We have investigated the charging behavior of a layer of self-assembled InAs quantum dots placed in close vicinity to a two-dimensional electron gas ͑2DEG͒. As the gate bias is changed, the number of electrons in each system is altered simultaneously. Based on the quantum capacitance of the involved layers we develop a general model to determine the charging state of coupled low-dimensional systems from capacitance-voltage ͑CV͒ spectroscopy. The model is then applied to the special case of a layer of self-assembled quantum dots coupled to a 2DEG. As a complementary method, we have employed Hall voltage measurements. We find that the measurement of the two-dimensional carrier density through lateral transport provides a direct insight into the vertical charging process of the quantum dot system. Six individual charging peaks related to the occupation of the s and p shells of the dots can be resolved. In agreement with results from CV spectroscopy, Coulomb blockade and quantization energies can be extracted. Moreover, the Hall measurement offers a higher peak-to-valley ratio and a better estimate for the number of simultaneously charged dots than the capacitance data.

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