Theory of thermionic emission from a two-dimensional conductor and its application to a graphene-semiconductor Schottky junction
Maxim Trushin




Centre for Advanced 2D Materials, National University of Singapore, 6 Science Drive 2, Singapore 117546






The standard theory of thermionic emission developed 
for three-dimensional semiconductors does not apply to
two-dimensional materials even for making qualitative predictions
because of the vanishing out-of-plane quasiparticle velocity.
This study reveals the fundamental origin of the out-of-plane charge carrier motion 
in a two-dimensional conductor due to the finite quasiparticle lifetime and huge uncertainty of
the out-of-plane momentum. The theory is applied to  
a Schottky junction between graphene and a bulk semiconductor
to derive a thermionic constant, which, in contrast to the 
conventional Richardson constant, is determined by the 
Schottky barrier height and Fermi level in graphene.



Theory of thermionic emission from a two-dimensional conductor and its application to a graphene-semiconductor Schottky junction
    Maxim Trushin
    December 30, 2023
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§ INTRODUCTION



Microelectronic devices employing thermionic emission over the Schottky barrier between graphene and silicon
<cit.>
or another bulk semiconductor
<cit.>
have experienced a boom over the last few years, see Refs. di2016graphene,xu2016interface,xu2016contacts for the most recent reviews.
Graphene-semiconductor Schottky junctions possess rectification properties 
<cit.>
and can be used in photodetection
<cit.>
as well as in solar energy harvesting.<cit.>
The Schottky barrier height, Φ_B,  is determined by the difference between the work function of graphene and semiconductor affinity,
see Fig. <ref>. Since the work function depends on the Fermi energy E_F (which is tunable in graphene by an external electric field),
the barrier height depends on the bias voltage across the junction.<cit.>

The thermionic current density through an ideal Schottky junction (i.e. no thermionic field emission, no series resistance etc.) then reads <cit.>

    j=j_0 (e^eV/k_B T - 1 ),

where e is the elementary charge, k_B is the Boltzmann constant, V is the bias voltage, and T is the electron temperature.
The reverse saturation current density, j_0, flows across the junction
when a reverse bias voltage (V<0) pushes the electrons from graphene over the Schottky barrier to the semiconductor side.
The goal of this Letter is to derive j_0 for a two-dimensional (2D) conductor and apply the obtained formula to the 
graphene-semiconductor junction shown in Fig. <ref>.

The theoretical difficulties in modeling thermionic emission from 2D materials have been recognized only recently.
<cit.>
Indeed, thermionic emission from a surface of a bulk material is due to
(i) electron energy high enough to overcome the work function difference at the interface,
and (ii) non-zero electron velocity normal to the surface.
Electron kinetic energy can be controlled equally well in three-dimensional (3D)
as well as in  2D conductors by heating.
However, electrons in a 2D conductor (like graphene, which consists of only a surface) do not possess
an out-of-plane velocity and it is not clear where it should come from.
In attempt to circumvent this issue, Sinha and Lee have introduced 
an empirical carrier injection rate <cit.> (see also Refs. wei2013electron,varonides2016combined) whereas
Liang and Ang <cit.> have assumed a certain energy dispersion for the out-of-plane carrier motion,
see also Refs. liang2016modified,ang2016current,ang2016current,upadhyay2017photo,misra2017thermionic,ang2017generalized for further elaboration of their approach.
Despite their reasonable agreement with experiments,<cit.> the models introduced until now
are not self-contained (i.e. they require an external parameter) and,
most importantly, do not explain why the 2D carriers do actually move out of plane.

Hence, the fundamental origin of the out-of-plane carrier velocity in thermionic emission from a 2D conductor is still unclear.
Here, I fill this gap using the concept of a hot electron liquid confined in a 2D plane.








§ QUASIPARTICLE LIFETIME AND OUT-OF-PLANE VELOCITY

The quasiparticle concept is a cornerstone of the electron liquid theory at finite temperatures:
Any given charge carrier does not stay in its state forever but may fall down or be excited to any other empty state
provided by energy and momentum conservation.
Hence, a quasiparticle state with energy near the Fermi level E_F
possess a finite lifetime given by<cit.> τ_E_F∼ħ E_F/(k_B T)^2.
This estimation is valid for any normal metal, no matter 2D or 3D,<cit.>
as long as  E_F≫ k_B T. In intrinsic graphene, however,
the Fermi surface shrinks to a single point that results
in a 2D Dirac liquid <cit.> with the quasiparticle lifetime τ_0 ∼ħ/(k_B T).
This lifetime can be seen as a mean time between the quasiparticle 
creation and annihilation events thanks to a time-independent perturbation.

Within this lifetime, τ, a quasiparticle may acquire a certain perturbation-independent energy difference Δ E
suggested by the energy uncertainty relation Δ E = ħ/τ.
In contrast to Δ E, the out-of-plane momentum uncertainty Δ p_z ∼ħ /Δ z is a constant determined solely
by the single atomic layer thickness Δ z being of the order of 1 A.
Thanks to the finite Δ p_z, a quasiparticle may acquire 
a finite out-of-plane velocity, v_z = Δ E/Δ p_z.<cit.>
Assuming that the initial out-of-plane velocity is zero one can write the following
formula for the velocity v_z that a quasiparticle acquires within the quasiparticle lifetime τ:

    v_z Δ p_z ∼ħ/τ.

Equation (<ref>) shows that the finite quasiparticle momentum uncertainty and lifetime are both necessary to correctly evaluate the out-of-plane velocity.
As long as the quasiparticle momentum may fluctuate within the uncertainty interval Δ p_z, the quasiparticle acquires the finite velocity v_z.
The shorter quasiparticle lifetime is the higher velocity the quasiparticle has.
The out-of-plane velocity vanishes only at absolute zero temperature when τ→∞.
In the case of zero momentum uncertainty (bulk limit, Δ z→∞),
the out-of-plane velocity becomes disentangled from the energy uncertainty<cit.> and is given by 
the standard relation v_z=p_z/m^* in terms of an effective mass m^*.




The major merit of Eq. (<ref>) is to evaluate the out-of-plane quasiparticle velocity 
without the ill defined quasiparticle injection time or out-of-plane  effective mass. 
For quasiparticles in intrinsic graphene with τ=τ_0 and T of a few hundreds of kelvins, one can estimate v_z∼ 10^6 cm/s,
i.e. it is two orders of magnitude lower than the in-plane Fermi velocity v_F∼ 10^8 cm/s.
If graphene is doped (say, E_F∼ 0.2 eV), then τ=τ_E_F and the out-of-plane velocity is even lower, v_z∼ 10^5 cm/s.



§ THERMIONIC EMISSION

Let us apply Eq. (<ref>) to an atomically thin conductor that forms a Schottky junction with a bulk semiconductor. 
The reverse saturation current density in Eq. (<ref>) can be calculated by integrating the out-of-plane velocity
over the quasiparticle states with energies above the barrier as

    j_0 ∼ eg_svv_z Δ p_z /2πħ∫dp_y/2πħ∫dp_x/2πħ f^(0)_E_p_xp_y,

where g_sv is the spin/valley degeneracy, f^(0)_E_p_xp_y is the hot Fermi-Dirac distribution, p_x,y (p_z) are
the in-plane (out-of-plane) components of the quasiparticle momentum, and, in contrast to the conventional approach,<cit.>
the integral over p_z has been substituted by its uncertainty Δ p_z.
Equation (<ref>) is approximate and becomes applicable once the characteristic electron momentum 
(e.g. the Fermi momentum in metals) gets comparable with its uncertainty, or, in other words,
the quasiparticle de Broglie wavelength becomes comparable with the conductor thickness.
The integral over p_z should be retained otherwise.




For a graphene-semiconductor Schottky junction shown in Fig. <ref>, we have f^(0)_E_p_xp_y∼exp(E_F-E_p_xp_y/k_BT)
at k_B T ≪Φ_B with E_p_xp_y=v_F √(p_x^2 + p_y^2) for electrons above the barrier, g_sv=4, and Eq. (<ref>) then reads

    j_0 ∼e/τ∫_Φ_B + E_F^∞dE E/π^2ħ^2 v_F^2e^E_F-E/k_BT,    k_B T ≪Φ_B,

where Φ_B=Φ_B0-E_F with Φ_B0 being the Schottky barrier height at E_F=0, see Fig. <ref>.
In the case of intrinsic graphene (|E_F|≪ k_B T, τ=τ_0) we have 

    j_0 ∼ A^*_G0 T^2e^-Φ_B/k_BT,    |E_F|≪ k_B T.

Here, A^*_G0=e(Φ_B0+ k_B T)k_B^2/(π^2 ħ^3 v_F^2) is the thermionic constant for intrinsic graphene.
Since  k_B T is much lower than Φ_B0, the former can be neglected in A^*_G0, and
the current density obeys the Richardson-Dushman law,<cit.> where the reverse saturation current is 
given by j_0=A^* T^2 exp[-Φ_B/(k_BT)] with A^*=em^*k_B^2/(2π^2ħ^3) being the Richardson constant.<cit.>
In contrast to the conventional Richardson constant,  A^*_G0 strongly depends on the barrier height Φ_B0.
For Φ_B0 of the order of 0.1 eV (see Table 1 in Ref. di2016graphene), we estimate A^*_G0 to be about
10 A/cm^2/K^2 that is comparable with the Richardson constant A^* = 120  (m^*/m_0) A/cm^2/K^2,
where the effective-to-free electron mass ratio m^*/m_0 is of the order of 0.1 for typical bulk metal-semiconductor junctions.<cit.>
One could define an out-of-plane effective mass for carriers in graphene as m^*v_F^2/2=Φ_B  
and utilize the standard formula for A^*. The Schottky barrier height plays therefore a role of the effective
electron mass in the conventional theory of thermionic emission. This makes sense because both 
the Schottky barrier and inertial mass resist the out-of-plane particle acceleration.

In the case of doped graphene (|E_F|≫ k_B T, τ=τ_E_F), Eq. (<ref>) reads

    j_0 ∼e(Φ_B0+k_B T)  k_B^3 T^3/π^2 ħ^3 v_F^2 |E_F|e^-Φ_B/k_BT,    k_B T ≪ |E_F|,

where E_F can be positive or negative depending on graphene doping, and
T^3-dependence indicates that the Richardson-Dushman law becomes invalid.
Nevertheless, one can formally define the temperature dependent thermionic constant as A^*_G = A^*_G0k_B T/|E_F|.
Hence, the thermionic constant is further reduced as compared with the intrinsic value A^*_G0
and could therefore partly explain why its actual value measured in graphene-silicon junctions is smaller than expected.<cit.>
The temperature dependence of j_0 given by Eq. (<ref>) at Φ_B0≫ k_B T is similar to what has been predicted by Liang and Ang<cit.>
but, in contrast to their model,  A^*_G is governed by Φ_B0 and E_F, not just by the fundamental constants and Fermi velocity in graphene.












§ DISCUSSION










The thermionic constant turns out to be strongly dependent on the Fermi energy
and offers an interesting opportunity to test the model predictions in real devices.
The constant increases while the Fermi level approaches the band crossing point in graphene
until it reaches the intrinsic value A^*_G0 at |E_F|≪ k_B T.
The trend is schematically shown in Fig. <ref> at Φ_B0∼ 0.5 eV.
The saturation current, however, does not follow this trend since
the inverse proportion to E_F is by far compensated for by the exponential dependence on E_F via Φ_B.
Anyway, the thermionic constant can easily be identified by plotting j_0(E_F) on a logarithmic scale
and subtracting the trivial Φ_B/T term.<cit.>

In contrast to conventional metal-semiconductor diodes, graphene-semiconductor junctions demonstrate 
a bias-driven increase of the reverse saturation current.<cit.>
To be specific, let us consider the current-voltage characteristics of a junction between graphene and an n-doped semiconductor
assuming that the Fermi level intercepts the band crossing point in graphene at zero bias (see Fig. 17 in Ref. [di2016graphene]).
A bias voltage redistributes the positive and negative charge across the junction, resulting in two different (quasi)-Fermi levels
for carriers on the graphene and semiconductor sides.
The forward bias (V>0) lowers the positive charge of the depletion layer in the semiconductor,
hence reducing the built-in potential Φ_i at the interface as well as the Fermi energy in graphene.
In contrast, the reverse bias (V<0) extends the depletion layer to the semiconductor side increasing the built-in potential.
At the same time, the Fermi level increases in graphene because of the higher charge density induced there 
to mirror the excessive immobile donor charge of the depletion layer. The induced charge density can be 
calculated using the Schottky-Mott relationship as<cit.>

    -e Δ n_ind   =   √(2ϵ_S N_D (Φ_i - eV - k_B T))
           - √(2ϵ_S N_D (Φ_i - k_B T)),

where ϵ_S∼ 10 is the relative permittivity of a typical semiconductor, N_D ∼ 10^16 cm^-3 is the 
donor concentration in the depletion layer, and Φ_i∼ 0.6 eV is the built-in potential.
The typical parameter values are taken from Ref. [PRX2012tongay] where various graphene/n-type semiconductor junctions have been studied.
The Fermi energy difference due to the reverse bias can be written for graphene as Δ E_F = ħ v_F √(-πΔ n_ind),
where the four-fold degeneracy has been taken into account in the relation between the Fermi wave vector and the carrier concentration.
The blue line in Fig. <ref> demonstrates that the reverse bias results in a transfer of negative charge 
to graphene large enough to increase Δ E_F by tens of meV at the voltages of a few volts, in accordance with the measurements of Ref. PRX2012tongay.
The upward shift of E_F in graphene causes a reduction of the barrier height Φ_B 
facilitating electron transport across the junction and preventing the reverse current from saturating.
If graphene behaved like a regular metal with a very high density of states,
then the Fermi energy would not change much under the bias voltage resulting in a nearly constant j_0
and perfect saturation of the reverse current (see the red dashed line in Fig. <ref>).
Retaining the exponential dependence j_0 on E_F but neglecting the substantial decrease
in the thermionic constant at higher |E_F|  
allows us to qualitatively reproduce the lack of the reverse current saturation observed in real graphene-semiconductor junctions<cit.>
(see the red dotted line in Fig. <ref>).
In fact, the inverse proportion of j_0 to E_F read out from Eq. (<ref>) partly compensates for this exponential trend.
The result is that the true reverse current values shown by the red solid line in Fig. <ref> lie between
the curves given by Eq. (<ref>) with E_F=0 assumed to be independent of V (the metal-like model, dashed curve) and
by the same equation with E_F=Δ E_F(V)  (the Richardson-Dushman law, dotted curve).

The experimental verification of the model proposed is obviously 
of the utmost importance for graphene-based device applications as the reverse saturation current 
is one of the most essential parameters to characterize a Schottky diode.
The previous transport measurements<cit.> have employed samples fabricated from graphene grown on copper by chemical vapor deposition
and subsequently transfered onto a semiconductor. Graphene has been found to be p-doped with the Fermi energy |E_F| ≳ 200 meV being much higher
than Δ E_F provided by the bias voltage (see Fig. <ref>).
It was therefore impossible to observe the qualitative change of the saturation current behavior due to
the crossover between intrinsic (|E_F| ≪ k_B T) and highly doped (k_B T ≪ |E_F|) regimes.
Moreover, the reverse current seen in Fig. <ref> is much higher than the measured one<cit.>
because the Schottky barrier is almost 0.3 eV lower in our case of intrinsic graphene.
To see the effects predicted above, one needs much cleaner samples with |E_F| ≲ 25 meV at zero bias.
Nevertheless, the T^3-dependence in the saturation current for doped graphene predicted by Eq. (<ref>)
can be verified by using existing experimental data<cit.> for graphene contacts with Si, MoS_2, GaAs, and GaN.
Indeed, the measured current density plotted as ln(j_0/T^3) vs. 1/T 
demonstrates a linear dependence for any of the four bulk semiconductors considered.
This is a clear indication of consistency between the theory and experimental data available at the moment. 

Besides possible applications, the experiments with nearly intrinsic graphene 
might shed some light on the fundamental properties of a Dirac liquid
confined in a 2D plane (see Ref. lucas2018hydrodynamics for review).
Indeed, the most important ingredient of the model introduced above is the quasiparticle lifetime which is τ_0 ∼ħ/(k_B T) in the intrinsic limit. 
Interestingly, an emission rate equal to k_B T/ħ has been obtained in Ref. wei2013electron
by fitting the thermionic constant for a 2D electron gas with the Richardson-Dushman law.
Here, we associate the emission rate with the inverted quasiparticle lifetime, which makes a lot of sense
as τ_0 can be seen as the average time needed to repopulate a given state above the barrier after emission.
Once the Fermi energy is shifted away from the band crossing point in graphene,
the quasiparticle lifetime decreases rapidly and, as a consequence,
the thermionic constant is reduced.  Since the Fermi energy depends on the bias voltage, 
this effect should be seen in the current-voltage measurements.
Alternatively, the Fermi energy of graphene can be varied through appropriate gating.<cit.>




§ SUMMARY AND OUTLOOK


The intimate relation between the quasiparticle emission rate and lifetime
is among the main findings of this work.
The model explains, above all, why the quasiparticles tightly confined
in a 2D plane still possess an out-of-plane velocity.
The thermionic emission mechanism in graphene-semiconductor junctions strongly differs from its bulk version.
The Richardson constant employed in the conventional model
is substituted by the thermionic constant A^*_G
that depends on the Fermi energy that is in turn tunable by an external electric field normal to the graphene plane. 

Graphene-semiconductor junctions should demonstrate efficient photocarrier injection with the excitation energy below the semiconductor bandgap.
The photocarriers excited in graphene are thermalized rapidly <cit.> creating a high-energy distribution tail above the Schottky barrier,
hence providing excess energy necessary for thermionic emission across the interface.
In contrast to the conventional heterojunctions,<cit.>
the out-of-plane momentum conservation is relaxed by the momentum uncertainty Δ p_z ∼ħ /Δ z,
which, thanks to the small layer thickness Δ z, is of the order of the first Brillouin zone size.
The interface disorder thereby does not much hamper the out-of-plane carrier transport between graphene and the semiconductor.
The injection rate is  governed  by the quasiparticle lifetime rather than the interface disorder.
Hence, the carriers do not behave like point-like particles along z-direction until they 
turn out to be on the semiconductor side with a well-defined momentum.
This peculiarity should be taken into account when designing optoelectronic devices
based on 2D materials.




I would like to thank Ilya Goykhman and Andrea Ferrari for turning my attention to the problem of thermionic emission from graphene.
Multiple discussions with Shi-Jun Liang are also acknowledged.
This work has been supported by the Director's Senior Research Fellowship from the Centre for Advanced 2D Materials at the National University of Singapore
(NRF Medium Sized Centre Programme R-723-000-001-281).




