Modeling temperature dynamic effects for high-power light-emitting diodes

Temperature is a dynamic variable in most electronic devices. As the device operates, it generates heat, which translates in a temperature increase. Available models commonly disregard these variations due to the fact that they manifest at very large time scales. However, temperature dynamic effects have profound implications on the device model and on our common understand-ing. This paper discusses implications of considering the temperature variations on the current – voltage characteristic curves of power light-emitting diodes. The main theoretical results establish that the current equation has a memristive nature when temperature is assumed as a dynamical state variable. This hypothesis is then validated experimentally.


| INTRODUCTION
Standard thermal analysis of power electronic circuits usually assumes static temperature conditions, that is, temperature evolution is considered a very slow process when compared to the operating frequencies. Under this simplifying assumption (and assuming one can neglect the transfer of heat through radiation and convection), the thermal analysis of a given circuit follows straightforward Ohm's law-like relations, where temperature increases linearly with the dissipated power with a slope equal to the thermal resistance of the device. 1 This simple relation is the basis of conventional heat sink (HS) selection procedures, where a series of multiple thermal resistances account for the heat conduction from the device's active region, where it is generated, to the environment, where it is dissipated.
Temperature is also a parameter that influences the device's characteristics. Current-voltage relations in diodes and transistors involve temperature in multiple instances: (i) the thermal voltage k B T=q (where k B is the Boltzmann constant and q is the charge of the electron) increases linearly with temperature; (ii) carrier concentrations follow Fermi-Dirac distributions, which are strongly affected by temperature; and (iii) material properties such as thermal conductivity vary with temperature. 2 These temperature-dependent device's current-voltage relations are generally oversimplified and mistreated in the available literature. For instance, it is a common procedure to depict the temperature dependence of the diode current-voltage relation as a collection of curves obtained for different fixed temperature values. 3 The term "fixed temperature" deserves a detailed reflection. What is the meaning of fixed temperature for a power device? Let us take the example of a power diode. As the forward voltage increases, the current that flows through the diode also increases, which leads to an increase in the generated electric power. In its turn, higher values of electric power imply a higher generation of heat-and as a consequence the temperature of the device may also increase. The current that flows through the diode will reflect this variation, reaching an equilibrium between the junction temperature and the external stimulus in a given amount of time, since temperature evolution is not an instantaneous process. As such the occurrence of static current or voltage and temperature conditions in a diode, or in any other electronic device, demands for special conditions. It is possible, by means of active heating and cooling techniques, to keep the temperature of a device at a particular level, but this process requires special apparatus and masks the normal "free" temperature evolution of the device. On the other hand, when temperature is considered a dynamic variable, ruled by a state equation, the DC current-voltage characteristic curves of the device acquire a very definite theoretical meaning, since they are defined for the stable equilibrium of the state equation. 4 The problem of the interdependence of the thermal and electrical characteristics of power diodes is not new, however, and has been addressed by different authors. Janicki et al., 5 for instance, proposed the derivation of compact thermal models in the form of Foster or Cauer RC ladders based on experimental results obtained with discrete power white light-emitting diodes (LEDs) and with modules containing several LEDs. 6 Other authors also considered the impact of the temperature on the luminous flux, thus obtaining a multidomain method that allows for the simultaneous electro-thermal-optical modeling of power LEDs. 7-10 However, the possible emergence of temperature-induced memory effects on the dynamics of these nonlinear devices was not taken into account. In general, when nonlinear dynamics is involved, memory effects or complex chaotic behaviors consequently emerge. For the case of a diode, as will be discussed in this paper, the complete nonlinear dynamics involving temperature evolution has a memristive nature, where temperature plays the role of a state variable. Consequently, the current-voltage plots feature the characteristic gullwing plots of extended memristors. 11,12 Memory effects in diodes and LEDs have been reported previously in the literature. [13][14][15] Chua and Tseng 13 proposed a memristive circuit model for p-n junction diodes, establishing the link between diodes and memristors. The model arises from the solution of the diffusion equation of the diode and can predict accurately the storage and fall times, as well as mimic various second order effects due to conductivity modulation. Mardani et al. 14 and Linnartz et al. 15 develop modeling perspectives on LEDs used for communication purposes, acknowledging the presence of memory effects due to the nonlinearities of the electron-hole recombination dynamics. In Alexeev et al., 16 thermal transient testing is used to characterize LEDs and for the development of compact LED models. However, the presented strategies overlook the presence of thermally induced memory effects as disclosed in the current paper. To the authors best knowledge, this contribution is the first one studying thermally-induced memory effects in power LEDs. However, justice should be paid to the original 1976's contribution on memristive devices by Chua and Kang,17 where the link to temperature is present on the thermistor model.
The memristive nature of the diode current-voltage characteristics implies that when the current that flows through these devices is high enough to induce heating, the use of the static characteristics may lead to erroneous results. This may be especially critical in the case of visible light communications, since the slow heating transients may impact the current levels and, consequently, the stability of the wavelength. For these applications, it may be necessary to anticipate the effects of the varying temperature on the output current. This paper describes a procedure that can be followed to do so.
In order to evaluate the memristive nature of a diode, high-power LEDs were assembled on FR4 and diamond boards, each featuring different thermal time constants. To model the effect of the slow heating transients, the temperature of the diodes was initially measured as a response to current steps. The time constants that were calculated from the transient response were fed into a first-order temperature state equation which was solved together with the SPICE equation provided by the diode manufacturer. It was theoretically predicted and experimentally observed that when the LED is driven by a triangular current wave with a period comparable to the thermal time constants of the system composed by the LED and the board, the current-voltage characteristics show hysteresis and the system features the characteristics of an extended memristor. This paper is organized into the following sections: Section 2 introduces the general heat equation and describes the expected temperature dependence of the diode current; Section 3 describes the experimental setup used for characterizing the LED; Section 4 presents and compares the theoretical and experimental results. Finally, Section 5 draws the final conclusions.

| THEORETICAL FRAMEWORK
The dynamic thermo-electrical characterization of an LED requires the simultaneous resolution of the equation that describes the transfer of the heat generated during the operation of the device, together with the equation that describes its current-voltage characteristics.

| Device heat equation
When the transfer of heat through convection and radiation can be neglected (such as the case of a power LED mounted on a large HS), the equation that describes the transfer of heat through a solid body can be written as 18 where ρ is the mass density, c θ is the specific heat capacity, κ is the thermal conductivity, g is the power density of the generated heat, and T is the temperature. In the case of an electronic device or circuit, the whole setup can usually be divided into different components, such as the chip, the chip carrier, the package, and the board. The thermal behavior of each of these elements can be described by a thermal impedance (an equivalent RC cell): where R θ,i and τ θ,i are the thermal resistance and the time constant of the element i, respectively. The thermal time constant is calculated as the product of the thermal resistance and the thermal capacitance of the element: The effective volume is smaller than the real element volume if the temperature distribution is nonuniform. The transfer function of the thermal equivalent circuit of the complete set can be computed by applying Laplace transforms to the response of the network in the time domain: If one considers a one-dimensional planar thermal model and a device with only one time constant, Equation (1) reduces to where T A is the ambient temperature, P D is the thermal power generated by the device, and R θ and C θ are the thermal resistance and thermal capacitance of the device, respectively.

| Device current-voltage characteristics
The ideal p-n diode current-voltage (i-v) relationship is given by the Shockley equation. 19 When the diode is driven by a current source i, this equation can be written as where I S is the saturation current, q is the electron charge, k B is the Boltzmann constant, and T is the absolute junction temperature. The deviations from the ideal v-i characteristics in real p-n diodes are usually accounted for by including two other parameters in Equation (5), the ideality factor (n) and the series resistance (R S ): In this last equation, n accounts for phenomena such as the generation and recombination of carriers in the depletion layer, the high-injection condition that may occur even at relatively small forward bias, the tunneling of carriers between states in the bandgap, and the surface effects, whereas R s reflects the parasitic resistances of the contacts and cladding layers.
In addition to the explicit temperature dependence of Equation (6) (due to the T term that is in the argument of the logarithm), the saturation current I S is also strongly temperature dependent. In the case of a homojunction diode (i.e., a p-n junction fabricated with the same semiconductor material), the temperature dependence of I S is due to the impact of the temperature on the generation/recombination effects within the depletion region 19 and I S is given by Equation (7).
where A D is the junction area, n i is the intrinsic carrier concentration, N D /N A is the concentration of donor/acceptor atoms, D n /D p is the diffusion coefficient for electrons/holes, and L n /L p is the diffusion length for electrons/holes. This expression can be further reduced to where γ is a constant, q is the electron charge, and E g is the gap energy in eV.
In the case of an ideal single heterojunction, generation/recombination is still the dominant effect. However, the GaN-based LEDs studied in this paper are composed of a double heterojunction and multiple quantum wells. Besides the generation/recombination of the carriers, other processes may play a role, such as thermionic emission, tunneling, surface states, 20 Poole-Frenkel emission, hopping, 21,22 and space-charge current limitation. 23 Each of these processes has its own temperature dependency and, as a consequence, the derivation of an equation that expresses accurately the temperature dependence of the saturation current is not an easy task. Given the nonexistence of suitable physical models that explicitly describe the temperature dependence of the saturation current of GaN power LEDs, the temperature-dependent saturation current was represented using the SPICE equations for a p-n diode 24 : where E g ðTÞ is the bandgap energy at a given temperature T; V t ðTÞ ¼ k B Á T=q,T n ¼ 27 C; and I 0 ,n,R S , and XTI are the SPICE model parameters provided by the device manufacturer. The parameters n and R S correspond to the ideality factor and to the series resistance in Equation (6), and I 0 refers to the proportionality in Equation (8). The parameter XTI has no direct physical meaning; however, if the ideality factor n is also included in the denominator of the term inside the exponential in Equation (8), Equations (8) and (9) can be shown to be formally equivalent.
The temperature dependence of the bandgap energy is further given by the Varshni equation: where E g ð0Þ is the bandgap energy at a temperature of 0 K and α and β are empirical parameters characterizing the particular semiconductor material. The full temperature dependence of the diode voltage vði,TÞ can be obtained by substituting Equations (9) and (10) into Equation (6). The thermal power dissipated by the LED in Equation (4) can be calculated as P D ðtÞ ¼ ð1 À ηÞ Á vðtÞ Á iðtÞ, where η is the efficiency of the LED.
Equations (4) and (6) show that the memristive character of the diode appears naturally when temperature is taken as state variable. Previous works have addressed memristive effects associated with diodes, but not related to temperature effects. 13 To the authors knowledge, this is the first time that these effects are theoretically disclosed. Moreover, given that the temperature dependence of the electrical characteristics exists for virtually all electrical devices, it is likely that memory effects (not necessarily memristive) may also emerge for transistors and other devices.

| EXPERIMENTAL
Cold white XLamp XB-D LEDs were purchased from Cree. The LEDs have a small footprint (2:45 Â 2:45 mm 2 ) and a maximum current rating of 1 A. The cross-section view of an individual LED is shown in Figure 1A. 25 The LED die (in blue color) is composed by a silicon carbide (SiC) substrate 26 on which the gallium nitride (GaN) active layers 27 were deposited. The LED die is attached to an AlN carrier (in gray); the LED junction terminals and the electrodes deposited on the top surface of the carrier are electrically connected via wire bonds (red shapes). Electrical vias (not shown) connect electrically the electrodes on the top surface of the carrier to the external anode and cathode terminals at the back of the LED package and a thermal pad in the middle facilitates the removal of heat generated during the device operation.
One LED was soldered with lead-free solder on a 1 cm 2 1.6-mm-thick FR4 printed circuit board (PCB) ( Figure 1B) and a second LED was mounted on a 1 cm 2 300-μm-thick diamond board ( Figure 1C); both boards were prepared  according to the electrode layout suggested in the device datasheet. 28 The material, thermal conductivity, and dimensions of the epoxy, die, carrier, and boards are listed in Table 1. In order to measure the i-v characteristics, the boards with the LED were mounted on a HS with a copper (Cu) core and aluminum (Al) fins by means of thermal tape (150 μm thickness, 1.6 W/(mÁK) thermal conductivity) ( Figure 1D). The height/outer diameter of the Cu core and Al fins was 2/3 and 2.5/8 cm, respectively, and two FR4 flaps were attached to the HS to facilitate the mounting and removal of the board-marked by red arrows in Figure 1D. The thermal pad temperature was measured by a type K thermocouple placed on the thermal pad of the PCB (green arrow in Figure 1D), according to the instructions provided by the manufacturer. 35 The evolution of the LED temperature and voltage as a response to different current signals (a current step for the measurement of the thermal constants and a triangular current waveform for the evaluation of the transient i-v characteristics) was recorded with a custom-developed electronic module, represented in Figure 2A. The drive and acquisition system is composed of three main blocks: the computer, the driver and acquisition module, and the LED + PCB mounted on the HS. The computer acts as an interface with the user, asking for the desired system parameters (target LED current, step time/step rate) and receiving/storing the measured data (LED current, voltage, and thermal pad temperature). All the control and acquisition tasks are performed in the driver and acquisition module, which comprises several submodules, such as the microcontroller (μC), the current driver, the differential voltage amplifier, the current sensor, and the thermocouple amplifier. The whole process is controlled by the μC (Texas Instruments MSP430FR5994) embedded in a development board (Texas Instruments MSP-EXP430FR5994). Figure 2B shows the flowchart of the software that performs the driver current control and data acquisition. Initially, the μC receives the parameters from the computer and sets the target LED current. A pulsed-width modulation (PWM) module, embedded in the μC, generates a PWM signal which is the input of the current driver module. In this module, the PWM signal is filtered with a lowpass filter, resulting in a DC voltage, proportional to the duty cycle of the PWM signal. This voltage drives a voltagecontrolled current source, which controls the LED current in a feedback loop. The LED current is sensed by an integrated circuit (TMCS1100A4QDR) which converts the measured current to a voltage signal with a fixed sensitivity. The LED voltage is measured using a precision difference voltage amplifier (INA2133UA). The voltage produced by the thermocouple is amplified with the AD595 instrumentation amplifier, precalibrated for type K thermocouples. All the three readings (current, voltage, and thermal pad temperature) are sequentially read by the μC built-in analog-to-digital converter (ADC) with time intervals of 0.1 s and sent to MatLab.
Using this setup, the thermal pad temperature was measured as a response to current steps between 100 and 700 mA (and a step of 100 mA) and to a 350 mA current step (nominal current). The thermal pad temperature and the LED voltage were also measured as a response to triangular current waveforms with a peak value of 700 mA and varying step rates (5,8,10,20,25,50,80,100,150,200,300, and 500 mA/s) corresponding to sweeping frequencies between F I G U R E 2 (A) Block diagram of the experimental setup driver and acquisition system. (B) Software flowchart for the driver and acquisition module 3.6 and 360 mHz. In both cases (current steps and triangular waveforms) the LED was allowed to cool down to room temperature between successive measurements. Figure 3A shows the evolution of the experimental thermal pad temperature (T SP ) as a response to different current steps. As the current begins to flow across the LED, T SP increases and then stabilizes, once thermal steady-state is reached. If one considers that the thermal mass of the PCB + HS set is much larger than the one of the LED case, the evolution of T SP with time can be described using the first-order equivalent thermal circuit represented in Figure 3B, and the following equation can be written:

| Theoretical current-voltage loops
where T SS is the steady-state temperature, T A is the ambient temperature, and τ θ is the thermal time constant of the PCB + HS set. τ θ can be further expressed as τ θ ¼ R θ Á C θ , with R θ and C θ the thermal resistance and thermal capacitance of the PCB + HS set, respectively. In steady-state conditions, the thermal resistance R θ can be calculated as where P D is the thermal power generated within the GaN active layers. The thermal parameters of the system were calculated for the nominal current value of 350 mA and considering an efficiency of 25% 36 ). The values of T SS and τ θ (71.6 C and 5.9 s, respectively) were determined by fitting the experimental data with Equation (11), the thermal resistance R θ (60.7 C/W) was obtained by solving Equation (12), and the thermal capacitance C θ (97.0 Â 10 À3 J/ C) was determined from τ θ .
Rough theoretical estimations for these parameters can be also obtained. The estimated thermal resistance, R θ ¼ h=ðκ Á AÞ ¼ 80 C/W, is found considering the thermal conductivity of the FR4 board κ ¼ 0:2 W/(mÁ C), the PCB thickness h ¼ 1:6 mm, and the board area A ¼ 1 cm 2 ( Table 1). The estimated thermal capacitance of the PCB, C θ ¼ ρ Á c θ Á h Á A 2 ¼ 92 Â 10 À3 J/ C, is obtained using the FR4 material data from Guenin 37 : ρ ¼ 1:91 Â 10 3 kg/m 3 and c θ ¼ 0:6 Â 10 3 J/(kgÁ C). The 1/2 factor before the PCB volume V ¼ h Á A takes into account the nonuniformity of the F I G U R E 3 (A) Evolution of thermal pad temperature as a response to different current steps. (B) First-order equivalent thermal circuit temperature distribution in the board. As one can see, the estimated values of R θ and C θ are comparable to the measured ones. The smaller value of the measured thermal resistance is due to the additional heat dissipation effects (convection, radiation, etc.) not taken into account in the rough estimations.
The evolution of the thermal pad temperature T SP and the diode voltage v as a response to a 20-period triangular current waveform i with a maximum current value of 700 mA and 5, 50, and 500 mA/s step rates (that correspond to a frequency of 3.6, 36, and 360 mHz, respectively) were calculated theoretically by solving Equations (4), (6), (9), and (10) with an ordinary differential equations solver available in Matlab© (ode15s) and considering the experimental values of thermal resistance and thermal capacitance extracted from the response to a 350 mA current step (R θ ¼ 60:7 C/W, C θ ¼ 97:0 Â 10 À3 J/ C) and the diode parameters in the SPICE model provided by the manufacturer (Table 2).
Although the assumption that the HS behaves as an infinite temperature reservoir holds for high step rates, for the lower step rates, as the i-v sweeps progress, the temperature of the HS below the LED begins to increase. In order to avoid the influence of this slow temperature transient, attention was focused on the 20 th cycle of excursion of i. The theoretical evolution of T SP À i during the 20 th cycle can be seen in Figure 4A for a step rate of 5, 50, and 500 mA/s. The i-v curves obtained in the same cycle and step rates are shown in Figure 4B, and the area of the theoretical hysteresis loops is represented as a function of the step rate in Figure 4C. The black dots correspond to the theoretical values, and the red line is the trend line obtained by fitting the experimental results to an eighth-order polynomial. For very small step rates, the temperature can follow the variation of the current, and the loop area is small. As the step rate increases, the temperature increases slower that the current and hysteresis becomes visible, reaching a maximum around 40 mA/s. It is interesting to note that, in the forward regime, the behavior of the LED is similar to the behavior of an extended memristor: for DC (or very low step rates) and very high frequencies, the memristor behaves as a pure resistor with two different values, while for intermediate frequencies (or step rates) the i-v curves show hysteresis and cross the origin. Similarly, for very low/high step rates the i-v curves resemble the regular characteristics of an LED with different R S values while for intermediate step rates the i-v curve shows hysteresis and crosses the same point in the i-v plane.

| Experimental current-voltage loops
The evolution of the experimental thermal pad temperature of the LED mounted on the FR4 PCB during the 20th current excursion cycle and the experimental 20 th i-v cycles obtained for a step rate of 5, 50, and 500 mA/s are shown in Figure 5A,B, respectively, proving that the effects that were predicted theoretically in the previous section effectively manifest under certain conditions. In the current case, the high thermal resistance of the PCB board, together with the high thermal mass of the HS, create the conditions required for the appearance of hysteresis in the i-v curves. The area of the loop is represented as a function of the step rate in Figure 5C. The black dots correspond to the experimental values, and the red trend line was obtained by fitting the experimental results to an eighth-order polynomial. The experimental loop area is maximum for a step rate of 50 mA/s, a value close to the 40 mA/s anticipated in the previous section.
However, and despite the striking similarity of the three curves, the experimental and theoretical curves show some differences. To begin with, the theoretical values of the thermal pad temperature are larger than the experimental ones. In the theoretical calculations, the thermal conductivity of the materials was assumed to be constant with temperature. For metals such as Cu, this assumption is well justified because the variation of conductivity with temperature is low within the relatively narrow range of temperatures we are interested in. Regarding the FR4 substrate-which is a composite material formed by fiberglass and epoxy resin-there is simply no reliable data on the temperature dependence of the thermal conductivity. Nevertheless, being an epoxy-based material, its thermal conductivity is expected to follow the same trend as the one of common epoxies. The thermal conductivity of an epoxy has an initial value near room temperature, then slowly increases with temperature until it reaches a maximum value, and then decreases or levels off with further temperature increases. 40 Taking a look at the experimental data in Chern et al., 40 it is reasonable to expect that the thermal conductivity of the FR4 board will increase with its temperature, meaning that the experimental R θ in Figure 3B will decrease for high current levels (that correspond to a higher dissipated power and hence temperature). As a consequence, the experimental temperature values obtained for higher currents will be lower than the theoretical ones. The overestimation of the theoretical thermal pad temperature results in larger hysteresis in the i-v curves, and hence, the theoretical values of the loop area are higher than the experimental ones ( Figures 4C and 5C)  Finally, and in order to prove that the memristive character of the i-v characteristics that was observed is related with the large thermal resistance of the LED + PCB board, the i-v characteristics of the LED mounted on the diamond board are shown in Figure 6. The experimental characteristics do not show hysteresis.

| CONCLUSIONS
The models that describe the current-voltage characteristics of a diode typically consider temperature as a constant parameter. While this approach holds valid in many situations, it may fail in particular cases-especially when large currents, large time constants, and slow current changes are involved.
In this paper, we propose a methodology that enables accurate determination of the dynamic current-voltage characteristics of the diode board under test with either constant or time-varying stimulae. To this end, an experimental setup that measures the case temperature of power diodes as a function of the applied current was initially fabricated. This setup was then used to record the evolution of the temperature of high-power Cold White XLamp XB-D Cree LEDs mounted on FR4 and diamond boards when driven with different steps of current. The first-order time constants and the thermal resistance of the diode board under test set were then obtained from the transient temperature curves and were shown to be comparable to the theoretical values. The first-order thermal state equation obtained with the experimental thermal parameters and the SPICE current-voltage-temperature model provided by the manufacturer were then solved simultaneously using a simple Matlab script.
The model that was developed predicted that the current-voltage characteristics show hysteresis when the LED is driven by a current triangular wave with a period comparable to the thermal time constants of the system composed by the LED and the board, and the maximum loop area was obtained for a current step rate of around 40 mA/s. The LED board were shown to feature the characteristics of an extended memristor when the temperature is considered as a state variable.
Finally, when driving the LED mounted on the FR4 board with adequate current steps, the experimental currentvoltage curves showed hysteresis and the loop area reached a maximum for a step rate of 50 mA/s, showing that the theoretical first-order model that was derived adequately describes the dynamics of the diode board under test.