This paper studies the finite-time $H_{\infty}$ control problem of Markov jump discrete-time silicon single crystal growth systems under incompletely measurable system states and external disturbances. By introducing the Markov jump process into the silicon single crystal growth model, a Markov jump discrete-time silicon single crystal growth systems are established. Considering that the system state in the actual silicon single crystal growth process is not completely measurable and affected by external disturbances, the state observer and controller are constructed using the measurement outputs. Based on this, according to the Markov jump control system theory and the finite-time $H_{\infty}$ control theory, a sufficient condition is obtained to ensure that the closed-loop discrete-time silicon single crystal growth systems are finite-time bounded with the $H_{\infty}$ performance. The solution method of controller and observer gains is given via using linear matrix inequality (LMI) technology. Finally, the effectiveness of the proposed control scheme is verified by the model parameters of actual silicon single crystal growth systems.