#  A General Impossibility Theorem on Pareto Efficiency and Bayesian Incentive
Compatibility111 We are thankful to Matías Núñez for the fruitful discussions
and useful comments. We are also grateful to two anonymous reviewers and the
Associate Editor for their thoughtful and insightful comments. Financial
support by Investissements d’Avenir, ANR-11-IDEX-0003/Labex
Ecodec/ANR-11-LABX-0047 and PHC Sakura program, project number 45153XK, is
gratefully acknowledged.

Kazuya Kikuchi  Tokyo University of Foreign Studies. E-mail:
<EMAIL_ADDRESS>Yukio Koriyama  CREST, Ecole Polytechnique,
Institut Polytechnique de Paris. E-mail<EMAIL_ADDRESS>
###### Abstract

This paper studies a general class of social choice problems in which agents’
payoff functions (or types) are privately observable random variables, and
monetary transfers are not available. We consider cardinal social choice
functions which may respond to agents’ preference intensities as well as
preference rankings. We show that a social choice function is ex ante Pareto
efficient and Bayesian incentive compatible if and only if it is dictatorial.
The result holds for arbitrary numbers of agents and alternatives, and under a
fairly weak assumption on the joint distribution of types, which allows for
arbitrary correlations and asymmetries.

##  1 Introduction

We consider a general class of social choice problems in which agents’ payoffs
are privately observed random variables and monetary transfer is not
available. Payoffs are drawn from a continuous space, so that the aggregation
of the cardinal preferences is at stake. We prove an impossibility theorem: a
social choice function is ex ante Pareto efficient and Bayesian incentive
compatible if and only if it is dictatorial, under fairly general conditions.

Our main contribution lies in the generality of the conditions under which the
impossibility holds. We provide a proof of the theorem for an arbitrary number
of agents and alternatives, under a mild assumption about the joint
distribution of preference types. In particular, our theorem does not hinge on
either independence or symmetry assumptions.

We would like to emphasize that allowing for correlation is relevant in many
applications. In voting in general, it is common for voters’ preferences for
each alternative to be correlated. Our theorem provides insight not only for
normative analysis in which assumptions such as independence or symmetry of
the preferences are imposed, but also for positive analysis in which presence
of correlations in the joint distribution is relevant.

Our proof proceeds by highlighting the incentive for agents to exaggerate the
intensity of their preferences on the best and the worst alternatives, which
we call extremization. Our result suggests that the incentive that agents are
willing to send an extreme message about their best and the worst alternatives
is robust, and the idea that such a motivation would lead to an impossibility
theorem is formalized in the proof.

Impossibility results have been obtained under various conditions in the
literature (Börgers and Postl, 2009; Ehlers et al., 2020). However, to the
best of our knowledge, existing models are either limited with respect to the
number of agents, the number of alternatives, or the conditions on the joint
distribution of preferences. We discuss our contribution to the literature in
more detail below.

###  1.1 Related literature

Our result concerns the efficiency and incentive properties of cardinal social
choice rules in a Bayesian environment without monetary transfers. Cardinal
rules take into account the preference intensities (or payoffs) of agents,
unlike ordinal rules which are based only on the preference rankings. We focus
on rules defined on the full domain in which each agent’s payoff from each
alternative can take any value in the unit interval $[0,1]$. Another important
domain studied in the literature is the vNM domain in which each agent has a
von-Neumann-Morgenstern preference, which can be represented by a payoff
function whose minimum and maximum values are fixed to 1 and 0, respectively.

Previous studies on this topic have already demonstrated the tradeoff between
efficiency and incentives under cardinal rules. The result closest to ours is
Ehlers et al. (2020, Theorem 8) which shows that in the full domain, if each
agent’s belief about the other agents’ types is independent of his own type,
then any nondictatorial weighted utilitarian rule is not Bayesian incentive
compatible. Börgers and Postl (2009) derives the same conclusion for the vNM
domain, in a model with two agents who have opposite preference rankings over
three alternatives, and whose payoffs from the middle-ranked alternative are
independent and identically distributed random variables. Our motivation in
this paper is to establish the impossibility result under a minimum
assumption. We only assume that the joint distribution of agents’ types has a
density with full support, while restricting attention to the full domain. In
particular, we allow for arbitrary correlations of types.

Earlier papers provide impossibility results for different notions of
efficiency and incentive compatibility. Hylland (1980) shows that if a
cardinal rule is strategy-proof and ex post Pareto efficient, then it is a
random dictatorship. Moreover, if ex post efficiency is replaced with ex ante
efficiency, the rule must be a deterministic dictatorship. Dutta et al. (2007)
strengthens Hylland’s random dictatorship result by replacing ex post
efficiency with unanimity (see also Dutta et al. 2008). Nandeibam (2013)
further strengthens these results by proving that strategy-proofness and
unanimity imply weak random dictatorship, which allows the support of the
socially chosen lottery to be a subset of the best alternatives for the random
dictator.

For ordinal rules, it is well-known that strategy-proofness implies
dictatorship, if there are at least three alternatives (Gibbard 1973;
Satterthwaite 1975). When the social choice is probabilistic as a function of
the rank order profile, Gibbard (1977) provides an impossibility theorem
showing that the unique probabilistic social choice function that satisfies
strategy-proofness (defined by first-order stochastic dominance) and ex-ante
efficiency is dictatorial (an alternative proof is given by Nandeibam (1998)).
Majumdar and Sen (2004) demonstrates robustness of the Gibbard-Satterthwaite
theorem, by studying ordinal Bayesian incentive compatibility (OBIC),
introduced by d’Aspremont and Peleg (1988), and showing that there is a
generic set of independent belief systems for which an ordinal rule is OBIC
only if it is dictatorial. However, for a certain class of positively
correlated belief systems, Bhargava et al. (2015) provides a sufficient
condition (much weaker than dictatorship) for an ordinal rule to be locally
robustly OBIC. As for the possibility results, Majumdar and Roy (2021)
considers 2-player cases and explicitly shows the functions which are not
random dictatorship satisfying both OBIC and ex-post efficiency. Aziz et al.
(2018) illustrates the trade-off between efficiency and strategy-proofness,
using four preference extensions on lotteries. Interestingly, when strategy-
proofness is weakened to the one induced by the extension based on the sure-
thing principle, a possibility result is obtained.

Other works in the literature investigate optimal rules among cardinal rules,
subject to incentive constraints. Azrieli and Kim (2014) considers the case
with two alternatives and independent types, and show that ex ante efficient
rules and interim efficient rules subject to Bayesian incentive compatibility
are characterized by certain classes of (ordinal) weighted majority rules.
Schmitz and Tröger (2012) also studies the case with two alternatives. They
assume that the distribution of types is symmetric with respect to both the
alternatives and agents, but allow the types to be correlated. Then some weak
majority rule, in which the two alternatives are chosen with equal
probabilities unless either alternative obtains a sufficiently large support,
is optimal among strategy-proof rules. Moreover, the standard majority rule is
optimal among Bayesian incentive compatible rules if types are independent,
but there can be some superior rule if types are correlated.

These results suggest that an incentive compatible rule cannot effectively
utilize the cardinal information of preferences. Indeed, Ehlers et al. (2020)
shows that if a social choice rule satisfies some continuity condition in
addition to Bayesian incentive compatibility, then it must be vNM-ordinal in
expectation, meaning that the rule responds to a change in an agent’s type
only if his expected-utility ranking of alternatives changes. Kim (2017) shows
that there is a Bayesian incentive compatible vNM-ordinal rule that achieves a
higher utilitarian welfare than any ordinal rule.

Finally, Jackson and Sonnenschein (2007) shows that for a general Bayesian
collective decision problem, any ex ante Pareto efficient social choice rule
can be approximately implemented by the linking mechanism, defined on a set of
independent copies of the original decision problem.

##  2 The impossibility theorem

Let $I$ be the set of agents with $|I|=n\geq 2$, and $X$ the set of
alternatives with $|X|=m\geq 2$, both finite. If alternative $x\in X$ is
chosen, agent $i\in I$ obtains a random payoff $U_{i}(x)$ taking values in
$[0,1]$. The payoff vector $U_{i}:=(U_{i}(x))_{x\in X}\in[0,1]^{m}$ is called
the type of the agent. Let $U=(U_{i})_{i\in I}\in[0,1]^{mn}$ denote the
profile of agents’ types. We assume the following:

######  Assumption 1.

The joint distribution of the type profile $U=(U_{i})_{i\in I}$ is absolutely
continuous (with respect to Lebesgue measure) and has full support over
$[0,1]^{mn}$.

A social choice function (SCF) is a Borel measurable function
$f:[0,1]^{mn}\to\Delta(X)$, where $\Delta(X):=\\{(p(x))_{x\in
X}\in[0,1]^{m}|\sum p(x)=1\\}$ is the set of probability distributions on $X$.
Given the realization $u$ of the type profile $U$, the alternative chosen by
the distribution $f(u)$ is a random variable on $X$. We denote this random
alternative also as $f(u)$. Agent $i$’s payoff under the SCF is thus the
random variable $U_{i}(f(U))$, whose randomness arises from both the random
type profile $U=(U_{j})_{j\in I}$ and the probabilistic SCF $f$. The ex ante
expected payoff is then:

$\pi_{i}(f):=\mathbf{E}\left[U_{i}(f(U))\right].$

Consider the direct mechanism in which the agents report types and SCF $f$
chooses an alternative based on the profile of reported types. If agent $i$’s
true type is $u_{i}$ and he reports a type $v_{i}$, while all other agents
report their types truthfully, then $i$’s interim expected payoff is:

$\pi_{i}(f,v_{i}|u_{i}):=\mathbf{E}\left[u_{i}(f(v_{i},U_{-i}))|U_{i}=u_{i}\right].$

Let $f$ be an SCF. The SCF is ex ante Pareto efficient if there is no SCF $g$
such that for all agents $i$, $\pi_{i}(g)\geq\pi_{i}(f)$, with strict
inequality for some $i$. It is dictatorial if there is an agent $i$ such that
${f}(U)\in\arg\max_{x}U_{i}(x)$ almost surely. It is Bayesian incentive
compatible if $\pi_{i}(f,u_{i}|u_{i})\geq\pi_{i}(f,v_{i}|u_{i})$ for all
agents $i$ and all types $u_{i},v_{i}\in[0,1]$.

######  Theorem 1.

A social choice function is ex ante Pareto efficient and Bayesian incentive
compatible if and only if it is dictatorial.222A restricted version of Theorem
1 for the case with two alternatives has appeared as Proposition 3 in the
working-paper version of Kikuchi and Koriyama (2023), available at
https://arxiv.org/abs/2206.09574.

##  3 Proof of the theorem

To prove Theorem 2, we first characterize ex ante Pareto efficient SCFs as
weighted utilitarian SCFs (Proposition 1). We then show that the only weighted
utilitarian SCFs that are Bayesian incentive compatible are dictatorial SCFs
(Proposition 2).

We call an SCF $f$ weighted utilitarian if there is a nonnegative vector
$\lambda=(\lambda_{i})_{i\in I}$ with $\sum\lambda_{i}=1$ such that
${f}(U)\in\arg\max_{x}\sum_{i}\lambda_{i}U_{i}(x)$ almost surely. Note that a
dictatorial SCF is a weighted utilitarian SCF that gives the whole weight 1 to
one agent.

######  Proposition 1.

A social choice function is ex ante Pareto efficient if and only if it is
weighted utilitarian.

###### Proof.

Let $\pi(f)=(\pi_{i}(f))_{i\in I}\in[0,1]^{n}$ denote the profile of expected
payoffs under SCF $f$. Let $F$ denote the set of all SCFs, and
$\pi(F)=\\{\pi(f)|f\in F\\}$ the set of all expected payoff profiles induced
by SCFs. The set $\pi(F)$ is a convex subset of $[0,1]^{n}$, since $F$ permits
SCFs that randomize over the alternatives.

“Only if” part. Let $f$ be an ex ante Pareto efficient SCF. Since $\pi(F)$ is
a convex set in $[0,1]^{n}$, by Mas-Colell et al. (1995, Proposition
16.E.2),333Part of Mas-Colell et al. (1995, Proposition 16.E.2) says that if
$Y\subset\mathbb{R}^{k}$ is a convex set and if an element $y^{\ast}\in Y$ is
in the Pareto frontier of $Y$, then there is a vector
$\mu=(\mu_{i})_{i=1}^{k}$ with $\mu_{i}\geq 0$ for all $i=1,\cdots,k$ and
$\mu\neq 0$ such that $y^{\ast}$ is a solution of the maximization problem
$\max_{y\in Y}\mu\cdot y$. there is a nonnegative vector
$\lambda=(\lambda_{i})_{i\in I}$ with $\sum\lambda_{i}=1$ such that the
expected payoff profile $p=\pi(f)$ is a solution to the following maximization
problem:

$\max_{p\in\pi(F)}\lambda\cdot p.$ (1)

Thus, $f$ must be a solution to the maximization problem:

$\max_{f\in F}\lambda\cdot\pi(f).$ (2)

By linearity of expectation, the objective function in (2) equals
$\mathbf{E}\left[\sum_{i}\lambda_{i}U_{i}({f}(U))\right]$, which is maximized
if and only if $f$ is a $\lambda$-weighted utilitarian SCF. This proves the
“only if” part.

Uniqueness of the solution to (1). The above argument shows that if $p$ is a
solution of (1), then $p=\pi(f)$ for some $\lambda$-weighted utilitarian $f$.
There are multiple $\lambda$-weighted utilitarian SCFs, as there can be
multiple alternatives $x$ maximizing the $\lambda$-weighted utilitarian sum
$\sum_{i}\lambda_{i}U_{i}(x)$. However, absolute continuity implies that such
ties occur with probability 0. This implies that all $\lambda$-weighted
utilitarian SCFs coincide almost surely, and hence induce the same expected
payoff vector $\pi(f)$. Therefore, the solution $p$ of (1) is unique.

“If” part. Suppose on the contrary that there is a nonnegative vector
$\lambda=(\lambda_{i})_{i\in I}$ for which a $\lambda$-weighted utilitarian
SCF $f$ is not ex ante Pareto efficient. Let $p=\pi(f)$ be the expected payoff
profile under this SCF. There is a Pareto-dominating expected payoff profile
$q\in\pi(F)$ with $q\neq p$ such that $q_{i}\geq p_{i}$ for all $i$. Then
$\lambda\cdot q\geq\lambda\cdot p$, which contradicts the fact that $p$ is the
unique solution to (1). ∎

We next show that the dictatorial SCFs are the only weighted utilitarian SCFs
that are Bayesian incentive compatible.

######  Proposition 2.

A social choice function is weighted utilitarian and Bayesian incentive
compatible if and only if it is dictatorial.

###### Proof.

Every dictatorial SCF is obviously Bayesian incentive compatible. It remains
to show that any nondictatorial weighted utilitarian SCF is not Bayesian
incentive compatible.

Fix an agent $i$ with positive weight $\lambda_{i}>0$. Let $u_{i}$ be a type
of the agent such that for some alternatives $a,b$,

$u_{i}(b)=\max_{x}u_{i}(x)<1\text{ and }0<u_{i}(b)-u_{i}(a)<\frac{\sum_{j\neq
i}\lambda_{j}}{\lambda_{i}}.$ (3)

That is, for agent $i$, $b$ is the best alternative but gives a payoff less
than 1. Alternative $a$ is strictly worse than $b$ but gives a payoff
sufficiently close to $b$. Since $\sum_{j\neq i}\lambda_{j}>0$ by the no-
dictator assumption, such types $u_{i}$ always exist and form a set of
positive measure.

We show that conditional on any type $u_{i}$ satisfying (3), agent $i$ can
profitably deviate from truth-telling to reporting the following false type:

$\hat{u}_{i}=(1,(u_{i}(x))_{x\neq b})$

which is the same as the true type $u_{i}$ except that it exaggerates the
payoff from the best alternative $b$ as 1 (we call such a strategy as an
extremization for the best alternative).

To do this, let $v_{i}(b)\in[0,1]$ be any payoff value, and suppose that agent
$i$ reports the type $(v_{i}(b),(u_{i}(x))_{x\neq b})$ when the true type is
$u_{i}$, i.e., if he reports truthfully except for the payoff of his best
alternative $b$, while all other agents report their true types. In this case,
the weighted sum of reported payoffs for each alternative $x$, denoted $S(x)$,
is given by

$S(x)=\begin{cases}\lambda_{i}v_{i}(b)+\sum_{j\neq
i}\lambda_{j}U_{j}(b)&\text{if $x=b$}\\\ \lambda_{i}u_{i}(x)+\sum_{j\neq
i}\lambda_{j}U_{j}(x)&\text{if $x\neq b$}.\end{cases}$

The interim probability that alternative $x$ is chosen, denoted
$f(x,v_{i}(b)|u_{i})$, is

$f(x,v_{i}(b)|u_{i})=\mathbf{P}\\{S(x)>S(y)\text{ for all $y\neq
x$}|U_{i}=u_{i}\\},$

where ties ($S(x)=S(y)$) can be ignored as they almost never occur by the
absolute continuity assumption. The interim expected payoff for agent $i$ is
(with slight abuse of notation):

$\pi_{i}(f,v_{i}(b)|u_{i})=\sum_{x\in X}u_{i}(x)f(x,v_{i}(b)|u_{i}).$

Now suppose that $i$ increases the reported value $v_{i}(b)$ from $u_{i}(b)$
to 1. Then the weighted sum $S(b)$ for $i$’s best alternative $b$ will
increase while the weighted sum $S(x)$ for all other alternatives $x\neq b$
will remain the same. Thus, the interim probability $f(b,v_{i}(b)|u_{i})$ for
alternative $b$ will weakly increase while $f(x,v_{i}(b)|u_{i})$ for all
$x\neq b$ will weakly decrease. That is, $i$’s deviation will only transfer
some portion (possibly zero) of the interim probabilities of alternatives
$x\neq b$ to his best alternative $b$. This implies that $i$’s interim
expected payoff will weakly increase:
$\pi_{i}(f,1|u_{i})\geq\pi_{i}(f,u_{i}(b)|u_{i})$. To show that the inequality
is strict, it suffices to check that for alternative $a$ satisfying (3), the
interim probability will strictly decrease:
$f(a,u_{i}(b)|u_{i})>f(a,1|u_{i})$.

To verify this, note that the probability $f(a,v_{i}(b)|u_{i})$ can be
decomposed into the product of two factors: the probability that $S(a)>S(x)$
for all $x\neq a,b$ (call this event $E$), and the conditional probability
that $S(a)>S(b)$ given $E$. That is,

$f(a,v_{i}(b)|u_{i})=\mathbf{P}\\{S(a)>S(b)|E,U_{i}=u_{i}\\}\mathbf{P}\\{E|U_{i}=u_{i}\\}.$

Since event $E$ is defined by inequalities involving only $S(x)$ for $x\neq
b$, its probability $\mathbf{P}\\{E|U_{i}=u_{i}\\}$ does not depend on $i$’s
reported payoff $v_{i}(b)$ of alternative $b$. It thus suffices to check that
$\mathbf{P}\\{S(a)>S(b)|E,U_{i}=u_{i}\\}$ strictly decreases as $v_{i}(b)$
increases from $u_{i}(b)$ to 1.

To see this, let us rewrite the inequality $S(a)>S(b)$ as

$\lambda_{i}[u_{i}(a)-v_{i}(b)]>\textstyle\sum_{j\neq
i}\lambda_{j}[U_{j}(b)-U_{j}(a)].$ (4)

By the full-support assumption, the conditional distribution of the right-hand
side of (4) has the support $[-\sum_{j\neq i}\lambda_{j},\sum_{j\neq
i}\lambda_{j}]$. Condition (3) implies that if $i$ tells the truth (i.e., if
$v_{i}(b)=u_{i}(b)$), then the left-hand side of (4) is an interior point of
this interval. Thus, as $i$ increases the value of $v_{i}(b)$ from $u_{i}(b)$
to $1$, the conditional probability of (4) strictly decreases. This
establishes that $f(a,u_{i}(b)|u_{i})>f(a,1|u_{i})$, and therefore
$\pi_{i}(f,1|u_{i})>\pi_{i}(f,u_{i}(b)|u_{i})$. ∎

##  4 Discussion and Conclusion

This paper focuses on aggregation of cardinal preferences without monetary
transfers. Such a situation is typical of voting, where monetary transfers are
generally forbidden, and voters may have heterogeneous preference intensities.
We briefly discuss the application of Theorem 2 to the voting problems before
we conclude.

A common practice for reflecting preference intensity is to use a cardinal
voting rule, in which voters assign a score to each alternative, and the
decision is made based on the (weighted) sum of the scores (see, e.g., Núñez
and Laslier 2014). This class includes a wide variety of evaluative voting
rules. However, cardinal voting may lead to inefficient outcomes due to
strategic behavior of voters. Kikuchi and Koriyama (2023) pointed out this
problem by showing the social dilemma in the context of collective decision
making among distinct groups. Each group chooses an internal rule that
specifies the allocation of weight to the alternatives as a function of the
preferences of group members, and the winner-take-all rule turns out to be
weakly dominant, while the equilibrium is Pareto dominated.

Theorem 2 shows that it is impossible to solve the social dilemma, even though
the decision rule is extended to an arbitrary social choice function, instead
of a fixed cardinal weighted voting. In this sense, the social dilemma is
inherent in cardinal voting. Notice that the winner-take-all strategy, shown
to be a dominant strategy in Kikuchi and Koriyama (2023), is a special case
for $m=2$ of the extremization strategy specified in Section 3. The fact that
the extremization strategy is ordinal for $m=2$ is consistent with the
insights obtained in Lemma 2 of Azrieli and Kim (2014), which shows that
incentive compatibility entails ordinal preference aggregation. When the model
is extended to an arbitrary number of alternatives, the above intuition is
generalized in Theorem 2.

Characterizing the second-best mechanism and evaluating the associated welfare
loss are natural questions which follow our results. Another possible
extension would be the consideration of interim or ex post Pareto efficiency,
instead of ex ante efficiency. However, they are beyond the scope of the
current paper and we leave them for future research.

We have proved an impossibility theorem for social choice functions defined on
the full domain in which agents’ payoffs from each alternative can take any
value in the interval [0, 1]. There is an important subdomain, the vNM domain,
in which each agent has a von-Neumann Morgenstern preference, represented by a
utility function whose maximum and minimum values are fixed to 1 and 0,
respectively. A natural question arises: does the impossibility theorem hold
when the domain is restricted to the vNM domain? Börgers and Postl (2009)
proves an impossibility theorem in a model with two agents and three
alternatives, assuming that the types are independent. However, it seems
fairly challenging to provide a proof for general numbers of players and
alternatives, without assuming independence of the joint distribution. Another
possibility is to use the proof techniques employed in Ehlers et al. (2020).
Unfortunately, it is not straightforward to adapt their proof technique (in
Theorems 7 and 8) to the vNM domain with a correlated distribution, due to the
critical role of the full domain and independence assumptions. They also
provide an informal, but very interesting, idea of proof using the revenue
equivalence theorem. However, extending it to the correlated case would also
be a major challenge, since the independence assumption is crucial in the
revenue equivalence theorem. Extending the analysis to the vNM domain with
correlated types is another important avenue that could be pursued in future
research.

Competing interests: The authors have no competing interests to declare that
are relevant to the content of this article.

## References

  * Aziz et al. (2018) Aziz, H., F. Brandl, F. Brandt, and M. Brill (2018): “On the tradeoff between efficiency and strategyproofness,” _Games and Economic Behavior_ , 110, 1–18.
  * Azrieli and Kim (2014) Azrieli, Y. and S. Kim (2014): “Pareto efficiency and weighted majority rules,” _International Economic Review_ , 55, 1067–1088.
  * Bhargava et al. (2015) Bhargava, M., D. Majumdar, and A. Sen (2015): “Incentive-compatible voting rules with positively correlated beliefs,” _Theoretical Economics_ , 10, 867–885.
  * Börgers and Postl (2009) Börgers, T. and P. Postl (2009): “Efficient compromising,” _Journal of Economic Theory_ , 144, 2057–2076.
  * d’Aspremont and Peleg (1988) d’Aspremont, C. and B. Peleg (1988): “Ordinal Bayesian Incentive Compatible Representations of Committees,” _Social Choice and Welfare_ , 5, 261–279.
  * Dutta et al. (2007) Dutta, B., H. Peters, and A. Sen (2007): “Strategy-proof cardinal decision schemes,” _Social Choice and Welfare_ , 28, 163–179.
  * Dutta et al. (2008) ——— (2008): “Strategy-proof cardinal decision schemes,” _Social Choice and Welfare_ , 30, 701.
  * Ehlers et al. (2020) Ehlers, L., D. Majumdar, D. Mishra, and A. Sen (2020): “Continuity and incentive compatibility in cardinal mechanisms,” _Journal of Mathematical Economics_ , 88, 31–41.
  * Gibbard (1973) Gibbard, A. (1973): “Manipulation of voting schemes: a general result,” _Econometrica: journal of the Econometric Society_ , 587–601.
  * Gibbard (1977) ——— (1977): “Manipulation of Schemes that Mix Voting with Chance,” _Econometrica_ , 45, 665–681.
  * Hylland (1980) Hylland, A. (1980): “Strategy proofness of voting procedures with lotteries as outcomes and infinite sets of strategies,” Tech. rep., mimeo.
  * Jackson and Sonnenschein (2007) Jackson, M. O. and H. F. Sonnenschein (2007): “Overcoming incentive constraints by linking decisions 1,” _Econometrica_ , 75, 241–257.
  * Kikuchi and Koriyama (2023) Kikuchi, K. and Y. Koriyama (2023): “The winner-take-all dilemma,” _Theoretical Economics_ , 18, 917–940.
  * Kim (2017) Kim, S. (2017): “Ordinal versus cardinal voting rules: A mechanism design approach,” _Games and Economic Behavior_ , 104, 350–371.
  * Majumdar and Roy (2021) Majumdar, D. and S. Roy (2021): “Ordinally Bayesian incentive compatible probabilistic voting rules,” _Mathematical Social Sciences_ , 114, 11–27.
  * Majumdar and Sen (2004) Majumdar, D. and A. Sen (2004): “Ordinally Bayesian incentive compatible voting rules,” _Econometrica_ , 72, 523–540.
  * Mas-Colell et al. (1995) Mas-Colell, A., M. D. Whinston, and J. R. Green (1995): _Microeconomic Theory_ , Oxford University Press.
  * Nandeibam (1998) Nandeibam, S. (1998): “An alternative proof of Gibbard’s random dictatorship result,” _Social Choice and Welfare_ , 15, 509–519.
  * Nandeibam (2013) ——— (2013): “The structure of decision schemes with cardinal preferences,” _Review of Economic Design_ , 17, 205–238.
  * Núñez and Laslier (2014) Núñez, M. and J. F. Laslier (2014): “Preference intensity representation: strategic overstating in large elections,” _Social Choice and Welfare_ , 42, 313–340.
  * Satterthwaite (1975) Satterthwaite, M. A. (1975): “Strategy-proofness and Arrow’s conditions: Existence and correspondence theorems for voting procedures and social welfare functions,” _Journal of economic theory_ , 10, 187–217.
  * Schmitz and Tröger (2012) Schmitz, P. W. and T. Tröger (2012): “The (sub-) optimality of the majority rule,” _Games and Economic Behavior_ , 74, 651–665.


# Atomic step disorder on polycrystalline surfaces leads to spatially
inhomogeneous work functions

Morgann Berg  Sean W. Smith  David A. Scrymgeour  Michael T. Brumbach  Ping Lu
Sara M. Dickens  Joseph R. Michael  Taisuke Ohta  Ezra Bussmann∗
<EMAIL_ADDRESS>Harold P. Hjalmarson  Peter A. Schultz  Paul G. Clem
Matthew M. Hopkins  Christopher H. Moore∗<EMAIL_ADDRESS>1Sandia National
Laboratories, Albuquerque NM, 87185 USA

###### Abstract

Structural disorder causes materials surface electronic properties, e.g. work
function ($\phi$) to vary spatially, yet it is challenging to prove exact
causal relationships to underlying ensemble disorder, e.g. roughness or
granularity. For polycrystalline Pt, nanoscale resolution photoemission
threshold mapping reveals a spatially varying $\phi=5.70\pm 0.03$ eV over a
distribution of (111) textured vicinal grain surfaces prepared by sputter
deposition and annealing. With regard to field emission and related phenomena,
e.g. vacuum arc initiation, a salient feature of the $\phi$ distribution is
that it is skewed with a long tail to values down to 5.4 eV, i.e. far below
the mean, which is exponentially impactful to field emission via the Fowler-
Nordheim relation. We show that the $\phi$ spatial variation and distribution
can be explained by ensemble variations of granular tilts and surface slopes
via a Smoluchowski smoothing model wherein local $\phi$ variations result from
spatially varying densities of electric dipole moments, intrinsic to atomic
steps, that locally modify $\phi$. Atomic step-terrace structure is confirmed
with scanning tunneling microscopy (STM) at several locations on our surfaces,
and prior works showed STM evidence for atomic step dipoles at various metal
surfaces. From our model, we find an atomic step edge dipole $\mu=0.12$ D/edge
atom, which is comparable to values reported in studies that utilized other
methods and materials. Our results elucidate a connection between macroscopic
$\phi$ and nanostructure that may contribute to the spread of reported $\phi$
for Pt and other surfaces, and may be useful toward more complete descriptions
of polycrystalline metals in models of field emission and other related vacuum
electronics phenomena, e.g. arc initiation.

##  I Introduction

Field emission (FE) of electrons from solid surfaces is an integral mechanism
to many electronic technologies, and a fundamental contributor to complex
phenomena, e.g. vacuum arcs. Jensen (2018); Boxman, Sanders, and Martin
(1996); Tang  _et al._ (2016) In cold FE, electron fluxes, $J$, depend on a
surface’s work function $\phi$, via the Fowler-Nordheim relation $J$ ($\phi$,
$\beta E$) $=(A\beta^{2}E^{2}/\phi)$ exp $(-B\phi^{3/2}/\beta E)$, where
$\beta E$ is the local effective electric field, and $A$ and $B$ are
constants. Fowler and Nordheim (1928); Forbes and Deane (2007); Jensen (2019)
For realistic surfaces, $\phi$, and consequently $J$, vary spatially owing to
near-surface disorder. Forbes (2001); Lecoeur, Bellier, and Koehler (1990);
Vanselow and Li (1992); Smoluchowski (1941); Strayer, Mackie, and Swanson
(1973); Besocke, Krahl-Urban, and Wagner (1977); Haas and Thomas (1969);
Schultz  _et al._ (2021) Since $J$ depends exponentially on local $\phi$, the
consequences of spatially patchy $\phi$ distributions are amplified, and
especially impactful for feedback-driven phenomena like vacuum arc initiation.

Some $\phi$ spatial variation across crystalline, and polycrystalline, metal
surfaces arises from inherent crystal anisotropy effects combined with
roughness likely to be present in any realistic scenario. A crystal’s work
function is facet (hkl)-dependent, i.e. $\phi=\phi$ (hkl). Smoluchowski (1941)
For some high-symmetry principal facets, Fig. 1 (a), of Pt, experiments give
values $\phi$ (111)$=5.6$-$6.4$ eV, $\phi$ $(110)=5.4$-$5.7$ eV
(reconstruction-dependent), and $\phi$ $(100)=5.4$-$5.9$ eV, with measurement
uncertainties $\sim 0.1$ eV. Derry, Kern, and Worth (2015) These ranges of
values are consistent with well-converged density functional theory (DFT)
calculations. Schultz  _et al._ (2021) Smoluchowski explained $\phi$
anisotropy to result from charge spreading and localization between the
differing densities of surface atoms on crystal faces. Smoluchowski (1941)
Recently, an extensive DFT study of this effect across many elements was
published in ref. [Tran  _et al._ , 2019]. Some other notable approaches to
relate $\phi$ to varying crystal orientations include work by Brodie et al.,
that relates anisotropy to various other properties (effective mass, surface
relaxation), and Fazylov, relating anisotropy to dielectric function and
plasmon dispersions. Brodie (1995); Fazylov (2014) We find that Smoluchowski’s
ansatz is an intuitively appealing approach that reasonably estimates $\phi$
anisotropy.

To measure $\phi$ variation owing to crystalline anisotropy, Besocke et al.
studied surfaces oriented between principal facets, using carefully machined
macroscopic monocrystalline surfaces curving smoothly away from (111), Fig. 1
(b). Besocke reported that $\phi$ decreased linearly by $\sim 0.05$ eV/degree
of surface slope for up to several degrees. Besocke explained the result using
a Smoluchowski-type model in which surface atomic steps (inferred from
electron diffraction) introduce electric dipoles ($\sim 0.6$ D/edge atom) that
counteract the overall surface dipole, thereby lowering $\phi$. Later,
providing microscopic data substantiating Besocke’s model, Jia et al., and
Park et al., used scanning probe methods to resolve individual step dipoles on
Pt and other metals, but they did not provide a magnitude for the Pt(111)
atomic step edge dipole moment. Jia  _et al._ (1998); Park  _et al._ (2005)

Here, we show how nanoscale surface roughness arising from polycrystalline
textures impacts the local $\phi$ of sputter-deposited polycrystalline Pt thin
films with (111) texture. Clean polycrystalline metal thin films with textured
granular structure serve as a controlled and conceptually straightforward
departure from Besocke’s ideal oriented single crystals toward ensemble
disorder by introducing roughness with varying small-angle surface slopes,
$S$, and granular tilts, $T$ as depicted in Fig. 1 (c). In effect, a sputter-
deposited polycrystalline film serves as an ensemble of tiny monocrystalline
samples, each with a few vicinal surfaces and varying atomic step densities
that depend upon both local slope and granular tilts distributed around the
mean surface normal, Fig. 1 (c) and (d). Using spatially-resolved
photoemission threshold mapping, we find a mean $\phi=5.70$ eV spread by $\pm
0.03$ eV over a spatial distribution of many grain surfaces with random small-
angle (few degree) (111)-vicinal orientations. Scanning tunneling microscopy
(STM) indicates that the surfaces are predominately (111) step-terrace
structure with widely-varying densities of atomic steps at numerous sites.
Therefore, leveraging Besocke’s result, we explain the spatially-varying work
function using a Smoluchowski-smoothing model in which local $\phi$ decreases
in direct proportion to the local density of atomic steps (electrical
dipoles). Besocke, Krahl-Urban, and Wagner (1977) Since measuring atomic step
densities over meaningfully large-area statistical samples of numerous
micrometer-sized grains is not practical, we estimate atomic step distribution
statistics from slopes, $S$, via atomic force microscopy (AFM) data, and
tilts, $T$, via electron backscatter diffraction (EBSD) data. To estimate the
atomic step dipole distributions resulting from convolved effects of $S$ and
$T$, we find that a surprisingly straightforward estimate treating $S$ and $T$
as independent random variables leads to a reasonable description of the
photoemission $\phi$ properties. Our model is a straightforward basis for
understanding the measured spatially inhomogeneous $\phi$, while capturing the
salient low $\phi$ values likely to be important for initial electron emission
and simultaneously matching the narrow peak ($\pm 0.03$ eV). The model
estimates the step dipole strength to be $0.12$ D/atom, which is reasonable
and similar to values from prior works. Besocke, Krahl-Urban, and Wagner
(1977) We show that Smoluchowski’s smoothing ansatz, substantiated by
Besocke’s mesocale studies of "ideal" specially-prepared surfaces and Jia and
Park’s microscopic atomic-step dipole measurements, create a basis to
understand the impact of surface roughness on local $\phi$ for "realistic"
surfaces prepared by commonplace industrial methods that yield ensemble
disorder.

Figure 1:  Schematic relationships between atomic steps and crystal surfaces
with increasing complexity. (a) An oriented monocrystalline surface or step-
free (hkl) facet. (b) A curved sample with varying slope, $S$, used by Besocke
et al. to investigate relationships between atomic step density and work
function. (c) The polycrystalline surfaces in our study have locally varying
slopes and nonzero varying tilt, $T$, angles with respect to a global (mean)
surface normal, $\hat{n}$. (d) In a surface region with slope and tilt, the
atomic step density is proportional to the difference between $S$ and $T$.

##  II  Experiment: materials synthesis and structure

We sputter-deposited polycrystalline Pt (poly-Pt) thin film samples following
the procedure in ref. [Shelton  _et al._ , 2012]. Samples are $\sim 90$ nm-
thick Pt on a $40$ nm-thick ZnO adhesion layer, on $400$ nm of thermal SiO2 on
Si ($100$). We annealed the samples for 1 hour in air at 900o C to create a
coarse polycrystalline microstructure, with several-hundred-nanometer diameter
grains as determined by AFM and transmission electron (TEM).

We begin by characterizing the structure, crystallographic orientation of
poly-Pt grains, and topography of grain faces, since these properties will be
used to explain $\phi$ spatial variation in a model presented in a subsequent
section. Figure 2 (a-b) shows cross-sectional TEM images of our sample. The
top-most layer is the Pt film, with a ZnO adhesion layer and thermal SiO2
layer beneath it. In Fig. 2 (a), each Pt grain displays internal structures
with slightly different intensities of grey. The high-resolution TEM image
[Fig. 2 (b)] shows periodic intensity variation due to electron beam
diffraction from crystal planes, evincing the crystallinity of individual
grains. Similar intensity variation observed across multiple grains suggests
that they may be textured similarly.

We measure ensemble distributions of grain crystal orientations (tilt) with
respect to the substrate normal using EBSD. Figure 2 (c) presents plan-view
scanning electron microscopy (SEM) images (in gray scale) of poly-Pt films.
Images based on band contrast show boundaries between crystalline Pt grains
from which individual grains can be identified. Figure 2 (c) (color panels
Z,X,Y) are inverse pole figure maps of local crystallographic orientations
with respect to the sample substrate normal (Z), and two in-plane, orthogonal
directions (X,Y). The predominantly blue color with minimal variation in Z
indicates that Pt grains are predominantly (111)-oriented. In Fig. 2 (d), a
histogram of surface-normal (Z) angles indicates a small median crystal tilt
angle of $3.1^{\circ}$ with a standard deviation $1.5^{\circ}$. In-plane
components of EBSD data (Fig. 2 (c): X,Y), on the other hand, show the
tendency for crystal domains to orient randomly in the azimuth.

Figure 2:  Grain structure and tilts of poly-Pt. (a) Cross-sectional TEM image
of the thin film stack. The topmost layer in (a) is the Pt film. (b) High-
resolution TEM image of the poly-Pt film. Periodic intensity in the image
(along the horizontal direction of the image) indicates the crystallinity of
Pt grains. (c) EBSD image based on Kikuchi band contrast displaying the
boundaries between crystalline Pt grains and inverse pole figure maps (color)
of orientations with respect to the sample substrate normal (Z) and two in-
plane, orthogonal directions (X,Y). The color corresponds to the
crystallographic orientations specified by the color key shown in the upper
right. (d) Grain tilt histogram showing tilt with respect to the surface
normal. The inset shows the histogram normalized to a probability density,
$P_{T}(T)$, and plotted on a semilog scale to show the extent of the
distribution. The units on the vertical axis are (degree)-1.

To measure surface topography and slope, $S$, we performed AFM, as well as
STM. Using STM, we resolved the surface step-terrace structure at up to the
few-hundred-nanometer distances, while AFM is used to sample topography of
grain ensembles over $\mu$m lengths. In Fig. 3 (a), we show an AFM surface
topography image containing many grains. At many locations on the surface, STM
topography images, Fig. 3 (b), reveal atomic step-terrace structure and step
heights ($2.3$Å) consistent with low-angle vicinal surfaces on Pt($111$)
grains. Stipe  _et al._ (1997) This observation is consistent with the AFM
topography and the TEM data (Fig. 2). Prior experiments on Pt and numerous
other fcc metals have shown that at small-angle ($\sim 1^{\circ}$) grain tilts
and slopes near a high-symmetry direction, the surface is dominated by
terraces of the nearby high-symmetry facet separated by single-height atomic
steps. Bott, Michely, and Comsa (1992); Jeong and Williams (1999) Consistent
with the rolling topography in AFM, and grain tilts, the terrace size varies
significantly with spatial location, as indicated even over the 10-nm-
distances in Fig. 3 (b). Figures 3 (c) and (d) show a map and histogram of the
slope angle, $S$, calculated from the AFM topography. The distribution has a
median of $3.1^{\circ}$ and a standard deviation of $2.2^{\circ}$, underlining
that a majority of surfaces are vicinal to (111), and unlikely to be other
facets. The $S$ histogram is shown normalized to a probability density,
$P_{S}(S)$, in the inset. The range of small slope angles is consistent with
step-terrace structure in STM.

Figure 3:  Poly-Pt surface topography via AFM and STM, along with processed
images showing surface slope distribution. (a) Surface topography of poly-Pt,
measured by AFM. (b) An example of the (111) atomic step structure over the
100-nm-scale resolved by STM. (c) A calculation of the local surface slope
angle, $S$ (c) and a slope histogram (d). The inset shows the histogram
normalized as a probability density, $P_{S}(S)$. The units on the vertical
axis are (degree)-1.

##  III  Experiment: photoemission work function maps

The grain-to-grain $T$ variation (Fig. 2 (c)), along with the variation of $S$
over granular vicinal surfaces (Fig. 3 b)), is anticipated to produce
spatially inhomogenous electronic property variations. To reveal the spatial
$\phi$ distribution, we utilize photoemission electron microscopy (PEEM)
equipped for spectroscopic imaging under ultrahigh vacuum ($\leq 10^{-9}$
Torr). Sharma  _et al._ (2021) A brief anneal ($900^{\circ}$, 3 minutes) in
ultrahigh vacuum conditioned the sample surface for PEEM measurement. Figure 4
shows a PEEM $\phi$ map, as well as its distribution. The $\phi$ map, Fig. 4
(a), varies spatially at sub-micron length scales, consistent with variation
with grains and sub-grain vicinal faces. Fig. 4 (a) also shows some areas with
relatively low $\phi$ (e.g. black arrows), which manifest after annealing in
ultrahigh vacuum (UHV). AFM of the surface after PEEM measurement confirmed
that these low $\phi$ regions corresponded to pits in the poly-Pt film, where
Pt had been removed via dewetting effects. Jahangir  _et al._ (2014)
Accordingly, we excluded pixels of dewetted areas from the PEEM $\phi$
distributions shown in Fig. 4 (b).

The $\phi$ distribution has a mean of $5.70$ eV and a standard deviation of
$\pm 0.03$ eV. To better estimate $\phi$, we accounted for the Schottky effect
originating from the electric field $\sim 10^{7}$ V/m, due to the measurement
apparatus, at the sample surface, which lowers the surface barrier to emission
by $\sim 0.098$ eV. Schottky (1914); Renault  _et al._ (2006) For comparison,
prior reports on precisely (111)-aligned single crystals give a range
$\phi=5.6-6.4$ eV using comparable photoemission and analysis techniques, with
uncertainties of $\sim\pm 0.1$ eV.

Figure 4:  Photoemission threshold maps of poly-Pt. (a) Local $\phi$ map. The
field of view is $24\mu$m. $\phi$ was obtained from local photoemission
threshold, taking into account the Schottky effect originating from the
instrument’s electric field. Black arrows denote pitted areas where Pt had
been removed during annealing and that appear as low-$\phi$ areas. (b)
Histograms of work function values, normalized as a probability density,
$P_{\phi}$, in which bin sizes correspond to uncertainties in measured values.
The black solid line is a best-fit generated by our model described in the
discussion. Inset: $P_{\phi}$ plotted on a semilog scale to emphasize the
notable long-tail skew of the distribution. The units of the vertical axis are
(eV)-1.

##  IV  Discussion

In relation to field emission, salient properties in Fig. 4, are: (1) that the
submicron-scale textures evident in the $\phi$ spatial map, Fig. 4 (a),
indicate that $\phi$ varies spatially in connection with varying local surface
properties, and (2) that the $\phi$ distribution is skewed with a tail toward
low $\phi$ far below the mean value. In this section, we develop a model that
simultaneously approximates both the $\phi$-spatial variation in connection
with surface structure, and the long-tail skew distribution to low
$\phi$-values.

Motivated by Besocke et al.’s work connecting mesoscale local $\phi$ with
local surface atomic step density, substantiated by microscopic observations
of individual atomic-step dipoles on Pt(111) and several other fcc metals, we
utilize a Smoluchowski-type model of local $\phi$, relative $\phi$ (hkl) (here
for flat Pt(111)) to approximate $\phi$ spatial variation due to surface
morphology. Besocke, Krahl-Urban, and Wagner (1977); Park  _et al._ (2005);
Renault  _et al._ (2006) Our model expresses the microscopic viewpoint that
work function variation $\Delta\phi=\phi-\phi$ (hkl) is proportional to the
local atomic step density, $N$, and the strength of the atomic step dipole,
$\mu$. Besocke’s expression relating local $\phi$ of a stepped vicinal surface
to $\phi$(hkl) of the nearby high-symmetry plane is $\Delta\phi=-300\times
10^{-18}$ $4$ $\pi$ $\mu$ $N$, in units of eV, where $N$ and $\mu$ are
expressed in units cm-1 and Debye (D)/cm, respectively. Besocke, Krahl-Urban,
and Wagner (1977) The local atomic-step density, $N(S,T)=\sin(T-S)/a_{111}$,
on a surface region with local tilt $T$ and slope $S$. Here,
$a_{111}=a_{\circ}/\sqrt{3}=0.226$ nm is the distance between (111) planes,
with Pt’s fcc lattice constant $a_{\circ}=0.392$ nm. At small $S$ and $T$
angles as in our measurements, it is very accurate ($\lesssim$ 10$\%$ errors)
to use the small-angle approximation, i.e. that N$\simeq(T-S)/a_{111}$.

A key motivation for our overall approach is that STM images, Fig. 3 (b),
indicate that many sites on our surfaces have distinctly resolved, smoothly
varying, (111) step-terrace structure. Practical experimental speed limits
restrict STM to mapping several sites with $\sim$100-nm-scale images, i.e.
only providing exact descriptions of faces on several individual grains.

To approximate mesoscale ensemble properties across many grains, we estimate
the atomic-step density distribution, $N$, by computing it from from the
slope, $P_{S}$, and tilt, $P_{T}$, distributions (panel (d) of Figs. 2 and 3).
To do so, we must make a choice of how to estimate the random variable $T-S$.
Although $S$ and $T$ have known underlying driving forces (e.g. surface energy
anisotropy) that favor correlation (facets), we make the approximation that
$S$ and $T$ are random, independent variables. Effectively, we are assuming
that surface energy anisotropy is not so large (relative to $k_{B}T$) as to
drive well-developed step-free facets, which is consistent with the
observation of densely-stepped low-angle (vs. zero-angle) vicinal surfaces in
many STM images (see Fig. 3 (b)). In the approximation that $S$ and $T$ are
uncorrelated random variables, the difference $d=T-S$ is calculated by
discrete convolution, and the atomic step density distribution is

$N(d)=d/a_{111}=a_{111}^{-1}\sum_{k=-\infty}^{\infty}P_{S}(k)P_{T}(d+k)$

,

such that,

$\Delta\phi=-300\times 10^{-18}4\pi\mu N$

$=-300\times 10^{-18}4\pi\mu
a_{111}^{-1}\sum_{k=-\infty}^{\infty}P_{S}(k)P_{T}(d+k),$

where $P_{S}(S)$ and $P_{T}(T)$ are the measured probability densities for $S$
and $T$, and $k$ ranges over all values of $T$ and $S$.

We fit our measured $\phi$ distribution using the $\Delta\phi$ expression
above using $\mu$ as the fit parameter to match the distribution width and
$\phi(111)$ to set the center position. The fit is shown in Fig. 4 by a solid
black line. Both qualitative and quantitative features of the $\phi$
distribution are reasonably captured by the fit. First, and most significant
for field emission, both the spatially patchy grain-to-grain $\phi$-variation
evident in Fig. 4 (a), and the peculiar $\phi$ distribution, Fig. 4 (b), with
the relatively narrow $\sim 0.03$ eV "head" and skewed long-tail to much lower
$\phi\sim 5.4$ eV can be understood to directly correspond to the spread of
atomic step densities anticipated from the skewed $P_{S}$ and $P_{T}$
distributions. The best-fit values for the distributions are $\mu=0.12$ D/atom
($\mu\simeq 4\times 10^{6}$ D/cm-1, weakly dependent upon step
orientation/atomic density), and $\phi(111)=5.69$ eV.

Our atomic step dipole is lower than the value ($0.6\pm 0.1$ D/edge atom)
determined by Besocke et al., using very different material preparations and
macroscopic area-averaging measurement methods. Besocke, Krahl-Urban, and
Wagner (1977) It is worth pointing out that Besocke’s $\mu$ value is too large
to reconcile with other data, e.g. for just $S\sim 7^{\circ}$ slope it gives
$\Delta\phi\sim 0.3$ eV reduction in work function, which is comparable to
reported variations between principal high-symmetry facets with a range $\sim
0.5$ eV between (110) and (111) facets. Our result ($\sim 0.06$ eV for
$7^{\circ}$ from (111)) is more straightforward to reconcile within the $0.5$
eV full range of facet-to-facet work function anisotropy. Our value also
compares reasonably in a broader context of various results obtained for other
materials, which have yielded fairly wide-ranging values for step-edge dipoles
for most fcc metals. Jia  _et al._ (1998); Besocke, Krahl-Urban, and Wagner
(1977); Park  _et al._ (2005) For example on Au(111), Jia et al. found a step
dipole of 0.16 D/edge atom, while Besocke reports a value of $0.2$-$0.27$
D/edge atom, and Park measure 0.45 D/edge atom. Jia  _et al._ (1998); Besocke,
Krahl-Urban, and Wagner (1977); Park  _et al._ (2005) The step dipole strength
produced by our model fit is reasonable in comparison to those reported
values.

##  V  Summary

We report and analyze $\phi$ for (111)-textured polycrystalline Pt thin films
with grain-to-grain ensemble structural disorder. We measure a mean
$\phi=5.70\pm 0.03$ eV, which is consistent with reported $\phi=5.6$-$6.4$ eV
for intentionally aligned (111) facets. To explain the spatially varying,
skewed $\phi$, we develop a Smoluchowski model that accounts for the influence
of atomic step edge dipoles $\mu$. Connecting work function with step dipoles
is motivated by prior studies indicating that Pt, like many other fcc metals,
has step dipoles, and our observation of spatially varying vicinal (111)
atomic step densities in STM images of our surfaces. Our analysis yields a
Pt(111) atomic step dipole $\mu=0.12$ D/edge atom. Toward models of field
emission and related phenomena, e.g. vacuum arc initiation, that employ
realistic, rough, polycrystalline electrodes, two salient traits of our $\phi$
map and model are: (1) $\phi$ spatial variations in connection with surface
grain/facet structure, and (2) skew to lower $\phi$ values down to 5.4 eV. Our
analysis provides a good explanation of the granular ensemble $\phi$
properties, including the spatial variation and long-tail skew to lower
$\phi$. Future work should include coincident-site measurements to directly
correlate between local grain tilts (T), film topographies (S), and local
$\phi$.

###### Acknowledgements.

This work is funded by Sandia Laboratory Directed R&D (LDRD) program and also
by an Air Force Office of Scientific Research (AFOSR) program. Part of the
work is performed at the Center for Integrated Nanotechnologies, a U.S.
Department of Energy (DOE) Office of Science User Facility. Sandia National
Laboratories is a multi-mission laboratory managed and operated by National
Technology and Engineering Solutions of Sandia, LLC., a wholly owned
subsidiary of Honeywell International, Inc., for the U.S. Department of
Energy’s National Nuclear Security Administration under contract DE-NA0003525.
This paper describes objective technical results and analysis. Any subjective
views or opinions that might be expressed in the paper do not necessarily
represent the views of the U.S. Department of Energy or the United States
Government.

## Data Availability Statement

The data that support the findings of this study are available within the
article.

## Conflict of Interest Statement

The authors have no conflicts to disclose.

## References

  * Jensen (2018) K. L. Jensen, “A tutorial on electron sources,” IEEE Transactions on Plasma Science 46, 1881–1899 (2018).
  * Boxman, Sanders, and Martin (1996) R. L. Boxman, D. M. Sanders, and P. J. Martin,  _Handbook of vacuum arc science & technology: fundamentals and applications_ (William Andrew, 1996).
  * Tang  _et al._ (2016) W. W. Tang, D. A. Shiffler, J. R. Harris, K. L. Jensen, K. Golby, M. LaCour, and T. Knowles, “Field emission characteristics of a small number of carbon fiber emitters,” AIP Advances 6, 095007 (2016).
  * Fowler and Nordheim (1928) R. H. Fowler and L. Nordheim, “Electron emission in intense electric fields,” Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 119, 173–181 (1928).
  * Forbes and Deane (2007) R. G. Forbes and J. H. Deane, “Reformulation of the standard theory of Fowler–Nordheim tunnelling and cold field electron emission,” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 463, 2907–2927 (2007).
  * Jensen (2019) K. L. Jensen, “A reformulated general thermal-field emission equation,” Journal of Applied Physics 126, 065302 (2019).
  * Forbes (2001) R. G. Forbes, “Low-macroscopic-field electron emission from carbon films and other electrically nanostructured heterogeneous materials: hypotheses about emission mechanism,” Solid-State Electronics 45, 779–808 (2001).
  * Lecoeur, Bellier, and Koehler (1990) J. Lecoeur, J. Bellier, and C. Koehler, “Comparison of crystalographic anisotropy effects on potential of zero charge and electronic work function for gold $\\{$111$\\}$,$\\{$311$\\}$,$\\{$110$\\}$ and $\\{$210$\\}$ orientations,” Electrochimica acta 35, 1383–1392 (1990).
  * Vanselow and Li (1992) R. Vanselow and X. Li, “The work function of kinked areas on clean, thermally rounded Pt and Rh crystallites: Its dependence on the structure of terraces and edges,” Surface science 264, L200–L206 (1992).
  * Smoluchowski (1941) R. Smoluchowski, “Anisotropy of the electronic work function of metals,” Physical Review 60, 661 (1941).
  * Strayer, Mackie, and Swanson (1973) R. Strayer, W. Mackie, and L. Swanson, “Work function measurements by the field emission retarding potential method,” Surface Science 34, 225–248 (1973).
  * Besocke, Krahl-Urban, and Wagner (1977) K. Besocke, B. Krahl-Urban, and H. Wagner, “Dipole moments associated with edge atoms; a comparative study on stepped Pt, Au and W surfaces,” Surface Science 68, 39–46 (1977).
  * Haas and Thomas (1969) G. A. Haas and R. E. Thomas, “Distribution of crystal orientation and work function in tungsten ribbons,” Journal of Applied Physics 40, 3919–3924 (1969).
  * Schultz  _et al._ (2021) P. A. Schultz, H. P. Hjalmarson, M. Berg, E. Bussmann, D. A. Scrymgeour, T. Ohta, and C. H. Moore, “First-principles calculations of metal surfaces. ii. properties of low-index platinum surfaces toward understanding electron emission,” Physical Review B 103, 195427 (2021).
  * Derry, Kern, and Worth (2015) G. N. Derry, M. E. Kern, and E. H. Worth, “Recommended values of clean metal surface work functions,” Journal of Vacuum Science & Technology A: Vacuum, Surfaces, and Films 33, 060801 (2015).
  * Tran  _et al._ (2019) R. Tran, X.-G. Li, J. H. Montoya, D. Winston, K. A. Persson, and S. P. Ong, “Anisotropic work function of elemental crystals,” Surface Science 687, 48–55 (2019).
  * Brodie (1995) I. Brodie, “Uncertainty, topography, and work function,” Physical Review B 51, 13660 (1995).
  * Fazylov (2014) F. Fazylov, “Macroscopic theory of the electron work function in solids,” Philosophical Magazine 94, 1956–1966 (2014).
  * Jia  _et al._ (1998) J. Jia, K. Inoue, Y. Hasegawa, W. Yang, and T. Sakurai, “Variation of the local work function at steps on metal surfaces studied with stm,” Physical Review B 58, 1193 (1998).
  * Park  _et al._ (2005) J. Y. Park, G. Sacha, M. Enachescu, D. Ogletree, R. Ribeiro, P. C. Canfield, C. J. Jenks, P. A. Thiel, J. Sáenz, and M. Salmeron, “Sensing dipole fields at atomic steps with combined scanning tunneling and force microscopy,” Physical review letters 95, 136802 (2005).
  * Shelton  _et al._ (2012) C. T. Shelton, P. G. Kotula, G. L. Brennecka, P. G. Lam, K. E. Meyer, J.-P. Maria, B. J. Gibbons, and J. F. Ihlefeld, “Chemically homogeneous complex oxide thin films via improved substrate metallization,” Advanced Functional Materials 22, 2295–2302 (2012).
  * Stipe  _et al._ (1997) B. Stipe, M. Rezaei, W. Ho, S. Gao, M. Persson, and B. Lundqvist, “Single-molecule dissociation by tunneling electrons,” Physical review letters 78, 4410 (1997).
  * Bott, Michely, and Comsa (1992) M. Bott, T. Michely, and G. Comsa, “The homoepitaxial growth of Pt on Pt (111) studied with stm,” Surface science 272, 161–166 (1992).
  * Jeong and Williams (1999) H.-C. Jeong and E. D. Williams, “Steps on surfaces: experiment and theory,” Surface Science Reports 34, 171–294 (1999).
  * Sharma  _et al._ (2021) P. A. Sharma, T. Ohta, M. T. Brumbach, J. D. Sugar, and J. Michael, “Ex situ photoelectron emission microscopy of polycrystalline bismuth and antimony telluride surfaces exposed to ambient oxidation,” ACS Applied Materials & Interfaces 13, 18218–18226 (2021).
  * Jahangir  _et al._ (2014) S. Jahangir, X. Cheng, H. Huang, J. Ihlefeld, and V. Nagarajan, “In-situ investigation of thermal instabilities and solid state dewetting in polycrystalline platinum thin films via confocal laser microscopy,” Journal of Applied Physics 116, 163511 (2014).
  * Schottky (1914) W. Schottky, “Über den einfluss von strukturwirkungen, besonders der thomsonschen bildkraft, auf die elektronenemission der metalle,” Physik. Zeitschr 15, 872–878 (1914).
  * Renault  _et al._ (2006) O. Renault, R. Brochier, A. Roule, P.-H. Haumesser, B. Krömker, and D. Funnemann, “Work-function imaging of oriented copper grains by photoemission,” Surface and Interface Analysis 38, 375–377 (2006).