Work Function and the electron escape energy
Work Function is often confused with escape energy of an electron.
Those two concepts are related but not the same.
Escape energy
The escape energy of an electron from bulk of a conductor through a given surface is the minimum energy that needs to be delivered to an electron near the Fermi level of the conductor to move it outside of the material.
This energy is affected by the surface potentials created by:
- surface dipole layers and/or by
- surface unbalanced charge densities
because both of those quantities do shift the Fermi level of the conductor with respect to the lowest vacuum level. One can see that the escape energy is a function of both: the properties of the conductor’s material and the conditions (extra charges) it is currently in. In particular, if the conductor is positively charged, the escape energy increases because the escaping electron must overcome the pull of the positive charge.
The Work Function
The Work Function, on the other hand, is a property of the conducting material alone. Namely, the Work Function is equal to the electron escape energy through a surface, only if the excess charge density of that surface is zero.
This distinction between Work Function and the escape energy has profound consequences for measurement techniques that can be employed to determine the Work Function of a material.
In order to perform such a measurement of a Work Functions through a surface, one needs to make sure that that surface is not charged. In order to remove excess charge from such a surface an appropriate electric voltage $V^0$ needs to be applied between the material and a ground. The voltage shifts the Fermi level of the material with respect to the vacuum energy levels. See the Electrochemical vs chemical potential post. Thus, the Work Function through a given surface can be defined as the distance in energy between the material’s Fermi level and the bottom of vacuum band
$$
\Phi = E_{\mbox{vac}} - E_F
$$ provided that the surface has no excess charge i.e., a proper potential is applied to the conductor. This way one maintains different Work Functions through different surfaces of the same body. It is possible because the Fermi level $E_F$ is a function of the voltage $V^0$ applied to remove the excess charge from the surface of interest
$$
\Phi = E_{\mbox{vac}} - E_F\left( V^0 \right ).
$$ If one assumes that the $E_{\mbox{vac}}=0$ convention then the last expression simplifies into
$$
\Phi = -E_F\left (V^0\right )
$$ and the Work Function is given by just the Fermi level.
This simple expression captures all the nuanced aspects of the Work Functions concept. It includes effects related to a double layers that might be present at the surface or adsorbates modifying the surface charge distribution.
The double layer itself has zero electric charge but it generates an electric potential that affects the surface charge distribution on conductors.
IUPAC Recommendations 1985
The zero charge density approach to the Work Function is honored in the old IUPAC “electron work function” definition
It was put together by Sergio Trassati and is a rare example of deep understanding of the subject.
Incorrect definitions of Work Function
C. Kittel Introduction to Solid State Physics
In the 8th edition of Charles Kittel’s famous Introduction to Solid State Physics, page 494 introduces the Work Function as
the difference in potential energy of an electron between the vacuum level and
the Fermi level. The vacuum level is the energy of an electron at rest at a point
sufficiently far outside the surface so that the electrostatic image force on the
electron may be neglected-more than 100 Å from the surface. The Fermi
level is the electrochemical potential of the electrons in the metal.
This approach admits that the Fermi level, defined as electrochemical potential, is sensitive to overall excess charge of a material. Thus it confuses the electron escape energy with the Work Function.
Wikipedia (as of March 31st, 2025)
Wikipedia’s definition follows that of Kittel’s from the section above.
The wikipedia authors are undeterred by the fact that the potential $\phi$ can be non-uniform across even a single crystallographic face and the Work Function, according to proposed definition, is non-uniform for that face as well. See the discussion under the next erroneous definition.
New IUPAC Recommendations 2021
Regrettably, IUPAC has regressed from correct approach in Recommendations 1985 still available as “electron work function” to the definition cited below under the term “work function”
According to this definition, the electron removed from the material is still interacting with its surface. That means that the attractive image charge force of the material has not been overcome yet and there is no equilibrium reached! So, Work Function is a non-equilibrium quantity all of a sudden. It its absurd.
Lets examine it closer.
The expression for the Work Function written in less confusing notation is
$$
\Phi = -e \phi_{out} - E_F,
$$ where $\phi_{out}$ is the electric potential just outside the surface.
Lets consider the simplest case of a uniform monocrystal conductor, like tungsten crystal in this post, . The crystal has three pairs of crystallographic faces, each pair with different Work Function.
In termodynamical equilibrium, the Fermi level is uniform in the bulk of the crystal and given by
$$
E_F = \text{chemical potential} - e\phi(q),
$$ where the $\phi(q)$ is the uniform electric potential in the bulk and $q$ is a total excess charge on the surfaces of the tungsten. If the total excess charge $q=0$ then also $\phi(0) = 0$ and the Fermi level coincides with the chemical potential. No excess charge on the tungsten does not imply the zero charge density on its faces! To the contrary, faces with higher Work Functions will have negative charge densities and faces with the lower Work Functions will have positive charge densities.
The Work Function now is
$$
\Phi(f) = -e(\phi_{out}(f) - \phi(q)) - \text{chemical potential}.
$$ Here $f$ denotes the dependence of the electric potential just outside of the surface on the particular face of the crystal.
Now one can adjust the total charge $q$ on the tungsten in such a way that a charge density on one pair of crystal faces vanishes and
$$
\Phi(f) = e\phi(q^0_f) - \text{chemical potential}
$$ with $q^0_f$ being the total charge that removes the charge density of both faces $f$ altogether.
If the face had positive charge density for $q=0$ then the negative excess charge should be added. The negative excess charge will generate a negative electric potential $\phi(q)$. The measured Work Function through the face $f$ is indeed lower than the Work Function through the faces with negative charge densities at $q=0$.
Thus, the Work Function formula in UIPAC 2021 definition might be accidentally correct if one
- forces the charge neutrality of the face of interest and
- removes the electron far away (to infinity indeed) from the material. 😄
However, the formula is nonsense as presented by UIPAC even if one forgives the “nearby the solid surface” condition.
The electric potential $\phi_{out}$ (denoted by $\Psi$ in the original definition) is non-uniform within a crystal face. The potential $\phi(q)$ in the bulk must be uniform. Thus, unless $q = q^0_f$, the difference
$$
\phi_{out}(f) - \phi(q)
$$ is non-uniform and so is the Work Function across a given face in the UIPAC definition.
