Diffusion equation-based study of thin ®lm semiconductor gas sensor-response transient Naoki Matsunaga a , Go Sakai b , Kengo Shimanoe b , Noboru Yamazoe b,* a Department of Molecular and Material Sciences, Graduate School of Engineering Sciences, Kyushu University, Kasuga-shi, Fukuoka 816-8580, Japan b Faculty of Engineering Sciences, Kyushu University, Kasuga-shi, Fukuoka 816-8580, Japan Abstract A diffusion±reaction equation has been formulated and solved under non-steady condition in order to simulate how the gas concentration pro®le develops inside a thin ®lm of semiconducting oxide after its exposure to a target gas. The gas concentration can be expressed by a polynomial function involving diffusion coef®cient D), rate constant k), ®lm thickness L), depth from the ®lm surface x), time t) and target gas concentration outside C s ). Remarkably, the gas concentration at a given x exhibits overshooting behavior before reaching a steady value, the magnitude and appearance time of the overshooting being very dependent on x, k and L/D 1/2 . The overshooting appears as a result of the competition between diffusion and reaction. Two types of overshooting are recognized, which are ascribable to the gas molecules having entered from the surface and to those having re¯ected by the wall of substrate, respectively. Re¯ecting such an overshooting in gas concentration, the response transient also exhibits an overshooting phenomenon. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Diffusion equation; Response transient; Gas sensor; Thin ®lm 1. Introduction It is well accepted that a semiconductor gas sensor detects an in¯ammable gas target gas) based on the mechanism that involves the reaction consumption) of the gas with the adsorbed oxygen on the surface of the semiconducting oxide used. The gas molecules diffuse inside the sensor element, while they are consumed at a certain rate during the diffusion by the surface reaction. This means that the gas sensing process is viewed as a phenomenon in which diffusion and reaction are coupled together. Because of this coupling, response behavior is strongly in¯uenced by the catalytic and micro-porous properties of sensor material used. Some theoretical or empirical approaches based on diffusion and reaction have been attempted [1±7]. Williams and Patt investigated the sensing characteristics of a thick ®lm 100 mm thick) sensor from viewpoints of electrode geo- metry and gas penetration depth and pointed out the impor- tance of diffusion±reaction effects [1,2]. Lu et al. discussed the recovery transients of thin ®lm sensors assuming a diffusion±reaction model [3]. Despite these efforts, however, the diffusion±reaction effects in semiconductor gas sensors are still far from being well understood. For analyzing the effects quantitatively, one needs those sensor elements which are stable and well de®ned in microstructure. It appears that a lack of such sensor elements have hindered the analysis. We found recently that thin ®lms of SnO 2 , fairly well de®ned in grain size and ®lm thickness, could be fabricated from a hydrothermally treated SnO 2 sol suspen- sion by spin-coating [8]. Based on a diffusion±reaction equation assuming Knudsen diffusion and ®rst-order kinetics of the surface reaction, the response sensitivity) of the ®lms at steady state could be simulated well as a function of ®lm thickness and operating temperature, as reported elsewhere [9]. On this background, we tried to analyze the response transients of the ®lm theoretically by solving the diffusion±reaction equation under non-steady state condition. This paper aims at describing the transient behavior of gas concentration pro®le and gas response. 2. General expression of gas concentration profile inside a film It is assumed that the thin ®lms possess uniform pore- structure, through which the target gas molecules undergo Sensors and Actuators B 83 2002) 216±221 * Corresponding author. Tel.: 81-92-583-7539; fax: 81-92-583-7539. E-mail address: yamazoe@mm.kyushu-u.ac.jp N. Yamazoe). 0925-4005/02/$ ± see front matter # 2002 Elsevier Science B.V. All rights reserved. PII:S0925-400501)01043-7