External Quantum Efficiency (EQE) Measurement: Principle and Calculation

1. Introduction

1.1 What Is External Quantum Efficiency

External quantum efficiency (EQE) is one of the most important performance metrics in the field of light-emitting devices. Its physical definition is the ratio of the total number of photons emitted by the device into free space to the total number of electrons injected into the device:

$$ EQE = \frac{\text{Output photons}}{\text{Injected electrons}} $$

EQE comprehensively reflects the overall efficiency with which a device converts injected charge into extractable photons. For LEDs (light-emitting diodes) and PeLEDs (perovskite light-emitting diodes), the EQE directly determines the device's luminous brightness and energy utilization at a given current, making it the most intuitive and important comprehensive metric for evaluating device performance.

1.2 Difference Between EQE and PLQY

EQE (external quantum efficiency) and PLQY (photoluminescence quantum yield) are two easily confused physical quantities that differ markedly in measurement object and method:

Dimension EQE (external quantum efficiency) PLQY (photoluminescence quantum yield)
Excitation mode Electroluminescence (electrical injection) Photoluminescence (optical excitation)
Measurement object The complete device (including electrodes, transport layers, etc.) The luminescent material itself (film, solution, etc.)
Physical meaning Overall efficiency of injected electron → output photon Intrinsic efficiency of absorbed photon → emitted photon
Limiting factors Injection efficiency, exciton radiative recombination efficiency, light extraction efficiency Competition between radiative and non-radiative recombination rates
Typical values (PeLED) 15%–26% > 80%

In short, PLQY measures the intrinsic ability of a luminescent material to "emit how much light after absorbing light," while EQE measures the actual performance of a complete device to "emit how much light after injecting electricity." A high-PLQY material does not necessarily yield a high-EQE device, because going from material to device still requires overcoming multiple bottlenecks such as charge-injection balance, exciton confinement, and light extraction efficiency. Therefore, EQE measurement holds an irreplaceable position in device development and performance evaluation.

2. Key Performance Metrics of LEDs/PeLEDs

Evaluating the performance of LED and PeLED devices usually involves three dimensions: luminous efficiency, spectral characteristics, and stability.

2.1 Luminous Efficiency Metrics

  • External quantum efficiency (EQE, %): the ratio of output photons to injected electrons, the core metric of electro-optical conversion efficiency. Currently, the EQE of near-infrared and red PeLEDs has surpassed 25%, green is approaching 20%, and blue is steadily improving.
  • Current efficiency (cd/A): the ratio of luminous intensity (cd) to injected current (A), reflecting the luminous intensity produced per unit current.
  • Power efficiency (lm/W): the ratio of output luminous flux (lm) to input electrical power (W), considering both light output and voltage drop.
  • Luminance (cd/m²): the luminous intensity per unit area perceived by the human eye, the most important practical metric for visible-light LEDs. High-efficiency PeLEDs can exceed 10⁵ cd/m² in luminance.
  • Radiance (W·sr⁻¹·m⁻²): the radiant power per unit area per unit solid angle, commonly used to characterize near-infrared emitting devices.

2.2 Spectral Metrics

  • Peak wavelength (λ_peak): the wavelength of maximum intensity in the electroluminescence (EL) spectrum, which determines the emission color.
  • FWHM (Full Width at Half Maximum): the wavelength width of the spectrum at half of the peak intensity, reflecting color purity. The EL FWHM of PeLEDs is typically 15–40 nm, and their narrow-band emission is an important advantage for lighting and display applications.
  • CIE 1931 chromaticity coordinates (x, y): quantitative coordinates that map the emission color onto the standard two-dimensional chromaticity diagram, used to objectively describe and compare emission colors (see Section 8 for details).

2.3 Lifetime Metrics

  • Operational lifetime: under constant-current or constant-voltage driving, the time required for the luminance to decay to a certain percentage of its initial value (e.g., T70 = decay to 70%, T50 = decay to 50%). Lifetime testing is a key evaluation method on the path toward practical devices (see Section 9 for details).

3. Input-Output Relationship of an EQE Measurement System

3.1 System Composition Overview

A typical EQE measurement system consists of the following core units:

  1. Electrical excitation unit — a precision source measure unit (SMU) that applies a constant voltage or current to the device while accurately measuring the current through it;
  2. Optical collection unit — an integrating sphere or a fiber-coupled collection head that collects the light emitted by the device efficiently and uniformly into the spectrometer entrance;
  3. Spectrum acquisition unit — a calibrated spectrometer (array spectrometer or scanning monochromator) that disperses the collected light by wavelength and detects it;
  4. Data calculation unit — host software that computes all intermediate and final quantities from the spectral data and electrical parameters.

3.2 Input: Source-Meter Voltage and Current

The source meter provides two key electrical input quantities during measurement:

  • Drive voltage V (V): the voltage drop across the device, used to calculate the device's power efficiency and to analyze its I-V characteristics.
  • Drive current I (A): the current flowing through the device, the key parameter for calculating the number of injected electrons and the direct source of the denominator in the EQE formula.

In actual measurements, constant-current driving is usually adopted because luminance is directly related to the injected current, and constant-current conditions characterize the device's luminescence performance more stably.

3.3 Intermediate Quantity: Spectral Radiant Flux Distribution

The raw data output by the spectrometer is a wavelength-intensity distribution curve, which is converted through radiometric calibration into the spectral radiant flux distribution Φ_e(λ) (unit: W/nm) — that is, the radiant power density emitted by the device into the full space at each wavelength λ.

Φ_e(λ) is the hub of the entire EQE calculation chain — luminance, radiance, photon number, and all other optical result quantities are obtained by integrating or weighted-summing Φ_e(λ).

3.4 Summary of Result Quantities

The following result quantities are jointly derived from the electrical and optical inputs:

Category Result quantity Unit
Efficiency metric EQE %
Efficiency metric Current efficiency cd/A
Efficiency metric Power efficiency lm/W
Photometric metric Luminance L cd/m²
Radiometric metric Radiance L_e W·sr⁻¹·m⁻²
Colorimetric metric CIE 1931 (x, y) dimensionless
Colorimetric metric Peak wavelength, FWHM nm
Colorimetric metric CIE 1976 (L*, a*, b*) dimensionless

4. Radiometric Calibration and Optical Power Acquisition

4.1 Significance of Radiometric Calibration

The raw signal directly acquired by the spectrometer is the count value (counts or ADU) of the detector at each pixel position. This value is affected not only by the incident optical power but also by the transmittance of the spectrometer's optical system, the spectral responsivity of the detector, the integration time, and other factors. Therefore, the raw signal itself is not a physical radiant power value and must be converted into an absolute radiometric quantity through radiometric calibration.

4.2 Radiometric Calibration Principle

The basic idea of radiometric calibration is: use a standard light source with known spectral radiant flux (such as a traceable tungsten-halogen standard lamp), acquire its spectral signal under the same geometric configuration, and establish the "counts–radiant power" mapping at each wavelength, i.e., the calibration coefficient K_cal(λ):

$$ K_{\text{cal}}(\lambda) = \frac{\Phi_{\text{std}}(\lambda)}{\text{Counts}_{\text{std}}(\lambda)} $$

where Φ_std(λ) is the spectral radiant flux of the standard source at the spectrometer entrance, and Counts_std(λ) is the raw count value measured by the spectrometer at that wavelength.

For the device under test, multiplying the raw spectral count value Counts_DUT(λ) obtained under the same measurement conditions by the calibration coefficient yields its absolute spectral radiant flux distribution:

$$ \Phi_e(\lambda) = K_{\text{cal}}(\lambda) \times \text{Counts}_{\text{DUT}}(\lambda) $$

4.3 Precautions

  • Calibration and measurement must use the same geometric configuration (integrating-sphere size, fiber position, slit width, etc.); any change to the optical path invalidates the calibration.
  • Aging of the integrating sphere's diffuse-reflective coating changes its spectral reflectance; it is recommended to periodically recheck the calibration coefficients and recalibrate after every change to the system configuration.
  • The lamp current of the standard source must be set strictly to the value on its traceability certificate and must not be adjusted arbitrarily.

5. Derivation of Luminance and Radiance

5.1 Luminance Calculation

Luminance is the photometric quantity describing how bright a visible-light emitting surface appears to the human eye. For an LED/PeLED device, when its spectral radiant flux distribution Φ_e(λ) (unit: W/nm) and effective emitting area S (unit: m²) are known, the luminance is calculated as follows:

Step 1: single-wavelength luminance contribution. For wavelength λ, under the ideal Lambertian assumption, the relationship between luminous intensity and spectral radiant flux is:

$$ L(\lambda) = \frac{683 \times \Phi_e(\lambda) \times K(\lambda)}{\pi \times S} $$

where:
- 683 lm/W is the maximum spectral luminous efficacy for photopic vision (corresponding to a wavelength of about 555 nm);
- K(λ) is the CIE photopic spectral luminous efficiency function V(λ), a dimensionless function normalized to 1;
- π arises from the geometric relationship between the exitance and luminance of an ideal Lambertian surface source;
- S is the effective emitting area of the device.

Step 2: total luminance summation. Sum the contributions of all wavelengths over the visible band (380–780 nm):

$$ L_t = \frac{683}{\pi \times S} \times \sum_{380}^{780} \Phi_e(\lambda) \times K(\lambda) \, \Delta\lambda $$

In actual calculations, the summation is performed over all wavelength channels with the spectrometer pixel width Δλ as the discrete step.

5.2 Radiance Calculation (Near-Infrared Domain)

For devices whose emission wavelength exceeds the visible range (such as near-infrared LEDs), luminance is no longer applicable; radiance should be used instead. Radiance does not depend on the human-eye visual function and is based directly on radiant power:

$$ L_e = \frac{1}{\pi \times S} \times \sum \Phi_e(\lambda) \, \Delta\lambda $$

Unit: W·sr⁻¹·m⁻².

5.3 Current Efficiency and Power Efficiency

After obtaining the total luminous flux Φ_v = 683 × Σ Φ_e(λ) × K(λ) Δλ (unit: lm), the following can be further calculated:

  • Current efficiency: η_C = Φ_v / I (cd/A is equivalent to lm/A; under the Lambertian assumption 1 cd = π lm/sr → note the conversion)
  • Power efficiency: η_P = Φ_v / (V × I) (lm/W)

6. Calculation of Photon Number and Electron Number

6.1 Photon Number Calculation

At each wavelength, the number of photons corresponding to the spectral radiant flux Φ_e(λ) is derived from the photon energy relation. The energy of a single photon is:

$$ E_{\text{photon}} = h\nu = \frac{hc}{\lambda} $$

Therefore, the number of photons output per unit time by radiation of wavelength λ is:

$$ N(\lambda) = \frac{\Phi_e(\lambda)}{hc} \lambda $$

Summing over the entire band yields the total number of output photons (unit: s⁻¹):

$$ N_t = \sum N(\lambda) \, \Delta\lambda = \frac{1}{hc} \sum \Phi_e(\lambda) \lambda \, \Delta\lambda $$

For visible-light LEDs, the summation range is usually 380–780 nm; for near-infrared LEDs, it is extended to their actual emission band.

6.2 Electron Number Calculation

The number of injected electrons is the current I measured by the source meter (unit: A = C/s) divided by the elementary charge:

$$ N_e = \frac{I}{q} $$

where $q = 1.6022 \times 10^{-19}$ C is the elementary charge. The physical meaning of this formula is the number of electrons flowing through the device per second.

6.3 Physical Constants Used

Constant symbol Name Value Unit
$q$ Elementary charge $1.6022 \times 10^{-19}$ C
$h$ Planck constant $6.6261 \times 10^{-34}$ J·s
$c$ Speed of light in vacuum $2.998 \times 10^{8}$ m/s
$K_m$ Maximum spectral luminous efficacy 683 lm/W

7. Full Derivation of the EQE Formula

7.1 Derivation Process

Starting from the basic definition of EQE:

Step 1: write the definition of EQE

$$ EQE = \frac{N_t}{N_e} $$

Step 2: substitute the photon-number expression

$$ N_t = \frac{1}{hc} \sum \Phi_e(\lambda) \lambda \, \Delta\lambda $$

Step 3: substitute the electron-number expression

$$ N_e = \frac{I}{q} $$

Step 4: rearrange to obtain the final EQE formula

$$ EQE = \frac{q}{hcI} \sum \Phi_e(\lambda) \lambda \, \Delta\lambda $$

7.2 Interpretation of the Formula

This formula directly reveals the core logic of EQE measurement:

  • Numerator — the sum of Φ_e(λ)λΔλ combines the device's radiant output at all wavelengths (each wavelength weighted by λ to convert to photon number), essentially a measure of the "total number of output photons";
  • Denominator — I is the injected current measured by the source meter, essentially a measure of the "total number of injected electrons";
  • Prefactor — q/(hc) is the unit-conversion factor that turns the radiant-power-to-current ratio into a photon-number-to-electron-number ratio.

In actual measurement software, this summation is performed over all wavelength sampling points covered by the spectrometer: compute Φ_e(λ) × λ × Δλ for each wavelength channel, accumulate, and multiply by q/(hcI) to obtain the EQE result.

7.3 Precautions for Use

  1. Accuracy of the emitting area: the EQE calculation does not explicitly include the emitting area S, but the absolute value of Φ_e(λ) depends on the geometric calibration of the integrating sphere or collection optics, which in fact incorporates the geometric factor of the emitting area. If the emitting area differs from that at calibration, a systematic EQE deviation will result.
  2. Spectral range coverage: the summation range must cover the device's entire emission band. If the spectrometer is calibrated only over the visible range (380–780 nm) while the device has a near-infrared emission tail, the EQE will be underestimated.
  3. Accuracy of the current: the current measured by the source meter must be the actual current flowing through the device's emitting region, excluding the influence of parasitic paths such as leakage current.
  4. Lambertian assumption: the above derivation assumes the device is an ideal Lambertian surface emitter. Real devices may have angle-dependent emission patterns; high-precision measurements require correction through angle-resolved measurements.

8. CIE Chromaticity Coordinate Calculation

8.1 CIE 1931 Chromaticity Coordinates

The CIE 1931 standard colorimetric system is the internationally universal standard for describing emission color. The calculation is as follows:

Step 1: from the spectral radiant flux distribution Φ_e(λ) and the CIE 1931 color-matching functions $\bar{x}(\lambda)$, $\bar{y}(\lambda)$, $\bar{z}(\lambda)$, calculate the tristimulus values:

$$ X = \sum \Phi_e(\lambda) \times \bar{x}(\lambda) \, \Delta\lambda $$

$$ Y = \sum \Phi_e(\lambda) \times \bar{y}(\lambda) \, \Delta\lambda $$

$$ Z = \sum \Phi_e(\lambda) \times \bar{z}(\lambda) \, \Delta\lambda $$

Note: Φ_e(λ) here can be a relative spectral distribution that has not been luminance-calibrated, because chromaticity coordinate calculation depends only on the relative proportions of the tristimulus values. The color-matching function values $\bar{x}(\lambda)$, $\bar{y}(\lambda)$, $\bar{z}(\lambda)$ are standard tabulated data; the summation range is usually 380–780 nm with a step of 1 nm or 5 nm.

Step 2: calculate the chromaticity coordinates (normalized):

$$ x = \frac{X}{X + Y + Z} $$

$$ y = \frac{Y}{X + Y + Z} $$

(x, y) is the CIE 1931 chromaticity coordinate of the light-emitting device, which can be plotted on the standard tongue-shaped chromaticity diagram to mark the exact position of the emission color.

Step 3 (optional): if luminance information is needed, calculate:

$$ z = \frac{Z}{X + Y + Z} = 1 - x - y $$

And the Y value itself (the Y tristimulus value), after appropriate calibration, is the luminance.

8.2 CIE 1976 L*a*b* Color Space

In scenarios requiring quantitative comparison of color differences (color difference ΔE), the perceptual non-uniformity of the CIE 1931 chromaticity diagram is inconvenient. In that case, one can convert to the CIE 1976 L*a*b* color space:

$$ L^* = 116 \cdot f\left(\frac{Y}{Y_n}\right) - 16 $$

$$ a^* = 500 \left[ f\left(\frac{X}{X_n}\right) - f\left(\frac{Y}{Y_n}\right) \right] $$

$$ b^* = 200 \left[ f\left(\frac{Y}{Y_n}\right) - f\left(\frac{Z}{Z_n}\right) \right] $$

where $X_n$, $Y_n$, $Z_n$ are the tristimulus values of the reference white point, and the function f(t) is defined as:

$$ f(t) = \begin{cases} t^{1/3} & \text{if } t > \left(\frac{6}{29}\right)^3 \\ \frac{1}{3}\left(\frac{29}{6}\right)^2 t + \frac{4}{29} & \text{otherwise}\end{cases} $$

9. LED Lifetime Testing Methods

9.1 Constant-Current Testing (Most Common)

Place the LED/PeLED device under constant-current driving and continuously monitor the change of its luminance over time. Since device degradation mainly manifests as a decline in quantum efficiency, the luminance decay under constant current directly reflects EQE degradation.

Typical test procedure:
1. Set the target operating current (e.g., corresponding to an initial luminance of 100 cd/m² or 1000 cd/m²);
2. Acquire the EL spectrum at fixed time intervals (e.g., every 1 minute or every 10 minutes);
3. Record the luminance decay curve L(t);
4. Extract the key lifetime parameters from the L(t) curve.

9.2 Constant-Voltage Testing

Keep the voltage across the device unchanged. As the internal resistance rises during device degradation, the current decreases accordingly, and the luminance decay reflects the combined effect of both the drive-current change and quantum-efficiency degradation. Constant-voltage testing is closer to some real driving scenarios, but because it is difficult to separate the degradation mechanisms, constant-current testing is more commonly used in research work.

9.3 T50/T70/T90 Lifetime Parameters

The lifetime parameter Tx is defined as the elapsed time for the luminance to decay to x% of its initial value:

  • T50: the time required for the luminance to decay to 50% of its initial value, i.e., the "half-life";
  • T70: the time required for the luminance to decay to 70% of its initial value, common in the PeLED literature because a 30% luminance drop is already close to the usage limit for display applications;
  • T90: the time required for the luminance to decay to 90% of its initial value, used to assess rapid degradation in the initial stage.

Since LED luminance decay usually follows a stretched-exponential or power-law behavior, longer lifetimes can be obtained by extrapolation from accelerated aging tests, but the reliability and methodology of extrapolation remain active research topics in the field.

9.4 Precautions

  • The device packaging state has an enormous impact on lifetime. An unencapsulated PeLED can fail completely within minutes in air, so lifetime testing must be performed in an inert atmosphere (such as a glove box) or with reliable thin-film encapsulation.
  • The test temperature must be strictly controlled; a rise in junction temperature accelerates device degradation and makes lifetime test results incomparable.
  • The constant-current drive should be periodically interrupted briefly to acquire complete I-V-L characteristics, thereby monitoring the evolution of electrical parameters such as series resistance and shunt resistance.

10. Summary

External quantum efficiency (EQE) is the most fundamental comprehensive performance metric for evaluating LED and PeLED devices. Starting from the physical definition of EQE, this article systematically explains its essential difference from PLQY, sorts out the key performance metric system of LED/PeLED devices, and constructs the complete data-flow diagram of an EQE measurement system from the "input → intermediate quantity → output" perspective.

At the calculation level, this article derives in detail the complete chain from obtaining the spectral radiant flux Φ_e(λ) through radiometric calibration, to the photometric/radiometric conversion of luminance and radiance, to the calculation of photon number and electron number, and finally to the EQE formula. Every quantitative relationship has a clear physical basis and mathematical expression. In addition, this article covers the standardized method for CIE chromaticity coordinate and color-space calculation, as well as the engineering practice of LED lifetime testing — together these form the complete picture of light-emitting device performance evaluation.

Understanding these principles and calculation logic not only helps in using an EQE measurement system correctly, but also helps researchers and engineers judge the reasonableness and limitations of measurement results, thereby more effectively guiding device optimization and system iteration.

This article was compiled by Pynect