Heat dissipation due to electrical current

The current density I_th = 7.3 kA/cm² corresponds to a dissipated thermal power of P_th = U(I_th)I_thπ R²_m ≈3.5 W. The cryostat we use has only a cooling power of≈1 W. Hence, we have to electrically drive the sample in pulsed mode with a duty cycle of at most 25 %. The question now is the sample heating within a single current pulse.

Thermal conductivity For an applied current pulse with current density I we want calculate the lattice temperature T of the sample as a function of time t. The relevant equation is the equation of thermal conductivity

∂ T(r, t)

∂ t = λT(r, t)

ρ c_pT(r, t)∆T(r, t) + 1

ρ c_pT(r, t)η(I, r)

whereλis the thermal conductivity,ρthe density,cp the heat capacity,ηthe dissipated heat per volume and ∆ the Laplace operator. Because of the temperature dependence of λ and ρ (see Fig. 5.4) there is no analytic solution of this equation. Here, we will just treat the two extreme cases at very long times and at very short times. This will allow us to give a sufficient estimation for the lattice temperature of our sample.

• In the first case we consider the (quasi-stationary) thermal equilibrium, i.e., ^{∂ T}_{∂ t}|t=∞ = 0, which is reached after “long”times. We calculate the average temperature in the mesa T_eq ≡ T(t = ∞). The equation of thermal conductivity then yields the Poisson equation

∆T(r) = −η(I, r)

λ (5.1)

A strong simplification of the problem can be achieved if we assume that the quantum cascade mesa (radius R_m) is a semi sphere with a surface which is equal to the surface

10^-1

Figure 5.4: (a) Thermal conductivity [135] and (b) heat capacity [136] of GaAs as a function of temperature.

of the circular mesa disc. The radius of this sphere is R_s = ^R√^m

2. In that way the thermal flux, which is the essential quantity for thermal conductivity, is the same in both geometries. For symmetry reasons we can consider a total spherical situation, where r is the distance from the center. The boundary condition is T(r=∞) = T_CF where T_CF is the temperature of the cold-finger of the cryostat. The solution of this spherical problem for λ=const. is

r≥R_s: T(r) = 1 We are interested in the average mesa temperature

T_m(I, R_m)≡ T(R_s) +T(0)

2 = 5

12

η(I)R_s²

λ +T_CF

The heat dissipation per volume in the semi-sphere due to the electric current in the quantum cascade laser reads

Note that the temperature in the thermal equilibrium depends on the mesa radiusR_m.

0 1 2 3

Figure 5.5: (a) Temperature at the quasi-stationary equilibriumT_eq (dashed line), tempera-ture dependence at early times T_s (dotted line) and an exponential behavior (solid line) as a function of time for a quantum cascade mesa with radius R_m = 60µm and for a current density of I = 5 kA/cm². τ_eq is the time constant reaching the quasi-stationary equilib-rium. (b) Temperature at the quasi-stationary equilibrium as a function of current density (R_m = 60µm).

• In the second case, at very short times, the term with the thermal conductivity can be neglected, i.e., ∆T = 0. The equation of thermal conductivity can be easily solved for c_p =const. these two regimes are depicted in Fig. 5.5 (a). The cold-finger temperature is T_CF = 77 K.

The thermal conductivity is taken to be λ = λ(250 K) = 0.5 W

K cm [cf. Fig. 5.4 (a)], the heat capacity as c_p(100 K) = 0.2 J

g K [cf. Fig. 5.4 (b)] and the density of GaAs is ρ = 5.3 g/cm³. The temperature in the mesa is shown as a function of time. The dashed line represents the temperature T_eq which will be reached after long times. The dotted curve indicates the temperature dependence for very short times T_s. These calculations are ”worst case”calculations and the true temperature will be somewhere below (solid line).

Nevertheless, with these calculations we can estimate a time constant τ_eq ≈700 ns reaching the quasi-stationary equilibrium. Comparing (5.2) and (5.3) one can see that this time constant does not depend on the current. In Fig. 5.5 (b) the current dependence of the temperature T_eq is shown. We get room temperature T_eq = 300 K for a current density of 6 kA/cm². The maximum working temperature of the laser based on the laser structure of our sample isT_th= 140 K. It is reached already at 2.5 kA/cm².

It is important to note that the temperatures calculated above describe the temperature of the lattice of the quantum cascade structure. In general, these temperatures are not

optical pulses

electrical pulses

2.5 µs

50 µs

Figure 5.6: Timing of electrical and optical pulses. The electrical pulses are applied with a repetition rate of 23.8 kHz and a duty cycle of 5 %. The optical pulses have a repetition rate of 2.1 MHz.

valid for the carrier system. The electron distribution in electrically driven quantum cascade structures will have a temperature far above the lattice temperature (cf. chapter 3).

Electrical pulse length For the experiment we use the 2 MHz cavity-dumped Ti:sapphire system as described in section2.2.1. To ensure a lattice temperature on the order of the cold-finger temperature the ideal electrical pumping would be to use very short pulses with pulse lengths on the order of τ_eq/10≈70 ns [Fig. 5.5 (a)] in combination with a synchronization of the current with the mid-infrared pulses. This is not possible, however, since high-current pulser providing current pulses with such elevated repetition rates are not available. For the experiments presented in this chapter we use a scheme as depicted in Fig. 5.6 with current pulses being longer than the inverse repetition rate of the optical pulses. Now, the transmission change signal is proportional to the electrical pulse length. We use a repetition rate for the electrical pulses of 23.8 kHz with a duty cycle of 5 % which corresponds to a pulse length of approximately 2.5µs. Within this time, quasi-stationary equilibrium of the lattice temperature is attained [Fig. 5.5 (a)]. From Fig. 5.5 (b) we see that Tth = 140 K is reached already at a current density of 2.5 kA/cm². Therefore it makes only sense to use this method applying low currents.