2016; Du  _et al._ , 2015; Museur  _et al._
, 2008) and in visible region from 1.6 to 2.2 eV (Tran  _et al._ , 2016b;
Sajid  _et al._ , 2020; Li  _et al._ , 2020; Tran  _et al._ , 2016a;
Gottscholl  _et al._ , 2020; Chejanovsky  _et al._ , 2021; Mendelson  _et al._
, 2021). Based on this, many kinds of point defects have been proposed and
analyzed theoretically through density functional theory (DFT) calculations.
Several of them could match the experimental result well, such as the boron
vacancy (VB) (Abdi  _et al._ , 2018; Ivády  _et al._ , 2020; Reimers  _et al._
, 2020), nitrogen vacancy (VN) (Sajid  _et al._ , 2018), and Stone-Wales
defect (Wang  _et al._ , 2016; Hamdi  _et al._ , 2020). VB have been
identified experimentally (Jin  _et al._ , 2009; Gottscholl  _et al._ , 2021;
Liu  _et al._ , 2021; Kianinia  _et al._ , 2020) and single spins of other
defects could be coherently manipulated (Chejanovsky  _et al._ , 2021; Stern
_et al._ , 2022). Besides, defect complexes and impurities that deliberately
incorporated during synthesis or ion implantation also could serve as
potential candidates (Sajid  _et al._ , 2018; Wu  _et al._ , 2019; Weston  _et
al._ , 2018; Gottscholl  _et al._ , 2020; Chejanovsky  _et al._ , 2021;
Mendelson  _et al._ , 2021). Among them, carbon impurity in hBN has been
extensively considered and investigated. Previous study indicated that the
recombination from CN as a donor-acceptor pair with VN donor (Du  _et al._ ,
2015) should not be related to 4.1-eV emission line due to the deep donor
level of VN whereas the CB might be a possible source with charge transition
level $(0/+)$ at 3.71 eV (Weston  _et al._ , 2018). Later carbon dimer CNCB
was also associated to this emission with a calculated ZPL at 4.3 eV (Mackoit-
Sinkevičienė  _et al._ , 2019; Li  _et al._ , 2022a). For the visible
emission, both VNCB and CNVB are considered depending on their charge states
and spin multiplicity. Our study already revealed the positively charged
VNCB$(+)$ has ZPL about 1.5 eV which could be classified to so called
“Group-2” SPE (Abdi  _et al._ , 2018). In the triplet electronic configuration
of its neutral charge state, the two defects have ZPL optical transitions at
1.58 eV and 1.54 eV, respectively (Sajid  _et al._ , 2018). A recent study has
indicated that the quartet state of the negatively charged CNVB$(-)$ is the
origin of the SPEs in carbon doped hBN (Mendelson  _et al._ , 2021).

The robust emission features from carbon defects in hBN samples grown with
different techniques and environment indicate that they come from very stable
configurations. However, considering the fact that the carbon atom in VNCB and
CNVB structures can migrate easily under external perturbation, it is
necessary to investigate the relative stability and transformation of the two
defects in detail. Formation energy and energy barrier calculation can help to
identify the possible structures. The experimental observed ZPL emission is
about 1.6 to 2.2 eV with Huang-Rhys factor less than 2. Here we use these
criteria to examine the possibility of VNCB and CNVB as single photon emitter
candidates. In this paper, we employ plane-wave supercell DFT calculations to
study the energetic and electronic properties of VNCB and CNVB at different
charge states and spin multiplicities. We find that the charge states
influence the relative stability of the two proposed defects. Although the ZPL
energy of CNVB$(-)$ with quartet spin multiplicity is in the experimental
range of interest, the calculated formation energy and charge transition level
(CTL) indicate that the quartet state is not stable. Our results imply that
the charge states of defects can dramatically affect their relative stability
and provide further evidence for the identification of quantum emitters in the
visible region.

Figure 1: Schematic view of carbon-vacancy complexes in hBN. The carbon atom
can migrate to the antisite-vacancy configuration. Grey, green, and brown
colored balls represent nitrogen, boron and carbon atoms, respectively. The
usual CNVB considered before is denoted as n-CNVB.

##  II Results

The ground state electronic structures are calculated for VNCB and CNVB in
$-1$, $0$ and $+1$ charged states with the high-spin (HS) and low-spin (LS)
states included. Previous results about the spin multiplicity of the ground
state varied with the applied calculation methods (Sajid  _et al._ , 2018;
Mendelson  _et al._ , 2021). For neutral VNCB$(0)$, Cheng et al. (Cheng  _et
al._ , 2017) predicted a triplet ground state, however, further comprehensive
calculations indicated that the ${}^{1}A_{1}$ singlet is the most favorable
one (Reimers  _et al._ , 2018). Our calculation shows that the singlet (LS)
state of VNCB$(0)$ has lower energy considering the symmetry reduction caused
by the pseudo-Jahn-Teller (JT) distortion in the singlet state while it does
not occur in the triplet state. The $C_{s}$ symmetry configuration is lower in
energy by 1.87 eV over that of $C_{2v}$ symmetry configuration. In the
following calculation, we remove the symmetry constrain and investigate the
electronic structure. We find that the $C_{2v}$ symmetry is the most stable on
in the positive charge state regardless the spin multiplicity, and VNCB(+)
prefers the low-spin doublet (LS). Meanwhile, VNCB($-$) exhibits doublet state
with $C_{s}$ symmetry.

For the CNVB defect (n-CNVB in this paper), the triplet state is predicted to
have lower energy than singlet for CNVB(0) with transition energy in
experimental value (Sajid  _et al._ , 2018). However, a recent report
indicates that this configuration suffers from multiple low-energy minima, and
none of them has significant oscillator strength (Mendelson  _et al._ , 2021).
Noteworthy, we find that the CNVB could form a Stone-Wales-like (sw-CNVB)
configuration, as shown in Fig. 1, which could have lower formation energy.
The multireference character still exists here which is manifested as the
unrestricted spin-polarized calculation yields different energy for spin-up
and spin-down channels. We speculate this phenomenon is due to the spin
contamination from the triplet ground state and indeed the calculated energy
of the singlet is almost identical to that of the triplet. Although these
states are not spin eigenstates, the unrestricted DFT orbitals provide more
realistic total energies. Except for HSs of CNVB($-$), other electronic states
prefer the sw-CNVB configuration. In the following discussion, always the most
stable configuration is discussed for CNVB defects in the context. LSs always
have lower energy whereas the CCSD methods pointed out the HS quartet for
CNVB($-$) could be the ground state (Mendelson  _et al._ , 2021). This
controversy motivated us to further investigate the relative stability of
carbon impurity in these two defects.

The energy level diagrams for the two defects are shown in Fig. 2. VNCB($-$)
at both HS and LS states do not have empty localized state therefore only
defect-to-conduction band (CB) excitation occurs which results in a relatively
dim emission. The experimental observed bright luminescence should come from
defect states in the fundamental band gap and neither of the vertical
transition energies is in line with the experimental PL energies. The VNCB(0)
HS state with $C_{2v}$ symmetry has been studied (Sajid and Thygesen, 2020),
the defect demonstrates suitable ZPL energy and 1.5 Huang-Rhys (HR) factor.
However, the HR factor of the more stable LS state in $C_{s}$ is 7.25, much
larger than the experimental value 1.45. The ZPL of VNCB(+) LS is 1.51 eV
(Abdi  _et al._ , 2018), which agree relatively well with the experimental
data for the Group-2 emitters. Despite of that, the corresponding HR factor is
about $24$ so we can also disregard as a good candidate for the observed
visible emitters. CNVB($-$) with both HS and LS states have defect-to-defect
transitions. The singlet CNVB(0) has ZPL of 1.98 eV and we believe this value
is still in the range of interest even taking into account the possible error
due to multireference character of the ground state, however, the HR factor is
8.17. There are no occupied defect states in the gap for CNVB(+) and the
allowed transition is between valence band maximum to defect states which
should result in again a relatively dim optical transition.

We summarized the calculated lowest vertical excitation energies in Tab. 1.
For defects in the negative charge state, the CNVB($-$) with HS state is the
only possible candidate for the bright visible emitters, and the defects in
other charge states have either too small transition energy or large phonon
side band. The calculated ZPL is 1.89 eV which is close to that obtained in
the flake model (Mendelson  _et al._ , 2021).

Figure 2: Energy level diagram of VNCB (a-f) and CNVB (g-l) with different
charge states and spin multiplicities. The filled and empty arrows indicate
the occupied and unoccupied defect states in the spin-up and spin-down
channels. Resonant defect levels below valence band maximum are not shown
here. The dash lines represent the allowed optical transition in the visible
wavelength region. We use the most stable configurations at every charge and
spin states. Here, the CNVB($-2$) and HSs of CNVB($-$) have n-CNVB
configuration and the others are calculated with sw-CNVB.

The defect formation energies $E_{f}$ are calculated to determine the charge
stability with the following equation,

$\begin{split}E^{q}_{f}=&E^{q}_{d}-E_{\text{per}}+\mu_{\text{C}}-\mu_{\text{B}}-\mu_{\text{N}}+q\left(\epsilon^{\text{per}}_{\text{VBM}}+\epsilon_{\text{Fermi}}\right)\\\
&+E_{\text{corr}}\left(q\right)\text{,}\end{split}$ (1)

where $E_{d}^{q}$ is the total energy of hBN model with defect at $q$ charge
state and $E_{\text{per}}$ is the total energy of hBN layer without defect.
$\mu_{\text{C}}$ is the chemical potential of carbon and can be derived from
pure graphite. The Fermi-level $\epsilon_{\text{Fermi}}$ represents the
chemical potential of electron reservoir and it is aligned to the valence band
maximum (VBM) energy of perfect hBN, $\epsilon^{\text{per}}_{\text{VBM}}$. The
$E_{\text{corr}}\left(q\right)$ is the correction term for the charged system
due to the existence of electrostatic interactions of the periodic images of
the defect. VNCB and CNVB always prefers LS state for different charge states.
As shown in Fig. 3, in the fundamental gap, the CTL of VNCB for $(+1/0)$ is
2.88 eV and it is 5.30 eV for $(0/-1)$ with respect to VBM. These CTLs are far
from conduction band minimum (CBM) and VBM, respectively, making VNCB a hyper
deep donor and acceptor. For CNVB, the $(+1/0)$ and $(0/-1)$ levels are at
1.51 eV and 3.19 eV, respectively. We notice that the $+2$ charge state of
VNCB and $-2$ charge state of CNVB are also stable as reported previously
(Maciaszek  _et al._ , 2022). VNCB($+2$) have $C_{2v}$ symmetry and the
CNVB($-2$) have n-CNVB configuration. However, the calculated CTLs are
relatively close to the band edge. $(+2/+1)$ is at 1.11 eV for VNCB while the
$(-1/-2)$ is at 4.49 eV for CNVB. Optical excitation results in
photoionization to other charge states and thus cannot have optical transition
energies in fluorescence close to the experimental values. VNCB is generally
more stable with lower formation energy than CNVB when
$\epsilon_{\text{Fermi}}$ < 4.75 eV. The CNVB($-2$) has lower formation energy
with Fermi-level close to CBM which is not a typical experimental condition.

Figure 3: The formation energies of VNCB (black) and CNVB (red) as a function
of the Fermi-level. The slope of line segment corresponds to the charge state.
The solid and dash lines indicate the LS and HS state, respectively. Crossing
of the lines denote the charge transition level. Here, the CNVB($-2$) and HSs
of CNVB($-$) have n-CNVB configuration and the others are calculated with sw-
CNVB.

Next, we calculated the defect migration with the climbing-image nudged-
elastic-band method (Henkelman  _et al._ , 2000; Henkelman and Jónsson, 2000)
as shown in Fig. 4. The n-CNVB and sw-CNVB are both considered here. The
carbon atom can migrate between boron site and nitrogen site and this
transition rate $\Gamma$ can be simply expressed as (Weston  _et al._ , 2018),

$\begin{split}\Gamma=\Gamma_{0}exp(-\frac{E_{b}}{k_{B}T}),\end{split}$ (2)

where $E_{b}$ is the reaction barrier energy, $k_{B}$ is the Boltzmann
constant, $T$ is the temperature and $\Gamma_{0}$ is typical phonon frequency
in hBN which is about $10^{14}s^{-1}$ (Geick  _et al._ , 1966). The calculated
$T$ can be regarded as annealing temperature at which the defect becomes
mobile as listed in Tab. 1. Usually the temperature is estimated when the jump
rate $\Gamma$ is 1/s (Janotti and Van de Walle, 2007). Above this temperature,
the reaction barrier can be passed to reach an equilibrium point. In the
negative charge state, the reaction barrier energies for LSs and HSs are 0.98
eV and 1.96 eV, respectively, for carbon migration from boron site to nitrogen
site. The large barrier energies indicate that the carbon atom cannot jump
from one site to another, therefore, the two defects could exist
simultaneously. The annealing temperature are 353 K and 705 K, therefore, VNCB
would transform to CNVB if the sample is grown or annealed above these
temperatures. However, the barrier energies are quite small in the neutral and
positive charge states. The barrier energies are generally less than 0.6 eV,
especially, for LSs in the neutral state, manifesting the carbon can move
freely from nitrogen site to boron site. The calculated annealing temperature
is lower than room temperature, hence the VNCB defect will dominate in these
two charge states, and the CNVB can only be stabilized through low-temperature
irradiation.

The reaction barrier energies and migration paths depend on the charge state.
We tentatively associate the low stability of CNVB(0) and CNVB(+) to the
unoccupied localized states in the gap. Filling these defect levels by
electrons could stabilize the defect.

##  III Discussion

We calculated the formation energies and the barrier energy of transformation
of carbon-vacancy complexes depending on the symmetry and spin multiplicities.
We note that the geometry distortions from the planar structure was found for
certain carbon-vacancy complexes as the common Jahn-Teller distortion effect
where the atoms farther the core of the defect structure remain in the sheet
of hBN layer. This is different from recent proposed geometry containing
dramatically out-of-plane warping calculated by CAM-B3LYP DFT functional
(Mendelson  _et al._ , 2021). This warping might be induced by the hydrogen
termination of the model or the freezing of atoms during optimization. In
addition, the large scale distortion might yield opposite result. In our
model, the relaxation energy of HS CNVB($-$) is 0.16 eV and HR factor is 2.2
which are close to the experimental data (Mendelson  _et al._ , 2021).
However, the formation energy difference between the HS and LS for n-CNVB($-$)
is 0.4 eV, where the LS configuration exhibits $C_{2v}$ symmetry. Our NEB
calculation shows there is no barrier for n-CNVB to transform to sw-CNVB at
LSs for three charge states which indicates the n-CNVB might not exist at all.
Recent work indicates that n-CNVB is dynamically unstable and it quickly relax
to VNCB defect (Ortigoza and Stolbov, 2022). This is consistent with our
present data for the neutral charge state. We show here that sw-CNVB occurs in
the other charge states too, thus this behavior is independent on the position
of the Fermi-level of hBN.

Similar to VB (Gottscholl  _et al._ , 2020), hyperfine interaction related
features may be a unique fingerprint of the CNVB($-$) defect observed by
electron spin resonance techniques. Recently, an electron paramagnetic
resonance (EPR) center has been observed in hBN (Toledo and Krambrock, 2020).
The experimental data clearly shows splitting pattern of five peaks with
relative intensities of 1:2:3:2:1. This is due to the hyperfine coupling
between one electron spin and two equivalent nitrogen nuclear spin of $I$ = 1,
therefore, the EPR center has been tentatively associated with the spin
doublet of n-CNVB($-$) defect (Toledo and Krambrock, 2020). Although, our
calculations indicate that n-CNVB($-$) does not exist in hBN, we still
simulated its EPR spectrum after calculation of the hyperfine tensors of the
defect (see Tab. 2). Indeed, the simulated EPR spectrum of n-CNVB($-$) agrees
with the experimental one (Toledo and Krambrock, 2020) (Tab. 2). We note that
the hyperfine signatures of the HSs configuration significantly differ. The
stable sw-CNVB($-$) has very different spin density distribution with
localized on two boron ions, thus its EPR spectrum significantly deviates from
that of the observed EPR center. However, n-CNVB($-$) is not a stable
structure, therefore, we conclude that it cannot account for the EPR center as
tentatively proposed in Ref. (Toledo and Krambrock, 2020), despite the
agreement between the simulated and observed hyperfine related features in the
EPR spectra.

We found that the most stable form of carbon-vacancy complex in hBN is VNCB
thus, we plot the simulated EPR spectrum for the paramagnetic VNCB(+) in Fig.
5(b) which may occur in hBN. The EPR spectrum shows 7 peaks due to two
neighboring boron atoms ($I=3/2$ for 11B). Further experimental data is needed
to confirm the existence of such kind of defect.

We note that there is a metastable triplet state for VNCB(0). The triplet
state may be accessed by optical pumping of the system from the singlet ground
state to the singlet excited state followed by an intersystem crossing from
the excited singlet state towards the metastable triplet state. However, the
large singlet-triplet energy gap makes this process inefficient, thus, we do
not consider the triplet of VNCB(0) to be observed by photo-EPR studies.

Figure 4: The reaction barrier for the carbon atom from initial state VNCB to
final state CNVB in different charge states. Transitions from VNCB to n-CNVB
(a-f) and from VNCB to sw-CNVB(g-k) are shown here. The red arrow indicates
the reaction barrier $E_{b}$. The HSs of CNVB does not have sw-CNVB
configuration. Figure 5: The simulated EPR spectrum of defects considered. (a)
The experimental data is from (Toledo and Krambrock, 2020) recorded at 550 ∘C
to clearly show the line splitting. sw-CNVB has relative small hyperfine
coupling so there is no line splitting. The experimental spectrum is shifted 4
mT towards left to match simulation result. (b) The EPR spectrum of VNCB(+).

##  IV Summary and conclusion

In this study, we performed density functional theory calculations on the
carbon-vacancy complexes in hBN. The analyzed relative stability of these
complexes could reveal the origin of the single photon emitters observed in
experiments in the visible wavelength region. sw-CNVB is always more stable
than n-CNVB and n-CNVB transforms to sw-CNVB without any barrier in three
charge states. Hence, previously reported n-CNVB is not the origin of visible
SPEs in hBN. sw-CNVB is metastable and can transform to VNCB with some barrier
energies depending on the charge states. In addition, neither the VNCB nor
CNVB is a potential candidate for the SPEs associated with the carbon
impurities. The formation energies for VNCB and CNVB complexes imply that
these complexes cannot account for the experiments claiming that the number of
C-B bond is larger than C-N bond in carbon contaminated hBN (Mendelson  _et
al._ , 2021). Further investigation is needed for finding the microscopic
origin of SPEs associated with the carbon impurities. Recent studies proposed
the C2C${}_{\text{N}}$ configuration (Auburger and Gali, 2021; Li  _et al._ ,
2022b; Golami  _et al._ , 2022) of which optical properties well reproduce the
experimental data. We also found that the recently reported EPR center (Toledo
and Krambrock, 2020) is not associated with the unstable VNCB defect. Further
investigation is needed to identify this EPR center in hBN. We provide the EPR
spectrum for the most stable carbon-vacancy complex in hBN that might be found
in future EPR studies of carbon contaminated hBN layers.

##  V Methods

Table 1:  The lowest vertical transition energy for two defects in hBN. The
energy unit is eV. The simulated annealing temperature (temp.) is also listed
as obtained from the calculated barrier energies (see text). For CNVB defect,
except for the CNVB($-$) HSs, other charge states have lower total energy with
sw-CNVB configuration. The $\Delta E_{\text{diff}}$ indicates energy
difference between these two configurations and defined as $\Delta
E_{\text{diff}}$ = E(n-CNVB) - E(sw-CNVB).

configuration | -1 HS | -1 LS | 0 HS | 0 LS | +1 HS | +1 LS
---|---|---|---|---|---|---
VNCB | 0.64 | 0.87 | 1.86 | 1.93 | 1.81 | 2.10
SYM. | $C_{2v}$ | $C_{s}$ | $C_{2v}$ | $C_{s}$ | $C_{2v}$ | $C_{2v}$
HR factor |  |  | $1.50^{\text{a}}$ | 7.25 |  | 24
n-CNVB | 2.05 | 1.02 | 1.75 | 1.97 | 0.81 | 0.91
SYM. | $C_{2v}$ | $C_{2v}$ | $C_{2v}$ | $C_{2v}$ | $C_{2v}$ | $C_{2v}$
HR factor | 2.2 |  |  |  |  |
sw-CNVB | - | 1.12 | 1.49 | 2.57 | 0.14 | 0.49
$\Delta E_{diff}$ | - | 0.37 | 0.70 | 0.90 | 2.0 | 0.78
HR factor |  |  |  | 8.17 |  |
TEMP.(K) | 705 | 353 | 173 | 25 | 198 | 40

a. Simulated data from Ref. Sajid and Thygesen (2020)

Table 2:  Hyperfine constants for carbon-vacancy complexes in hBN. The unit is in MHz. | configuration | $A_{xx}$ | $A_{yy}$ | $A_{zz}$
---|---|---|---|---
14N | Exp. Toledo and Krambrock (2020) | 90-98 | 90-98 | 145-148
14N | n-CNVB($-$) LS | 86.92 | 83.12 | 166.41
14N | n-CNVB($-$) HS | 34.05 | 33.04 | 59.54
14N | sw-CNVB($-$) LS | 6.75 | 6.53 | 13.25
11B | VNCB($+$) LS | 101.47 | 100.75 | 126.87

We performed spin-polarized DFT calculation within the Kohn-Sham scheme as
implemented in the VASP code (Kresse and Furthmüller, 1996a, b). Standard
projector augmented wave (PAW) formalism (Blöchl, 1994; Kresse and Joubert,
1999) is used to separate the valence electrons from nuclei. The convergence
threshold is 0.01 eV/Å for force acting on each atoms and energy cutoff for
the expansion of the plane-wave basis set is 450 eV. The screened hybrid
density functional of Heyd, Scuseria, and Ernzerhof (HSE) (Heyd  _et al._ ,
2003) is used to calculate the electronic structure and localize bound states.
In this approach, we could mix part of nonlocal Hartree–Fock exchange to the
generalized gradient approximation of Perdew, Burke, and Ernzerhof (PBE) with
fraction $\alpha$. $\alpha$ = 0.32 can reproduce the experimental band gap
about 6 eV. We embedded the carbon defects in a $9\times 5\sqrt{3}$ monolayer
supercell with 162 atoms which is sufficient to avoid the periodic defect-
defect interaction, a vacuum layer of 12 Å is applied to separate the periodic
layer images. The single $\Gamma$-point scheme is converged for the k-point
sampling for the Brillouin zone. The excited states were calculated by
$\Delta$SCF method (Gali  _et al._ , 2009). For the formation energy
calculation, a bulk model with two hBN layers is used to include the
interlayer interaction and decreases the artificial influence induced by
vacuum layer. The charge correction term is computed by SXDEFECTALIGN code
from Freysoldt method (Freysoldt and Neugebauer, 2018). During the NEB
calculation, for VNCB to CNVB transition, the threshold of force is set to
0.02 eV/Å whereas for n-CNVB to sw-CNVB transition it is 0.1 eV/Å due to the
numerical challenges. The electron paramagnetic resonance (EPR) simulation is
performed with EASYSPIN software at X band region (9.45 GHz) (Stoll and
Schweiger, 2006).

The total HR factor is defined as the number of an effective phonon
participating in the optical transition which is a key parameter of the
absorption and fluorescence spectra. The total HR factor can be readily
calculated within Franck-Condon approximation which assumes that the
vibrational modes in the ground and excited states are identical. The
associated phonon overlap spectral function can be derived from the overlap
between the phonon modes in the electronic ground and excited states
(Alkauskas  _et al._ , 2014; Gali  _et al._ , 2009).

## Author contribution

All authors contributed to the discussion and writing the manuscript. AG led
the entire scientific project.

## Competing interests

The authors declare that there are no competing interests.

## Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

###### Acknowledgements.

AG acknowledges the Hungarian NKFIH grant No. KKP129866 of the National
Excellence Program of Quantum-coherent materials project and the support for
the Quantum Information National Laboratory from the Ministry of Innovation
and Technology of Hungary.

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Pascal Michaillat, Emmanuel Saez
Pascal Michaillat: Brown University. Emmanuel Saez: University of California–Berkeley. We thank David Baqaee, Daniel Bogart, Varanya Chaubey, Olivia Lattus, Romain Ranciere, Guillaume Rocheteau, and Pierre-Olivier Weill for helpful comments and discussions.

Most governments are mandated to maintain their economies at full employment. We propose that the best marker of full employment is the efficient unemployment rate, $u^*$. We define $u^*$ as the unemployment rate that minimizes the nonproductive use of labor—both jobseeking and recruiting. The nonproductive use of labor is well measured by the number of jobseekers and vacancies, $u + v$. Through the Beveridge curve, the number of vacancies is inversely related to the number of jobseekers. With such symmetry, the labor market is efficient when there are as many jobseekers as vacancies ($u = v$), too tight when there are more vacancies than jobseekers ($v > u$), and too slack when there are more jobseekers than vacancies ($u > v$). Accordingly, the efficient unemployment rate is the geometric average of the unemployment and vacancy rates: $u^* = \sqrt{uv}$. We compute $u^*$ for the United States between 1930 and 2022. We find for instance that the US labor market has been over full employment ($u < u^*$) since May 2021.

§ INTRODUCTION

Full-employment mandate Policymakers in many countries are mandated to maintain the economy at full employment. In the United States this legislative mandate comes from the 1978 Full Employment and Balanced Growth Act—itself an amendment of the 1946 Employment Act.[See <https://perma.cc/N4N5-BGJD> and <https://perma.cc/QK6U-C73K> for the history of these laws.] Despite the long existence of the full-employment mandate, there are no employment or unemployment rates that governments can use to capture full employment. In this paper, we produce an unemployment rate $u^*$ that characterizes a state of full employment.

Full employment does not mean no unemployment Reaching full employment should be interpreted not as reaching zero unemployment. This is because it is not physically possible to reduce unemployment to zero. For instance, during World War 2, <cit.> noted that “Even under full employment, there will be some unemployment …as there is some unemployment even in Britain at war to-day.”

So what is full employment Rather, reaching full employment should be interpreted as reaching a socially desirable amount of unemployment. We define the socially efficient amount of unemployment as the amount that minimizes the nonproductive use of labor—both recruiting and jobseeking. It is because people constantly move in and out of jobs that unemployment cannot be reduced to zero, as noted by <cit.>: “However great the unsatisfied demand for labor, there is an irreducible minimum of unemployment, a margin in the labor force required to make change and movement possible.” Hence, while labor market flows are inevitable, the goal is that workers spend as much time as possible producing useful things and waste as little time as possible searching for employees or jobs.

Measuring jobseeking activity The number of jobseekers is well measured by the number of unemployed workers. In theory, unemployed workers might also produce useful things at home beside looking for jobs. But in practice home production is minimal, so we count unemployed workers full-time jobseekers <cit.>.

Measuring recruiting activity The number of recruiters is well measured by the number of vacancies. Indeed, it takes about one full-time worker to service a vacancy, so there are as many recruiters as vacancies <cit.>.

Social objective The social objective is to minimize the time spent on jobseeking and recruiting at the expense of producing. The amount of jobseeking is well measured by the number of unemployed workers, and the amount of recruiting by the number of job vacancies. Hence, the objective is to minimize the sum of the unemployment and vacancy rates, $u + v$.

Beveridge curve Because of the Beveridge curve, it is not possible to reduce the numbers of jobseekers and vacancies at the same time. When the economy is depressed, there are many jobseekers and few vacancies; conversely, when the economy is booming, there are few jobseekers and many vacancies. The Beveridge curve is approximately a rectangular hyperbola:  $vu = A$, where $A$ is a constant <cit.>. Hence, unemployment rate $u$ and vacancy rate $v$ are inversely related.

Efficiency criterion Because of the symmetrical roles played by unemployed workers and vacant jobs, we find that the nonproductive use of labor is minimized when there are as many unemployed workers as vacant jobs ($u = v$). The labor market is inefficiently tight when there are more vacancies than jobseekers ($v > u$), and inefficiently slack when there are more jobseekers than vacancies ($u > v$).

Efficient labor-market tightness The efficiency criterion can be reformulated in terms of labor-market tightness. The labor-market tightness is the ratio of vacancies to unemployment, $\t = v/u$. It measures the number of vacancies per unemployed worker. Our analysis implies that the efficient labor-market tightness is $\t^* = 1$. The labor market is inefficiently tight when $\t > 1$ and inefficiently slack when $\t < 1$.

Efficient unemployment rate Because the unemployment and vacancy rates play a symmetric role on the labor market, the efficient unemployment rate is the geometric average of the unemployment and vacancy rates: $u^* = \sqrt{uv}$. This formula arises by combining the Beveridge curve, $uv = A$, and the result that the efficient unemployment and vacancy rates are equal.

Measuring the efficient unemployment rate the United States Because it only requires unemployment and vacancy rates, the formula $u^* = \sqrt{uv}$ is easy to apply. In the United States, we can combine the historical unemployment and vacancy series produced by <cit.> and <cit.> with the data produced by the Bureau of Labor Statistics to assemble monthly unemployment and vacancy series from 1930 and 2022. We use these series to compute the efficient unemployment rate $u^*$ between 1930 and 2022. We find that $u^*$ averages $4.2\%$ over the period and is very stable.

Application to the post-pandemic labor market We also apply our formula in the post-pandemic period in the United States. There has been more vacant jobs than unemployed workers since May 2021, so the labor market has been too tight since then. In March 2022, there is twice as many vacant jobs as unemployed workers. Looking at historical data going back to 1930, we see that the last time that the labor-market tightness was so high, and the last time that the unemployment rate was so much below its efficient level, was at the end of World War 2. The post-pandemic labor market is in a very unusual situation: by any measure, 2022 is experiencing the tightest labor market since World War 2.

Application to modern monetary policy During a press conference in May 2022, Jerome Powell, the chair of the Federal Reserve, discussed labor-market tightness at length.[See pp. 12–13 at <https://perma.cc/4EGU-XZYG>.] In response to a question from journalist Howard Schneider asking which tightness the Fed might target, he responded: “So in terms of the vacancy-to-unemployment ratio, we don't have a goal in mind. There's no specific number that we're saying, `We've got to get to that.'  …I think when we got to one-to-one in the, you know, in the late teens, we thought that was a pretty good number. But, again, we're not shooting for any particular number.” We find that a vacancy-to-unemployment ratio of $1$ is indeed what the Fed should target.

§ DERIVATION OF THE EFFICIENT-UNEMPLOYMENT FORMULA

This section derives the formula for the efficient unemployment rate: $u^* = \sqrt{uv}$.

§.§ Definition of efficiency

From a social perspective, it is optimal to maximize the number of people who spend time producing useful things rather than shuffling employees or jobs. This perspective is consistent with the view expressed by <cit.> that “The material end of all human activity is consumption. Employment is wanted as a means to more consumption …as a means to a higher standard of life.” We therefore define the socially efficient amount of unemployment as the amount that minimizes the nonproductive use of labor—both jobseeking and recruiting. Because people constantly move in and out of jobs, it is impossible to have zero unemployment, and it is impossible to have all workers devoted to production. While labor market flows are inevitable, the goal is that workers waste as little time as possible searching for jobs or employees.

§.§ Measuring jobseeking activity

The number of jobseekers is well measured by the number of unemployed workers. In theory, beside looking for jobs, unemployed workers might also produce useful things at home. Such home production would be included into aggregate production and contribute to social welfare. But in practice home production is minimal, as was already noted by <cit.>: “The most important aspect of unemployment is its wastefulness. It is the existence of unused productive resources side by side with unsatisfied human needs that is the intolerable condition.” <cit.> measure the fraction of nonwork time devoted to home production in the United States. The measure is based on the results by <cit.>. Using administrative data from the US military,  study how servicemembers choose between reenlisting and leaving the military. The choices allow them to estimate the difference between market production and the sum of home production and public benefits during unemployment. Subtracting the value of public benefits from these estimates, <cit.> find that the value of home production relative to market production could be as low as $0.03$. Given such low value, we assume that unemployed workers do not engage in home production at all, and we count them as full-time jobseekers.

§.§ Measuring recruiting activity

The number of recruiters is well measured by the number of vacancies. In theory it might take more or less than one full-time worker to service a vacancy. But in practice it takes about one full-time worker to service a vacancy, so the numbers of recruiters and vacancies are about the same. In the United States, the amount of labor required to service a vacancy can be measured from the National Employer Survey, which was conducted by the Census Bureau in 1997 <cit.>. <cit.> estimate that servicing a vacancy requires $0.92$ worker at any point in time. So it takes about $1$ worker to service a vacancy, which is what we assume here by equating the number of recruiters to the number of vacancies.

§.§ Shape of the Beveridge curve

First observations of the Beveridge curve Looking at labor-market statistics for the United Kingdom, <cit.> first noted that the number of vacancies and the number of jobseekers move in opposite directions. When the economy is in a slump, there are lots of jobseekers and few vacancies. Conversely, when the economy is in a boom, there are few jobseekers and many vacancies. This negative relationship between unemployment and vacancies—known as the Beveridge curve—was first plotted by <cit.> using UK data for 1946–1956. Moreover, <cit.> noted that the Beveridge curve looked like a rectangular hyperbola in an unemployment-vacancy plane.

Regular scaleLog scale
Unemployment and vacancy rates in the United States, 1951–2019
The unemployment rate is constructed by the <cit.>. For 1951–2000, the vacancy rate is constructed by <cit.>; for 2001–2019, the vacancy rate is the number of job openings divided by the civilian labor force, both measured by the <cit.>. Unemployment and vacancy rates are quarterly averages of monthly series. The gray areas are NBER-dated recessions.

Unemployment and vacancy rates move in opposite directions The Beveridge curve holds remarkably well in the United States as well <cit.>. Figure <ref> depicts the unemployment rate $u$ (all jobseekers divided by the labor force) and the vacancy rate $v$ (all vacancies divided by the same labor force) in the United States for 1951–2019. The unemployment rate is constructed by the <cit.>. For 1951–2000, the vacancy rate is constructed by <cit.>; for 2001–2019, the vacancy rate is the number of job openings divided by the civilian labor force, both measured by the <cit.>. The figure shows clearly that unemployment and vacancy rates move in opposite directions.

Unemployment and vacancy rates are inversely related In fact, unemployment and vacancy appear to be the inverse of each other, so that doubling the unemployment rate cuts the vacancy rate in half, and inversely, doubling the vacancy rate cuts the unemployment rate in half. Figure <ref> displays again unemployment and vacancy, but now in log scale. The fluctuations of the unemployment and vacancy rates are almost the mirror image of each other. This empirical regularity is particularly striking because the two series are constructed independently of each other.

1951Q1–1961Q1: elasticity $= -0.85$1961Q2–1971Q4: elasticity $= -1.02$1972Q1–1989Q1: elasticity $= -0.84$1989Q2–1999Q2: elasticity $= -0.94$1999Q3–2009Q3: elasticity $= -1.00$2009Q4–2019Q4: elasticity $= -0.84$
Estimated Beveridge curve in the United States, 1951–2019
This figure is constructed from figures 5 and 6 in <cit.>. Unemployment and vacancy rates are the same as in figure <ref>. The number of structural breaks, dates of structural breaks, and elasticities of the Beveridge curve between breaks are estimated with the algorithm of <cit.>.

The Beveridge curve is a rectangular hyperbola Mathematically, the property that unemployment and vacancy rates are inversely related implies that in an unemployment-vacancy plane, the Beveridge curve is a rectangular hyperbola:

\begin{equation*}
vu = A,
\end{equation*}

where $A>0$ is a constant.

Estimates of the Beveridge elasticity We can formally establish that the Beveridge curve is a rectangular hyperbola by estimating the elasticity of the vacancy rate with respect to the unemployment rate, $\odlx{v}{u}$. An elasticity of $-1$ corresponds to an hyperbola. <cit.> use the algorithm of <cit.> to estimate the structural breaks of the US Beveridge curve, and the elasticity of the Beveridge curve between these breaks. They find that the Beveridge elasticity remains between $-0.84$ and $-1.02$, so never far from $-1$ (figure <ref>). This finding confirms that the Beveridge curve is close to an hyperbola.

Informal foundation for the hyperbolic Beveridge curve Why is the Beveridge curve an hyperbola? Empirically, the number of new matches depends symmetrically on the number of jobseekers and vacancies. Doubling jobseekers increases new matches much the same as doubling vacancies. This makes sense as both jobseekers and vacancies are striving to find each other. This is obvious for jobseekers who spend time looking for jobs, applying, doing interviews. But this is also the case for vacancies as recruiters have to advertise vacancies, and screen and interview applicants. Therefore, as new matches have to meet the number of job separations, cutting in half the number of jobseekers requires doubling the number of vacancies, explaining why unemployment and vacancy rates are the inverse of each other.

Formal foundation for the hyperbolic Beveridge curve In matching models the Beveridge curve says labor market flows are balanced: the number of job separations equals the number of jobs found. The employment rate $1-u$ is approximately constant around 1 since the unemployment rate $u$ is an order of magnitude less than 1. The job-separation rate $\l$ is also constant, so the number of job separations $\l (1-u)$ is approximately constant around $\l$. So at any point of the business cycle, the number of jobs found is approximately constant at $\l$.  With the standard symmetric Cobb-Douglas matching function, $m = \o \cdot\sqrt{uv}$, the number of jobs found at any point in time is proportional to $\sqrt{uv}$. [The US matching function appears to have a Cobb-Douglas form with exponents of $0.5$ on unemployment and vacancies. See <cit.> for a survey of the US estimates based on aggregate data, and <cit.> for a broader survey.] Hence $\sqrt{uv}$ is constant along the Beveridge curve, which implies $uv = \text{constant}$ along the Beveridge curve.

§.§ Efficiency criterion

The social objective is to minimize the time spent on jobseeking and recruiting at the expense of producing. Since the amount of jobseeking can be measured by the number of unemployed workers, and the amount of recruiting by the number of vacancies, the social objective is to minimize the sum of the unemployment and vacancy rates, $u + v$. This minimization is subject to the Beveridge curve constraint, $u v = \text{constant}$. Because of the Beveridge curve, it is not possible to reduce unemployment and vacancies at the same time. Therefore, the social planner must trade off unemployment and vacancies.

Symmetry argument Because of the symmetrical roles played by unemployed workers and vacant jobs, the nonproductive use of labor is minimized when there are as many unemployed workers as vacant jobs ($u = v$). Indeed, we would like to minimize wasted resources from unemployment and from vacancies, subject to the Beveridge curve.  Since this problem is perfectly symmetric in $u$ and $v$, the solution to minimize waste is to have $u = v$.

Departure from efficiency When the number of jobseekers is not equal to the number of vacancies, the labor market is operating inefficiently. The labor market is inefficiently tight when there are more vacancies than jobseekers ($v > u$). In that case, increasing $u$ and reducing $v$ would reduce waste. The labor market is inefficiently slack when there are more jobseekers than vacancies ($u > v$). Then, reducing $u$ and increasing $v$ would reduce waste.

First-order condition We can also establish these results mathematically. The planner minimizes nonproduction $u + v$ subject to Beveridge curve $uv = A$, or $v = A/u$. This is equivalent to minimizing $u+A/u$, which is convex in $u$. A first-order condition is necessary and sufficient to find the minimum. We take the derivative of $u+A/u$ with respect to $u$ and set it to $0$. We obtain $1-A/u^2 = 0$. This condition implies that the minimum occurs when $u = \sqrt{A}$. By the Beveridge curve we have $v = A/u$, so at the minimum $v = A/ \sqrt{A} = \sqrt{A}$. Accordingly, the efficient unemployment and vacancy rates satisfy $u^* = v^* = \sqrt{A}$.

Formulation in terms of labor-market tightness The efficiency criterion can be reformulated in terms of labor-market tightness. The labor-market tightness is the ratio of vacancies to unemployment, $\t = v/u$. It measures the number of vacancies per unemployed workers. The analysis implies that the efficient labor-market tightness is $\t^* = 1$; that the labor market is inefficiently tight when $\t > 1$; and that the labor market is inefficiently slack when $\t < 1$.

Unemployment and vacancy ratesLabor-market tightness
State of the labor market in the United States, 1951–2019
The unemployment and vacancy rates come from figure <ref>. The labor-market tightness is the vacancy rate divided by the unemployment rate. The gray areas are NBER-dated recessions. The labor market is inefficiently slack whenever the unemployment rate is above the vacancy rate (purple shade); it is inefficiently tight whenever the unemployment rate is below the vacancy rate (orange shade). Between 1951 and 2019, the US labor market is always inefficiently slack except during three episodes: the Korean War (1951Q1–1953Q3), the Vietnam War (1965Q4–1970Q1), and the end of the Trump presidency (2018Q1–2019Q4).

§.§ Efficient unemployment rate

We have seen that the efficient unemployment rate is given by $u^* = \sqrt{A}$ where the parameter $A$ determines the location of Beveridge curve, $uv = A$. Accordingly, the efficient unemployment rate is the geometric average of the unemployment and vacancy rates: $u^* = \sqrt{uv}$. Intuitively, the efficient unemployment rate is the average of the unemployment and vacancy rates because unemployment and vacancy rates play a symmetric role on the labor market.

§.§ Similarity with previous markers of full employment

Our criterion provide a theoretical underpinning for the old idea that the state of full employment could be assessed by comparing the number of vacant jobs to that of unemployed workers. For example,  <cit.> states that “Full employment …means having always more vacant jobs than unemployed men,” and before his report, the prevailing view was that full employment was “a state of affairs in which the number of unfilled vacancies is not appreciably below the number of unemployed persons.” <cit.> applies Beveridge' s verbal criterion to the United States. He computes the ratio between number of vacancies and number of unemployed and examines when the ratio crosses $1$. In the same spirit, the <cit.> scrutinizes the jobseeker-per-job-opening statistic and flags when it crosses $1$.

Efficient unemployment rateUnemployment gap
Efficient unemployment in the United States, 1951–2019
The unemployment rate $u$ and vacancy rate $v$ come from figure <ref>. The efficient unemployment rate is $u^* = \sqrt{uv}$. The unemployment gap is the difference between the actual unemployment rate and the efficient unemployment rate, $u-u^*$. The gray areas are NBER-dated recessions.

Comparison with efficient unemployment from <cit.>
The Michaillat & Saez (2021) line represents the efficient unemployment rate constructed by <cit.>. The $\sqrt{uv}$ line reproduces the efficient unemployment rate from figure <ref>. The gray areas are NBER-dated recessions.

§ EFFICIENT UNEMPLOYMENT IN THE UNITED STATES

We compute the efficient unemployment rate in the United States in three periods: the standard postwar period, 1951–2019; the coronavirus pandemic, 2020–2022; and the Great Depression and World War 2, 1930–1950.

§.§ Postwar period

State of the labor market We assess the state of the US labor market between 1951 and 2019. We use the unemployment and vacancy rates from figure <ref>. The labor market is inefficiently slack whenever the unemployment rate is above the vacancy rate; it is inefficiently tight whenever the unemployment rate is below the vacancy rate. Between 1951 and 2019, the US labor market is always inefficiently slack except in three episodes (figure <ref>): the Korean War (1951Q1–1953Q3), the Vietnam War (1965Q4–1970Q1), and the end of the Trump presidency (2018Q1–2019Q4).

Visualization with labor-market tightness The state of the US labor market can also be visualized by plotting the labor-market tightness (figure <ref>). The labor market is inefficiently slack whenever tightness is below $1$. It is inefficiently tight whenever tightness is above $1$. And it is efficient when the tightness equals $1$—when there is just one vacancy per jobseeker. The tightness averages $0.62$ between 1951 and 2019, and it is sharply procyclical. The tightness peaked at $1.49$ in 1953Q1, during the Korean War, and it bottomed at $0.16$ in 2009Q3, during the Great Recession.

Efficient unemployment rate We then compute the efficient unemployment rate between 1951 and 2019 using the formula $u^* = \sqrt{uv}$. The efficient unemployment rate is fairly stable: it remains between $3.0\%$ and $5.3\%$, with an average value of $4.2\%$ (figure <ref>).

Unemployment gap We also compute the unemployment gap, which is the difference $u-u^*$ between the actual unemployment rate and the efficient unemployment rate. The unemployment gap indicates the distance from full employment at any given time. The unemployment gap is sharply countercyclical and generally positive. The unemployment gap peaked at $+6.0$ percentage points in 2009Q4, during the Great Recession, and bottomed at $-0.6$ percentage points in 1969Q1, during the Vietnam War. The unemployment gap averages $1.6$ percentage points over the 1951–2019 period. Hence, the economy is generally not at full employment, and it is especially far from full employment in recessions.

Comparison with the formula from <cit.> In a previous paper, <cit.> compute a more sophisticated formula for the efficient unemployment rate:

\begin{equation}
u^* = \bp{\frac{\k \e}{1-\z}\cdot\frac{v}{u^{-\e}}}^{1/(1+\e)},
\label{e:ms16}\end{equation}

where  $\e$ is the Beveridge elasticity, $\z$ is the social value of nonwork, and $\k$ is the recruiting cost. The formula requires to keep track of three statistics in addition to the unemployment and vacancy rates ($$, $$, $$̨), so it is harder for policymakers to compute than the simple $u^* = \sqrt{uv}$. But $u^* = \sqrt{uv}$ is a special case of (<ref>) when $\e=1$, $\k=1$, and $\z=0$. The two formulas are so closely related because the model of the labor market presented here is a special case of the model in <cit.>. By specializing the model and therefore the values of the sufficient statistics, we obtain an extremely simple and user-friendly formula. In the United States, however, the two formulas yield almost identical unemployment rates (figure <ref>). Over 1951–2019, the two efficient unemployment rates depart by $0.2$ percentage points on average, and they never depart by more than $0.5$ percentage points.

Unemployment and vacancy ratesLabor-market tightness
State of the labor market in the United States, 2020M1–2022M3
The unemployment rate is constructed by the <cit.>. The vacancy rate is the number of job openings divided by the civilian labor force, both measured by the <cit.>. The labor-market tightness is the vacancy rate divided by the unemployment rate. The gray area is the NBER-dated pandemic recession. The labor market is inefficiently slack whenever the unemployment rate is above the vacancy rate (purple shade); it is inefficiently tight whenever the unemployment rate is below the vacancy rate (orange shade). The US labor market is inefficiently slack from 2020M3 to 2021M4; it is inefficiently tight in 2020M1–2020M2 and from 2021M5 to 2022M3.

Efficient unemployment rateUnemployment gap
Efficient unemployment in the United States, 2020M1–2022M3
The unemployment rate $u$ and vacancy rate $v$ come from figure <ref>.  The efficient unemployment rate is $u^* = \sqrt{uv}$. The unemployment gap is the difference between the actual unemployment rate and the efficient unemployment rate, $u-u^*$. The gray area is the NBER-dated pandemic recession.

Effect of the pandemic on the US Beveridge curve
Each dot represents the unemployment and vacancy rates in a month between 2001M1 and 2022M3. The unemployment rate is constructed by the <cit.>. The vacancy rate is the number of job openings divided by the civilian labor force, both measured by the <cit.>. The labor market is inefficiently slack when the unemployment rate is above the vacancy rate (purple shade); it is inefficiently tight when the unemployment rate is below the vacancy rate (orange shade); it is efficient when the unemployment rate equals the vacancy rate (pink line).

§.§ Coronavirus pandemic

We now apply our efficiency criterion and efficient-unemployment formula to the coronavirus pandemic in the United States. We focus on the period between January 2020 and March 2022.

Unemployment and vacancy rates The unemployment rate is constructed by the <cit.>. The vacancy rate is the number of job openings divided by the civilian labor force, both measured by the <cit.>. Both series are displayed on figure <ref>.

State of the labor market The labor market is inefficiently slack whenever the unemployment rate is above the vacancy rate; it is inefficiently tight whenever the unemployment rate is below the vacancy rate. We find that the US labor market is inefficiently slack from March 2020 to April 2021; it is inefficiently tight from May 2021 to March 2022 (figure <ref>).

Visualization with labor-market tightness The state of the US labor market can also be visualized by plotting labor-market tightness (figure <ref>). The labor market is inefficiently slack whenever tightness is below $1$. And it is inefficiently tight whenever tightness is above $1$. The tightness cratered to $0.20$ in April 2020, at the beginning of the pandemic. It then steadily recovered to reach $2.0$ in March 2022—the highest tightness recorded in the postwar period.

Efficient unemployment rate We then compute the efficient unemployment rate between 2020 and 2022 using the formula $u^* = \sqrt{uv}$. The efficient unemployment rate was at $4.0\%$ at the onset of the pandemic, but it sharply increased above $6.0\%$ in the first months of the pandemic (figure <ref>). It hovered around $6.0\%$ during the rest of the 2020–2022 period.

Unemployment gap We also compute the unemployment gap, which is the difference $u-u^*$ between the actual unemployment rate and the efficient unemployment rate (figure <ref>). The unemployment gap was initially positive and large: the labor market was much too slack in the first pandemic year. The unemployment gap reached $+8.1$ percentage points in April 2020—its highest level in the postwar period. But the economy recovered quickly, and the unemployment gap turned negative in May 2021. The unemployment gap reached $-1.5$ percentage points in March 2022. This is the lowest unemployment gap in the postwar period. So in 2022, the labor market is well beyond full employment.

Why did the efficient unemployment rate increase during the pandemic? The efficient unemployment rate increased by more than 2 percentage points at the onset of the pandemic. Such a sharp increase is unprecedented. It can be explained by the gigantic outward shift of the Beveridge curve that took place in the spring 2020. Graphically, the efficient unemployment rate appears at the intersection of the Beveridge curve and the pink identity line (figure <ref>). In February 2020, at the onset of the pandemic, the labor market was close to efficient, and the unemployment rate was around $4\%$. In May 2021, about a year later, the labor market had returned to efficiency, but the unemployment rate was now $6\%$. The increase of the efficient unemployment rate from $4\%$ to $6\%$ was caused by the outward shit of the Beveridge curve that occurred between March and April 2020. Mathematically, the efficient unemployment rate is determined by the location of the Beveridge curve ($u^* = \sqrt{A}$), so only a sharp outward shift of the Beveridge curve (increase in $A$) can increase the efficient unemployment rate.[The previous shifts of the Beveridge curve during the postwar period (figure <ref>) were not large enough to produce noteworthy changes in the efficient unemployment rate (figure <ref>).]

Regular scaleLog scale
Unemployment and vacancy rates in the United States, 1930–1950
The unemployment and vacancy rates are constructed by <cit.>. Unemployment and vacancy rates are quarterly averages of monthly series. The gray areas are NBER-dated recessions.

Unemployment and vacancy ratesLabor-market tightness
State of the labor market in the United States, 1930–1950
The unemployment and vacancy rates come from figure <ref>.  The labor-market tightness is the vacancy rate divided by the unemployment rate. The gray areas are NBER-dated recessions. The labor market is inefficiently slack whenever the unemployment rate is above the vacancy rate (purple shade); it is inefficiently tight whenever the unemployment rate is below the vacancy rate (orange shade). Between 1930 and 1950, the US labor market is always inefficiently slack except during World War 2 (1942Q3–1946Q3).

Efficient unemployment rateUnemployment gap
Efficient unemployment in the United States, 1930–1950
The unemployment rate $u$ and vacancy rate $v$ come from figure <ref>. The efficient unemployment rate is $u^* = \sqrt{uv}$. The unemployment gap is the difference between the actual unemployment rate and the efficient unemployment rate, $u-u^*$. The gray areas are NBER-dated recessions.

§.§ Great Depression and World War 2

Finally, we apply our efficiency criterion and efficient-unemployment formula to the historical period 1930–1950, which covers both the Great Depression and World War 2.

Unemployment and vacancy rates The unemployment and vacancy rates are constructed by <cit.> from NBER macrohistory files. Between 1930 and 1950 it remains true that unemployment and vacancy rates move in opposite directions (figure <ref>). Using a log scale, it also appears that unemployment and vacancy rates are inversely related (figure <ref>). These fluctuations indicate that just as in the postwar era, the hyperbolic Beveridge curve holds in 1930–1950.[See also figure 3 in <cit.>.]

State of the labor market The labor market is inefficiently slack whenever the unemployment rate is above the vacancy rate; it is inefficiently tight whenever the unemployment rate is below the vacancy rate. The US labor market is always inefficiently slack between 1930 and 1950 except at the end World War 2, during 1942Q3–1946Q3 (figure <ref>).

Visualization with labor-market tightness The state of the labor market can also be visualized by plotting labor-market tightness (figure <ref>). The labor market is inefficiently slack whenever tightness is below $1$; it is inefficiently tight whenever tightness is above $1$. Tightness is extremely volatile during the 1930–1950 period: it plunged to $0.03$ in 1932Q3, during the Great Depression, and climbed all the way to $6.8$ in 1944Q4, at the end of World War 2. These are the most extreme tightness fluctuations on record.

Efficient unemployment rate We then compute the efficient unemployment rate between 1930 and 1950 using the formula $u^* = \sqrt{uv}$. The efficient unemployment rate is again fairly stable: it remains between $2.5\%$ and $4.6\%$, with an average value of $3.5\%$ (figure <ref>). This value of the efficient unemployment rate is actually pretty close from the full-employment rate of unemployment advocated by <cit.> just after World War 2, which was $3\%$.

Unemployment gap Finally, we compute the unemployment gap, which is the difference $u-u^*$ between the actual unemployment rate and the efficient unemployment rate (figure <ref>). The unemployment gap was of course positive and very large during the Great Depression: the labor market was much too slack then. The unemployment gap reached $+20.9$ percentage points in 1932Q3—its highest level on record. The economy recovered only slowly from the Great Depression . The unemployment gap finally turned negative during World War 2, in 1942Q3. The unemployment gap reached $-1.6$ percentage points in 1945Q1—its lowest level on record. The unemployment gap turned positive again during the 1948–1949 recession.

Unemployment and vacancy rates, $u$ and $v$Labor-market tightness, $v/u$Efficient unemployment rate, $u^* = \sqrt{uv}$
State of the US labor market, 1930Q1–2022Q1
The unemployment and vacancy rates are obtained by splicing the unemployment and vacancy rates from figures <ref>, <ref>, and <ref>. Labor-market tightness and efficient unemployment rate are constructed from the unemployment and vacancy rates. The gray areas are NBER-dated recessions.

§.§ Summary

To summarize, we combine the US data for 1930Q1–2022Q1.

State of the labor market The unemployment rate is generally above the vacancy rate, and this gap is exacerbated in recessions (figure <ref>). This means that the labor market does not operate efficiently, unlike what economists often assume. It also means that monetary and fiscal policies are needed to reduce unemployment in bad times, and adhere more closely to the full-employment mandate. Before 2018, the labor market is only inefficiently tight in wartime—World War 2, Korean War, Vietnam War. Since then the labor market has been inefficiently tight at the end of the Trump presidency and during the recovery of the coronavirus pandemic. The state of the labor market in 2018–2019 and 2021–2022 is a rarity: it is the only peacetime labor market that is exceedingly tight. <cit.> doubted that the labor market could reach full employment in peacetime. It seems that he was essentially right: before 2018 the US labor market never reached full employment in peacetime.

Labor-market tightness The state of the labor market can also be visualized by plotting labor-market tightness (figure <ref>). Over 1930Q1–2022Q1, labor-market tightness averages $0.68$. In 2022, tightness reached a value of $2.0$, which it had last reached in 1945. In the recovery of the pandemic, the US labor market has reached historical levels of tightness.

Full employment Over 1930Q1–2022Q1, the efficient unemployment rate averages $4.2\%$ (figure <ref>). The efficient unemployment rate is stable over time. It hovered around $4\%$ between 1930 and 1970. It raised to about $5\%$ in the 1970s and stayed there in the 1980s. It then remained around $4\%$ again between 1990 and 2020. Finally, it rose to $6\%$ during the pandemic. This means that in the United States, the full-employment unemployment rate is stable over time, around $4\%$.

§ FREQUENTLY ASKED QUESTIONS

To conclude, we address questions that are frequently asked about the analysis. We cover questions about the analysis, about the policy implications of the results, about the theoretical underpinnings of the results, and about possible extensions.

§.§ Shouldn't unemployment dynamics play a role?

Informal argument Unemployment perpetually evolves through a dynamic process driven by differences in inflow into and outflow from unemployment. Yet, our analysis of labor-market efficiency is static. But because gross flows are so large relative to net flows—$4\%$ of employment per month compared to a fraction of a percent of employment per month—this dynamic unemployment process converges extremely quickly to the Beveridge curve. This explains why the static analysis is an excellent approximation to the dynamic process.

Formal argument in a matching model <cit.> In a matching model the unemployment rate evolves according to the following differential equation:

\begin{equation}
\dot{u}(t) = \l [1-u(t)] - f u(t),
\label{e:uDot}\end{equation}

where $\l$ is the job-separation rate and $f$ is the job-finding rate. The term $\l [1-u(t)]$ gives the number of workers who lose or quit their jobs and enter unemployment during a unit time. The term $f u(t)$ gives the number of workers who find a job and leave unemployment during a unit time. The difference between inflow into and outflow from unemployment determines the change in the unemployment rate, $\dot{u}(t)$. The solution to the differential equation (<ref>) is

\begin{equation}
u(t) - u^b = [u(0) - u^b] e^{-(\l+f)t},
\label{e:uSolution}\end{equation}

\begin{equation*}
u^b = \frac{\l}{\l+f}.
\end{equation*}

The unemployment rate $u^b$ is the critical point of differential equation (<ref>), so it corresponds to the unemployment rate given by the Beveridge curve when the job-finding rate is $f$. Equation (<ref>) shows that the distance between the unemployment rate $u(t)$ and the Beveridgean unemployment rate $u^b$ decays at an exponential rate $\l+f$. In the United States, the rate of decay is really fast: between 1951 and 2019, the job-finding rate averages $f = 55.8\%$ per month, the job-separation rate averages $\l = 3.2\%$ per month, so the rate of decay averages $\l+f = 59.0 \%$ per month. Accordingly, the half-life of the deviation from the Beveridgean unemployment rate, $u(t)-u^b$, is $\ln(2)/0.590 = 1.17$ month. Since about $50\%$ of the deviation evaporates within one month, about $90\%$ of it evaporates within one quarter, which implies that the unemployment rate is always close to the Beveridgean unemployment rate.

§.§ Shouldn't workers out of the labor force be part of the calculation?

If the labor-force participation rate systematically responded to the unemployment rate, then it would make sense to endogenize labor-force participation and include it in the welfare analysis. And in theory, two kinds of effects are possible. It would be possible that additional workers enter the labor force in bad times to supplement their household's income. It would also be possible that jobseekers become discouraged in bad times and leave the labor force. In practice, however, the labor-force participation rate appears acyclical, so variations in unemployment do not lead to systematic changes in participation <cit.>. Shimer's result is based on US data for 1960–2006. It is in line with earlier evidence from the United States. Using data for 1946–1954: <cit.> found that “There is no substantial evidence upholding the view that labor-force participation declines when there are moderate decreases in the demand for labor.”

§.§ Shouldn't the hardship from unemployment be included in the calculation?

Beyond production lost, unemployment can also create psychological hardship for unemployed workers. This psychological toll has been understood for a long time, even though it is often neglected in economics. <cit.> for instance noted that “The most striking aspect of unemployment is the suffering of the unemployed and their families—the loss of health and morale that follows loss of income and occupation.” In contrast, unemployed workers might enjoy more leisure than employed workers. <cit.> account for such considerations and find that psychological costs from unemployment offset the value of home production and leisure by unemployed workers. In net, it is as if unemployed workers do not engage in home production—which is what we assume here.

§.§ Are recruiting and jobseeking useless activities?

Of course recruiting and jobseeking are necessary for workers and firms to match with each other. But they do not generate any social welfare by themselves. In academia, faculty members have to spend time recruiting new colleagues. But during that time they do not teach or do research. Similarly, when graduate students are on the job market looking for a faculty position, they cannot teach or do research. In this paper we are looking for the unemployment rate that minimizes disruptions caused by recruiting and jobseeking.

§.§ Shouldn't we eliminate unemployment to almost zero?

Unemployment is clearly a waste of economic resources as people who would like to work and produce are forced to look for jobs and are not able to be productive. Yet, reducing the unemployment rate to zero is not desirable because, by the logic of the Beveridge curve, it would require a huge number of vacancies. Servicing vacancies takes work— posting advertisements, screening and interviewing candidates—work that has to be diverted away from other productive tasks. And machines cannot do recruiting for firms. Therefore, vacancies also consume economic resources. It turns out that job recruiting is as labor intensive as job seeking is: one vacancy consumes as much resources as what is wasted when one person is looking for a job instead of working. As a result, it is efficient to reduce the sum of the unemployment and vacancy rates, and not simply the unemployment rate.

§.§ Don't people move from job to job without going through unemployment?

It is true that many workers change jobs without going through unemployment. This does reduce the waste on the jobseeking side of the labor market. Conceptually, such moves can be considered as unemployment of duration zero. While it would be ideal if all unemployment spells were of zero duration, empirically they are not: this is why the Beveridge curve exists. Our analysis follows from the Beveridge curve, and it applies even with job-to-job moves.

§.§ What do the results imply for monetary policy?

In practice In the United States, the government is mandated to stabilize the economy at full employment. We have argued that our efficient unemployment rate $u^* = \sqrt{uv}$ is the appropriate measure of full employment. Hence, the Fed could use it as their full-employment target—especially since $\sqrt{uv}$ can be measured in real time.

In theory Moreover, targeting $u^*$ is optimal in certain monetary models with unemployment. This happens for instance in the model developed by <cit.>. In that model, it is optimal to eliminate the unemployment gap by setting the nominal interest rate appropriately. As the unemployment gap is countercyclical, and a reduction in interest rate lowers unemployment, it is optimal to lower the interest rate in slumps, when the unemployment gap is high, and to raise it in booms, when the unemployment gap turns negative.

Sufficient-statistic formula In fact, <cit.> show that starting from a nominal interest rate $i>0$ and an inefficient unemployment rate $u \neq u^*$, the Fed should set the federal funds rate to $i^*$ such that

\begin{equation}
i - i^* =  \frac{u-u^*}{du/di}.
\label{e:monetary}\end{equation}

The statistic $i-i^*$ indicates the decrease in interest rate required to reach the optimal policy. The statistic $u-u^*$ is the prevailing unemployment gap. And the statistic $du/di>0$ is the monetary multiplier: the percentage-point decrease in unemployment achieved by lowering the nominal interest rate by 1 percentage point. This sufficient-statistic formula ensures that the unemployment rate is efficient, and therefore also that the economy is at full employment.

Monetary multiplier A large literature estimates the effect of the federal funds rate on unemployment <cit.>. A midrange estimate of the US monetary multiplier is $du/di = 0.5$ <cit.>. Combined with a monetary multiplier of $0.5$, formula (<ref>) says that the Fed should cut its interest rate by $2$ percentage points for each positive percentage point of unemployment gap, and raise its interest rate by $2$ percentage points for each negative percentage point of unemployment gap. For instance in March 2022, the economy is exceedingly tight: the unemployment gap is $-1.6$ percentage points. The Fed should raise its interest rate by $1.6\times 2 = 3.2$ percentage points in total to cool the economy back to full employment.

At the zero lower bound If the zero lower bound on nominal interest rates becomes binding, conventional monetary policy will not be able to sustain full employment. But other policies, such as government spending, can bring the economy closer to full employment. In a fiscal model with unemployment, <cit.> find that optimal government spending deviates from the <cit.> rule to reduce—but not eliminate—the unemployment gap. Since the unemployment gap is countercyclical, and an increase in government spending reduces unemployment <cit.>, optimal government spending is countercyclical.

§.§ Shouldn't redistributive considerations be included in the analysis?

Our simple efficiency criterion minimizes the nonproductive use of labor. However, it is reasonable to think that waste on the jobseekers' side from looking for a job is more harmful than waste on the employer side for spending resources to fill vacancies. In other words, for distributive reasons, we might prefer having more waste if this waste is pushed toward employers. It is indeed the case that labor likes tight labor markets. Under some conditions, it is theoretically preferable to have direct redistribution (for example through more generous unemployment insurance funded by taxes on the wealthy) rather than through inefficiently tight labor markets. Practically, if taxes and transfers are set and cannot be made more generous, then an excessively tight labor market is an indirect way to redistribute toward labor.

§.§ Is the hyperbolic Beveridge curve consistent with a matching function?

<cit.> relate the matching elasticity $\h$ to the Beveridge elasticity $\e$ in a matching model:

\begin{equation*}
\h = \frac{1}{1+\e}\bp{\e-\frac{u}{1-u}}.
\end{equation*}

An hyperbolic Beveridge curve implies a Beveridge elasticity $\e=1$. With an unemployment rate around $u=5\%$, we obtain a matching elasticity of $\h = 0.47$. Such matching elasticity falls squarely in the range of estimates obtained with aggregate US data, which span $0.30$–$0.76$ <cit.>. So the hyperbolic Beveridge curve is consistent with US estimates of the matching function.

§.§ What causes the observed fluctuations along the Beveridge curve?

Over the business cycle, unemployment and vacancy rates move along the Beveridge curve (such as in figure <ref>). Different models offer different explanations for these fluctuations? In the DMP model, shocks to workers' bargaining power lead to fluctuations along the Beveridge curve <cit.>. In the variants of the DMP model by <cit.>, <cit.>, and <cit.>, real wages are rigid, and shocks to labor productivity generate realistic fluctuations along the Beveridge curve. In the mismatch model by <cit.> and stock-flow matching model by <cit.>, shocks to labor productivity also generate sizable fluctuations along the Beveridge curve. Finally, in the business-cycle model by <cit.>, prices are rigid and shocks to aggregate demand generate fluctuations along the Beveridge curve. Such shocks may be caused by changes in monetary policy, or by changes in people's willingness to save (modeled as changes in people's marginal utility of wealth).

§.§ Should the government really take the Beveridge curve as given?

Empirical argument Empirically the Beveridge curve is very stable over the business cycle. It does not seem to respond to stabilization policies (either monetary or fiscal policy) in any systematic way. Given this stability, it makes sense that the Fed takes the Beveridge curve as given.

Theoretical argument Most Beveridgean models of monetary and fiscal policy are built around a matching function <cit.>. In these models the Beveridge curve is determined by the matching function and job-separation rate. Neither respond to monetary or fiscal policy, so the Beveridge curve is unaffected by policy. In the context of business-cycle stabilization, it therefore seems appropriate for the government to take the Beveridge curve as given.[Stabilization policies do not seem to affect the Beveridge curve, but some structural policies might. For instance, Germany implemented labor-market reforms in 2003–2005 that were designed to improve how the labor market operates (Hartz reforms). Such policies might shift the Beveridge curve inward.]

§.§ What explains the inefficiency of the labor market?

In most Beveridgean models, unlike in Walrasian models, the market economy does not overlap with the planning economy. This is because most price and wage mechanisms do not ensure efficiency. For instance, in matching models of the labor market, the wage is determined in a situation of bilateral monopoly, so a range of wages is theoretically possible. A wage mechanism picks one wage within the range. There is only an infinitesimal chance that this wage is the one ensuring efficiency <cit.>. Accordingly, theory does not guarantee that unemployment is efficient—or that full employment occurs.

§.§ What does labor-market tightness imply for inflation?

In the recovery of the coronavirus pandemic in the United States, the labor market has reached a tightness never seen since World War 2 (figure <ref>). At the same time, wage inflation has also been very high. The Employment Cost Index for wages and salaries of workers in the private sector grew at an annual rate of $6.3\%$ in 2021Q3 and $5.0\%$ in 2022Q1, the highest rates recorded since the Employment Cost Index was created in 2001 <cit.>. It is therefore quite possible that wage inflation accelerates when the labor market is exceedingly tight. <cit.> provide some support for this hypothesis. They find that both unemployment and vacancies are good predictors of wage inflation. Since labor-market tightness is the vacancy-to-unemployment ratio, it could be a good predictor of wage inflation as well.