Nonlocal Hydrodynamic Drude Model

• NLHDM is a semiclassical model that extends the Drude framework by treating conduction electrons as a compressible charged fluid to capture spatial dispersion.
• The model introduces a Laplacian correction in Maxwell’s equations, yielding accurate predictions of resonance blueshifts, charge smearing, and field regularization in nanoscale structures.
• Applications include managing singularities in plasmonic gaps and modeling resonance shifts in metallic nanoparticles, proving crucial for advanced nanoplasmonic designs.

The Nonlocal Hydrodynamic Drude Model (NLHDM) is a semiclassical extension of the local-response approximation and the local Drude model for conducting media, developed to describe spatially dispersive free-electron dynamics in metallic nanostructures when characteristic dimensions, radii of curvature, or gap sizes approach the true nanoscale. In this framework, the conduction electrons are treated as a compressible charged fluid rather than as a pointwise Ohmic medium, so the induced current depends not only on the local electric field but also on spatial derivatives associated with electron-gas pressure and charge compression. The model introduces an intrinsic electronic length scale of order $v_F/\omega$, more precisely $\beta/\omega$, and becomes practically relevant for features in the $1$–$10\ \mathrm{nm}$ range, where local Drude electrodynamics can miss charge smearing, resonance shifts, and the regularization of singular near fields (Mortensen, 2013).

1. Historical placement and conceptual scope

The NLHDM is the modern nanoplasmonic form of the hydrodynamic model for the jellium electron gas, originally due to Bloch, and it is commonly presented as the standard or canonical nonlocal extension of Drude electrodynamics in plasmonics (Mortensen, 2013). Its central premise is that the local constitutive law

$$\mathbf J(\mathbf r)=\sigma(\mathbf r)\mathbf E(\mathbf r)$$

contains no intrinsic electronic length scale and therefore cannot represent the finite spatial range of the electron-gas response when structures become extremely small (Mortensen, 2013).

The need for such an extension is geometric as well as microscopic. In sharply curved tips, narrow dimers, ultrasmall particles, and strongly confined plasmonic waveguides, the electron gas cannot behave as an infinitely compressible local medium, and induced charge can no longer be confined to an infinitesimally thin mathematical surface layer (Mortensen, 2013). This is why the NLHDM is often described as an intermediate-scale theory: it is intended for structures small enough that nonlocal screening matters, but still large enough that a semiclassical hydrodynamic description remains meaningful (Mortensen, 2013).

A central physical consequence is that plasmonic systems acquire a characteristic nonlocal range set by the Fermi velocity. In the hydrodynamic model this range is of order $\beta/\omega$, and in the Thomas–Fermi model $\beta^2=\frac{3}{5}v_F^2$, so one obtains the widely quoted scaling $\xi_{\rm NL}\sim v_F/\omega$ (Mortensen, 2013). This provides an intrinsic short-distance cutoff absent in the local-response approximation.

2. General nonlocal formalism and hydrodynamic specialization

A general spatially nonlocal linear medium is described არა by a local dielectric function but by a two-point dielectric kernel $\boldsymbol{\varepsilon}(\mathbf r,\mathbf r')$, leading to

$$\mathbf\nabla\times\mathbf\nabla\times\mathbf E(\mathbf r) = \left(\frac{\omega}{c}\right)^2 \int d\mathbf r'\, \boldsymbol{\varepsilon}(\mathbf r,\mathbf r')\,\mathbf E(\mathbf r').$$

Within this formulation, the local Drude model is recovered when the kernel collapses to $\beta/\omega$0, with

$\beta/\omega$1

For a weakly nonlocal, short-range correction $\beta/\omega$2, the leading effect is generically a Laplacian correction to the wave equation, so that

$\beta/\omega$3

This is one of the model’s central conceptual results: short-range nonlocality can often be reduced from an integro-differential description to a PDE with an additional $\beta/\omega$4 operator (Mortensen, 2013).

In the hydrodynamic specialization, the electromagnetic field is coupled to an induced current density $\beta/\omega$5 obeying

$\beta/\omega$6

$\beta/\omega$7

The operator $\beta/\omega$8 is the mathematical locus of nonlocality in the standard pressure-only NLHDM: it modifies Ohm’s law by coupling the current to the compressional behavior of the electron fluid, and therefore introduces longitudinal electron-gas dynamics absent in local Drude theory (Mortensen, 2013).

The same structure can be written as a single effective field equation,

$\beta/\omega$9

This identifies the hydrodynamic model as an exact realization of the earlier phenomenological Laplacian form, with a coefficient determined by electronic parameters rather than by a generic short-range kernel (Mortensen, 2013).

3. Longitudinal modes, boundary conditions, and electromagnetic consequences

The hydrodynamic correction acts only through the longitudinal sector. In the standard formulation, transverse electromagnetic response remains Drude-local, while longitudinal charge-density oscillations become possible because the electron fluid has finite compressibility (Mortensen, 2013). This distinction is also explicit in generalized analyses: in standard HDM the nonlocal correction enters through $1$0, and therefore affects compressional currents rather than purely transverse ones (Svendsen et al., 2020).

Because the NLHDM introduces $1$1 or, equivalently, a free-electron polarization $1$2 as an additional field, standard Maxwell boundary conditions are not sufficient to close the problem. Under the hard-wall confinement assumption, the operative additional boundary condition is that electrons cannot leave the metal,

$1$3

or equivalently $1$4, while tangential current remains unrestricted in the basic hard-wall picture (Mortensen, 2013). The physical assumptions behind this closure are explicit: the equilibrium free-electron density is taken to be constant inside the metal and to vanish abruptly at the surface, so spill-out and quantum leakage are neglected (Mystilidis et al., 2023).

The necessity of such additional boundary conditions is not merely heuristic. A frequency-domain uniqueness theorem for Maxwell–HDM systems shows that the nonlocal constitutive operator produces additional surface terms in the standard energy argument, and these terms vanish only if suitable hydrodynamic boundary conditions are imposed (Mystilidis et al., 2023). For real $1$5, uniqueness follows under the usual passive sign conditions on $1$6 and $1$7; for complex hydrodynamic parameters, including GNOR-type extensions, the theorem yields the additional condition

$1$8

for $1$9 in passive media (Mystilidis et al., 2023).

The presence of longitudinal fields also changes standard electromagnetic identities. In the hard-wall hydrodynamic model, the classical local absorption density $10\ \mathrm{nm}$0 is no longer valid, and the generalized local absorption density becomes

$10\ \mathrm{nm}$1

where $10\ \mathrm{nm}$2 is the transverse component of the electric field (Sakat et al., 2021). Reciprocity is likewise reformulated in terms of $10\ \mathrm{nm}$3, not the total $10\ \mathrm{nm}$4, inside the nonlocal medium (Sakat et al., 2021).

4. Physical manifestations in nanoplasmonics

The most characteristic physical effect of the NLHDM is charge smearing: induced charge is no longer localized on an infinitesimally thin interface, but spreads over a thickness of order $10\ \mathrm{nm}$5 (Mortensen, 2013). This directly regularizes structures that local theory would render singular. In local electrodynamics, ideally sharp tips and vanishing gaps can drive divergent or arbitrarily increasing fields because the equations contain no microscopic cutoff. In the NLHDM, finite compressibility smooths the charge distribution and therefore regularizes both the charge and the field (Mortensen, 2013).

This smoothing also reduces extreme field enhancement in narrow plasmonic gaps. In dimers and closely spaced metallic structures, local theory predicts ever stronger capacitive coupling as the separation decreases, whereas the hydrodynamic model broadens the induced charge profile and thereby weakens coupling and enhancement (Mortensen, 2013). In nanosphere-on-mirror structures, a dedicated T-matrix implementation of the NLHDM finds systematic resonance blueshifts relative to the local model and reduced electric-field enhancement in $10\ \mathrm{nm}$6–$10\ \mathrm{nm}$7 gaps, with the difference diminishing as the gap increases (Yan et al., 2023).

A further standard signature is the blueshift of localized surface plasmon resonances, for which the hydrodynamic scaling is

$10\ \mathrm{nm}$8

for a spherical particle of diameter $10\ \mathrm{nm}$9 (Mortensen, 2013). In multilayer hyperbolic metamaterials, the same nonlocal screening renormalizes the effective perpendicular permittivity and can change the topology of the isofrequency contour from hyperbolic to elliptic or nearly flat, shifting the canalization condition and altering lens focusing (Yan et al., 2013). In metal slabs, the hydrodynamic response supports additional higher-frequency surface plasma wave modes and removes the local-model $$\mathbf J(\mathbf r)=\sigma(\mathbf r)\mathbf E(\mathbf r)$$0 divergence in the dipole-to-slab energy-transfer rate as the emitter approaches the surface (Brown et al., 2021).

The same nonlocal corrections propagate into spectroscopy and strong-coupling problems. In few-nanometer core–shell plexcitonic particles, the hydrodynamic Drude model produces the expected screening-induced blueshift of plasmonic and hybrid plasmon–exciton modes, while preserving the anticrossing width; GNOR then adds linewidth broadening without eliminating strong coupling in the cases studied (Zouros et al., 2020). This suggests that NLHDM corrections are often decisive for resonance placement and near-field structure even when the gross modal taxonomy remains recognizable.

5. Extensions, refinements, and multi-fluid generalizations

The most common extension of the NLHDM is the Generalized Nonlocal Optical Response (GNOR) model, which augments hydrodynamic pressure by diffusion. Its constitutive law can be written

$$\mathbf J(\mathbf r)=\sigma(\mathbf r)\mathbf E(\mathbf r)$$1

or, equivalently, as the replacement $$\mathbf J(\mathbf r)=\sigma(\mathbf r)\mathbf E(\mathbf r)$$2 (Svendsen et al., 2020). In this interpretation, standard HDM captures nonlocal screening and resonance shifts, whereas GNOR adds effective surface-enhanced Landau damping and therefore linewidth broadening (Svendsen et al., 2020).

A different refinement is Halevi’s extension of the Euler–Drude model, which preserves the hydrodynamic differential structure but replaces the constant $$\mathbf J(\mathbf r)=\sigma(\mathbf r)\mathbf E(\mathbf r)$$3 by a frequency-dependent complex parameter,

$$\mathbf J(\mathbf r)=\sigma(\mathbf r)\mathbf E(\mathbf r)$$4

This interpolation recovers $$\mathbf J(\mathbf r)=\sigma(\mathbf r)\mathbf E(\mathbf r)$$5 at low frequency and $$\mathbf J(\mathbf r)=\sigma(\mathbf r)\mathbf E(\mathbf r)$$6 at high frequency, and introduces a nonlocal collisional damping term that affects the width and amplitude of cylindrical plasmon resonances (Wegner et al., 2022).

In ultrapure metals in the genuine electron-hydrodynamic regime, viscosity can become a central nonlocal mechanism. A viscous hydrodynamic optical theory then gives

$$\mathbf J(\mathbf r)=\sigma(\mathbf r)\mathbf E(\mathbf r)$$7

so that even the transverse response becomes nonlocal and a second transverse “hydrodynamic mode” appears, together with a no-slip additional boundary condition rather than the hard-wall current condition familiar from standard nanoplasmonics (Toshio et al., 2019).

The single-fluid NLHDM has also been generalized to systems with more than one mobile carrier species. In semiconductors, a two-fluid hydrodynamic model introduces separate currents $$\mathbf J(\mathbf r)=\sigma(\mathbf r)\mathbf E(\mathbf r)$$8 and $$\mathbf J(\mathbf r)=\sigma(\mathbf r)\mathbf E(\mathbf r)$$9 and predicts two longitudinal branches—an acoustic mode and an optical mode—where ordinary HDM supports only the optical branch (Maack et al., 2017). Nonlinear hydrodynamic extensions, formulated in density–velocity form with convective, Lorentz-force, and Thomas–Fermi pressure terms, have likewise been used to analyze nanowire dimers and second-harmonic generation (Moeferdt et al., 2018).

6. Mathematical formulations, numerical methods, and limits of validity

The NLHDM has generated a substantial mathematical and computational literature because the hydrodynamic current equation contains a grad-div operator that must be discretized consistently. A rigorous finite-element formulation places the electric field in $\beta/\omega$0 and the hydrodynamic current in $\beta/\omega$1, leading naturally to Nédélec-type curl-conforming elements for $\beta/\omega$2 and div-conforming elements for $\beta/\omega$3. This avoids the older curl-free current approximation, which was shown to generate spurious resonances (Hiremath et al., 2012). A subsequent mathematical analysis established existence and uniqueness of weak solutions for the coupled frequency-domain Maxwell–hydrodynamic system and derived convergent Galerkin schemes based on Raviart–Thomas and Nédélec elements (Ma et al., 2019).

Alternative discretizations include hybridizable discontinuous Galerkin formulations for both NHD and GNOR. In that setting, two hybrid variables are introduced on the mesh skeleton, and global conservativity conditions enforce continuity of the tangential electric field and the normal component of the current density; the reported convergence rate is optimal (Li et al., 2016). Contour-integral eigensolvers have also been applied to the open NLHDM resonance problem, treating the Maxwell–hydrodynamic system as a nonlinear eigenvalue problem in complex frequency and enabling modal decompositions of extinction for structures such as sodium nanowires (Binkowski et al., 2019).

Semi-analytical methods remain particularly important in canonical geometries. Exact or quasi-exact hydrodynamic formulations have been developed for layered spheres, core–shell particles, multilayer hyperbolic metamaterials, nanosphere-on-mirror structures, and sphere aggregates, typically by augmenting local transverse bases with longitudinal waves and enforcing hydrodynamic additional boundary conditions at every interface (Yan et al., 2013). A plausible implication is that the computational complexity of NLHDM lies less in Maxwell’s equations themselves than in the need to track an additional longitudinal sector and its interface closure.

The model’s domain of validity remains sharply circumscribed. Standard NLHDM assumes a homogeneous equilibrium electron density, a semiclassical hydrodynamic treatment, hard-wall confinement, and no explicit spill-out unless the boundary treatment is modified (Mortensen, 2013). It neglects tunneling in very small gaps, discrete quantum confinement in ultrasmall particles, finite-width surface charge profiles due to the work function, and Friedel oscillations near surfaces (Mortensen, 2013). For that reason, it is generally regarded as appropriate for nanometer-scale plasmonics where local Drude theory is no longer sufficient, but sub-nanometer quantum effects have not yet become dominant (Mortensen, 2013).