The model performance vary between different train-test split? I fit my dataset to the random forest classifier and found that the model performance would vary among different sets of train and test data split. As what I have observed, it would jump from 0.67 to 0.75 in AUC under ROC curve (fitted by the same model under same setting of parameters) and the underlying range may be wider than that. So what is the issue behind this phenomena and how to deal with this problem? As my understanding, cross validation is used for a specific split of train and test data.
What you are experiencing is not a problem, but rather an inherent attribute of all classifiers. The performance of a classifier depends on the training set, therefore the performance will vary with different training sets.

To find the best parameters for a specific classifier you will therefore want to vary training and test split (such as in crossvalidation) and choose the parameter set which achieves the best average accuracy or AUC. 

Finally you will want to test the trained classifier on another dataset - evaluation set - which has not been part of the dataset used in crossvalidation.