Varshni Model Overview

Updated 13 November 2025

Varshni Model is a dual-framework theory that describes both analytical quantum potentials and an empirical equation for temperature-dependent band-gap shifts.
It applies solution methods such as the Nikiforov–Uvarov technique to solve Schrödinger, Dirac, and Klein–Gordon equations with short-range potentials.
Its empirical Varshni equation is widely used to fit temperature-dependent band-gap data in semiconductors, aiding material characterization and device design.

The Varshni model encompasses a set of related analytical frameworks, all originating with Y. P. Varshni, and is used extensively in physical chemistry, atomic/molecular/optical (AMO) physics, and semiconductor physics. In the quantum-mechanical context, it fundamentally refers to a short-range potential that enables analytical or quasi-analytical calculations in the Schrödinger, Dirac, Klein–Gordon, or Salpeter equations, and various generalizations (including supersymmetric partners and multi-dimensional analogs). Separately, an unrelated "Varshni equation" acts as the empirical standard for band-gap (or related excitation energy) temperature shifts in crystalline semiconductors.

1. The Varshni Potential: Definition and Parametrization

The original Varshni potential is a two- or three-parameter short-range interaction introduced to model diatomic molecular vibrations and, by analogy, other finite-range quantum systems. Its canonical forms include:

Two-parameter version

$V(r) = a\,\left[1 - b\,e^{-\beta r}\right]$

where

$a$ (energy, e.g., eV or GeV): the depth/height controlling the asymptotics,
$b$ (dimensionless, typically 0 ≤ b < 1): range parameter controlling the strength of the exponential screening,
$\beta$ (inverse length): the screening parameter dictating potential range,
$r$ : radial coordinate or inter-particle separation.

Three-parameter (diatomic-molecule) version

$V(r) = a\left[1 - \frac{b}{r}e^{-\beta r}\right]$

where $b$ acquires units of length (e.g., Å).

In the literature, this potential serves as a smooth, purely repulsive (if $a > 0$ ), or smoothly screened attractive interaction, interpolating between a hard core and exponentially decaying tail, generally saturating to a constant at large $r$ .

DKV and Related Potentials (Dutt–Khare–Varshni)

The “DKV potential” is a rational-exponential potential expressible in elementary functions via a change of variable associated to so-called DRtTP (Double Root Tangent-Polynomial) reductions. For $\eta = 1 + \tanh x$ : $V_{\mathrm{DKV}}(x) = \frac{A}{\eta^2} - \frac{B}{\eta} - \frac{3}{4}$ with $A, B$ derived from polynomial invariants of the associated hypergeometric operator (Natanson, 2015).

2. Analytical Solution Techniques: Schrödinger, Dirac, Klein–Gordon Equations

The Varshni potential is analytically tractable in non-relativistic and various relativistic wave equations if the centrifugal barrier is approximated (Pekeris or Greene–Aldrich schemes), reducing the problem to hypergeometric or Jacobi polynomial forms:

Schrödinger/N-dimensional Schrödinger:

Transformation to $x = e^{-\beta r}$ , followed by the Nikiforov–Uvarov (NU) method, yields energy eigenvalues and normalized eigenfunctions:

$R_{n\ell}(r) = x^{\lambda} (1 - x)^{\nu} P_{n}^{(2\lambda,2\nu)}(1-2x)$

with explicit expressions for $\lambda, \nu$ in terms of angular momentum, dimensionality, and potential parameters (Inyang et al., 2020, Inyang et al., 2021).

Relativistic equations (Dirac, Klein–Gordon, Salpeter):

With equal scalar and vector Varshni potentials, the radial equations reduce to hypergeometric type after the centrifugal term is approximated (Arda et al., 2014, 1705.01577, Oluwadare et al., 2017). Both spin and pseudospin symmetries in the Dirac case yield nearly degenerate doublets, with closed-form eigenvalues and two-component spinors constructed from Jacobi polynomials.

Scattering and Phase Shifts:

The analytic continuation of hypergeometric functions to $r \rightarrow \infty$ gives closed-form scattering phase shifts:

$\delta_l^{\mathrm{Var}} = \frac{\pi}{2}(l+1) + \arg\Gamma\left(\frac{2ik}{\beta}\right) - \arg\Gamma(a_1^*) - \arg\Gamma(a_2^*)$

where $a_1, a_2$ are complex parameters determined by the matching of boundary conditions (1705.01577, Oluwadare et al., 2017).

3. Empirical Varshni Equation for Temperature-Dependent Energy Shifts

Independently of the quantum potential, the "Varshni equation" is the de facto empirical standard for modeling the temperature dependence of band-gap energies in semiconductors and related materials: $E_g(T) = E_g(0) - \frac{\alpha T^2}{T + \beta}$ with:

$E_g(0)$ : band gap at 0 K,
$\alpha$ : (eV/K or eV/K $^2$ ), magnitude of electron–phonon coupling or phonon population effects,
$\beta$ : (K), typically related to the Debye temperature.

This model is widely validated for band-gaps in 4H–SiC, hBN, multinary oxides, perovskites, and more (Misiara et al., 12 Oct 2024, Liu et al., 2021, Huang et al., 2017, Lysak et al., 16 Feb 2024, Nishiwaki et al., 2023, Sarswat et al., 2011, Liu, 2015, Palmer et al., 1 Nov 2024). Fitting protocols typically involve nonlinear least-squares regression of experimentally extracted transition energies (from PL, PR, EELS, cathodoluminescence, etc.) at known cryostat temperatures to this formula, returning optimized $\alpha$ and $\beta$ .

Key physical interpretations:

$\alpha$ encodes chiefly the electron–phonon interaction strength.
$\beta$ tracks a phonon-characteristic temperature, closely paralleling the Debye temperature for the material.

Examples: | Material | Eg(0) (eV) | α (eV/K) | β (K) | Reference | |------------------------------------|------------|---------------|--------|---------------| | 4H–SiC (V1 ZPL) | 1.4393 | 29 μeV/K | 300 | (Misiara et al., 12 Oct 2024) | | hBN (VB $^-$ ZFS) | 3584 MHz | 1.06 MHz/K | 559 | (Liu et al., 2021) | | CH $_3$ NH $_3$ PbI $_3$ (band edge)| 1.495 | 4e-4 | 220 | (Huang et al., 2017) | | ZnCdO/ZnO:Eu | 3.274 | 8.2e-4 | 1029 | (Lysak et al., 16 Feb 2024) |

4. Practical Applications: Thermometry, Spectroscopy, and Device Physics

Thermometric Probes

The Varshni formalism underpins optical thermometry by relating measured redshifts in zero-phonon lines, band-edges, or core-loss edges to local temperature:

For SiC color centers, sharp ZPL redshifts calibrated by the Varshni relation yield microkelvin-resolved temperature maps within a photonic device (Misiara et al., 12 Oct 2024).
In hBN, the temperature-dependent zero-field splitting of spin defects, parametrized by the Varshni model, enables nanoscale temperature sensing in 2D platforms over 5–600 K (Liu et al., 2021).

Thermal Tuning in Light-Emitting Diodes

The Varshni equation quantifies the nearly linear shift of LED emission wavelength with temperature ( $\lambda(T) \approx \lambda_0 + mT$ ), enabling color tuning and precise device characterization with minimal measurement (Ocaya, 2020).

Heavy-Quark Spectroscopy and Molecular Physics

The quantum mechanical Varshni potential supports construction of analytic eigenvalues and eigenfunctions for heavy meson spectroscopy (charmonium, bottomonium) in the non-relativistic regime (Inyang et al., 2021). The formula: $E_{n\ell} = a(1+ba) - \frac{3a^3b}{8\mu} + \frac{\hbar^2 a^2}{8\mu} \left[2n+1+\sqrt{(2\ell+1)^2+\frac{8\mu B}{\hbar^2 a}}\right]^2$ closely matches experimental data for low-lying states. Analytical bound-state and phase-shift calculations under the Varshni potential are routine in the paper of semi-relativistic and relativistic equations.

DKV/Darboux–Heun Extensions

The Dutt-Khare-Varshni and related potentials introduce the possibility for constructing further exactly- and conditionally-exactly-solvable models, where spectra (or parts thereof) remain accessible through classical polynomials (Jacobi, Heun) (Natanson, 2015).

5. Extensions, Generalizations, and Model Limitations

Varshni–Hulthén and Superposition Potentials

Composite models combining the Varshni form with other short-range potentials (e.g., Hulthén, Hellmann, Shukla) are analytically tractable, using the same transformation and approximation schemes (Inyang et al., 2020, 1705.01577). These generalizations are useful for modeling more complex screened interactions.

Limits of Applicability

The empirical Varshni equation fails to capture high- $T$ (multi-phonon) behavior above $\sim 1.2\times$ the Debye temperature or in systems with non-canonical electron–phonon coupling; e.g., the Bose–Einstein model may better fit wide- $T$ data (Sarswat et al., 2011, Lysak et al., 16 Feb 2024). In quantum-mechanical applications, the main limitation is the Pekeris or similar approximation for the centrifugal barrier, affecting accuracy at high angular momentum or large screening parameter, and the exclusion of spin–orbit and relativistic effects for precision spectroscopy.

The Varshni potential bridges the gap between pure Coulomb, Yukawa (exponential Coulomb), and Hellmann (Coulomb + Yukawa) forms. Unlike the Coulomb or Yukawa, the Varshni interaction saturates to a finite constant at large $r$ . Analytical techniques for the Varshni model are directly transferrable to these related forms, as they all permit reduction to hypergeometric or polynomial eigenfunction equations with appropriate approximations (Oluwadare et al., 2017, 1705.01577, Inyang et al., 2020).

For temperature-dependent energy shifts, the Varshni phenomenology is nearly universal in bulk semiconductors but is sometimes supplanted by more rigorous Bose–Einstein-type expressions for cases with strong nonlinearity or over broad $T$ ranges.

7. Summary Table: Representative Varshni Model Applications

Subfield	Model Type	Main Application	Analytical Structure	Reference
Semiconductor optics	Empirical Varshni equation	Band-gap temp. dependence, thermometry	Nonlinear least-squares fit	(Misiara et al., 12 Oct 2024, Palmer et al., 1 Nov 2024)
Molecular/AMO	Varshni potential in Schrödinger/Dirac/KG	Spectroscopy, scattering, bound states	NU-method, hypergeometric/Jacobi	(Inyang et al., 2021, Arda et al., 2014)
SUSY quantum mechanics	DKV/elementary potential	Exactly/conditionally solvable SUSY partners	Heun polynomials, Darboux transforms	(Natanson, 2015)

The Varshni model and equation, encompassing both quantum potential and temperature-dependent energy shift paradigms, persist as standard tools in both theoretical and experimental studies requiring tractable, parameterizable models for short-range interactions and thermally-driven spectral features. These frameworks allow for the extraction of meaningful material/interaction parameters from spectra, the design of thermometric sensors at micro/nanoscale, and the construction of analytical benchmarks for sophisticated many-body and relativistic quantum systems.